physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaMécanique analytique''. Lagrangian mechanics describes a mechanical system as a pair consisting of a configuration space ''M'' and a smooth function

L

within that space called a ''Lagrangian''. For many systems, , where ''T'' and ''V'' are the kinetic and

potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...

energy of the system, respectively. The stationary action principle requires that the action functional of the system derived from ''L'' must remain at a

stationary point In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of a function, graph of the function where the function's derivative is zero. Informally, it is a point where the ...

(specifically, a

maximum In mathematical analysis, the maximum and minimum of a function (mathematics), function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given Interval (ma ...

,

minimum In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the ''local'' or ''relative ...

, or

saddle A saddle is a supportive structure for a rider of an animal, fastened to an animal's back by a girth. The most common type is equestrian. However, specialized saddles have been created for oxen, camels and other animals. It is not know ...

point) throughout the time evolution of the system. This constraint allows the calculation of the equations of motion of the system using Lagrange's equations.

Introduction

Newton's laws and the concept of forces are the usual starting point for teaching about mechanical systems. This method works well for many problems, but for others the approach is nightmarishly complicated. For example, in calculation of the motion of a

torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...

rolling on a horizontal surface with a pearl sliding inside, the time-varying constraint forces like the

angular velocity In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i ...

of the torus, motion of the pearl in relation to the torus made it difficult to determine the motion of the torus with Newton's equations. Lagrangian mechanics adopts energy rather than force as its basic ingredient, leading to more abstract equations capable of tackling more complex problems. Particularly, Lagrange's approach was to set up independent

generalized coordinate In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.p. 397 ...

s for the position and speed of every object, which allows the writing down of a general form of Lagrangian (total kinetic energy minus potential energy of the system) and summing this over all possible paths of motion of the particles yielded a formula for the 'action', which he minimized to give a generalized set of equations. This summed quantity is minimized along the path that the particle actually takes. This choice eliminates the need for the constraint force to enter into the resultant generalized system of equations. There are fewer equations since one is not directly calculating the influence of the constraint on the particle at a given moment. For a wide variety of physical systems, if the size and shape of a massive object are negligible, it is a useful simplification to treat it as a

point particle A point particle, ideal particle or point-like particle (often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take ...

. For a system of ''N'' point particles with

mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...

es ''m''₁, ''m''₂, ..., ''m_N'', each particle has a

position vector In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point ''P'' in space. Its length represents the distance in relation to an arbitrary reference origin ''O'', and ...

, denoted r₁, r₂, ..., r_''N''.

Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

are often sufficient, so , and so on. In

three-dimensional space In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values ('' coordinates'') are required to determine the position of a point. Most commonly, it is the three- ...

, each position vector requires three

coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as ...

to uniquely define the location of a point, so there are 3''N'' coordinates to uniquely define the configuration of the system. These are all specific points in space to locate the particles; a general point in space is written . The

velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...

of each particle is how fast the particle moves along its path of motion, and is the

time derivative A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t. Notation A variety of notations are used to denote th ...

of its position, thus

\mathbf_1 = \frac,
 \mathbf_2 = \frac,
 \ldots,
 \mathbf_N = \frac.

In Newtonian mechanics, the

equations of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathem ...

are given by

Newton's laws Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body re ...

. The second law "net

force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...

equals mass times

acceleration In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...

",

\sum \mathbf = m \frac,

applies to each particle. For an ''N''-particle system in 3 dimensions, there are 3''N'' second-order

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...

s in the positions of the particles to solve for.

Lagrangian

Instead of forces, Lagrangian mechanics uses the energies in the system. The central quantity of Lagrangian mechanics is the Lagrangian, a function which summarizes the dynamics of the entire system. Overall, the Lagrangian has units of energy, but no single expression for all physical systems. Any function which generates the correct equations of motion, in agreement with physical laws, can be taken as a Lagrangian. It is nevertheless possible to construct general expressions for large classes of applications. The ''non-relativistic'' Lagrangian for a system of particles in the absence of an electromagnetic field is given by

L = T - V,

where

T = \frac \sum_^N m_k v^2_k

is the total

kinetic energy In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...

of the system, equaling the sum Σ of the kinetic energies of the

N

particles. Each particle labeled

k

has mass

m_k,

and is the magnitude squared of its velocity, equivalent to the

dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...

of the velocity with itself. Kinetic energy is the energy of the system's motion and is a function only of the velocities v_''k'', not the positions r_''k'', nor time ''t'', so ''V'', the

potential energy In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...

of the system, reflects the energy of interaction between the particles, i.e. how much energy any one particle has due to all the others, together with any external influences. For

conservative force In physics, a conservative force is a force with the property that the total work done by the force in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the total work don ...

s (e.g.

Newtonian gravity Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the sq ...

), it is a function of the position vectors of the particles only, so For those non-conservative forces which can be derived from an appropriate potential (e.g. electromagnetic potential), the velocities will appear also, If there is some external field or external driving force changing with time, the potential changes with time, so most generally As already noted, this form of ''L'' is applicable to many important classes of system, but not everywhere. For relativistic Lagrangian mechanics it must be replaced as a whole by a function consistent with special relativity (scalar under Lorentz transformations) or general relativity (4-scalar). Where a magnetic field is present, the expression for the potential energy needs restating. And for dissipative forces (e.g.,

friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of t ...

), another function must be introduced alongside Lagrangian often referred to as a " Rayleigh dissipation function" to account for the loss of energy. One or more of the particles may each be subject to one or more holonomic constraints; such a constraint is described by an equation of the form If the number of constraints in the system is ''C'', then each constraint has an equation ..., each of which could apply to any of the particles. If particle ''k'' is subject to constraint ''i'', then At any instant of time, the coordinates of a constrained particle are linked together and not independent. The constraint equations determine the allowed paths the particles can move along, but not where they are or how fast they go at every instant of time. Nonholonomic constraints depend on the particle velocities, accelerations, or higher derivatives of position. Lagrangian mechanics ''can only be applied to systems whose constraints, if any, are all holonomic''. Three examples of nonholonomic constraints are: when the constraint equations are non-integrable, when the constraints have inequalities, or when the constraints involve complicated non-conservative forces like friction. Nonholonomic constraints require special treatment, and one may have to revert to

Newtonian mechanics Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body r ...

or use other methods. If ''T'' or ''V'' or both depend explicitly on time due to time-varying constraints or external influences, the Lagrangian is ''explicitly time-dependent''. If neither the potential nor the kinetic energy depend on time, then the Lagrangian is ''explicitly independent of time''. In either case, the Lagrangian always has implicit time dependence through the generalized coordinates. With these definitions, Lagrange's equations of the first kind are where ''k'' = 1, 2, ..., ''N'' labels the particles, there is a

Lagrange multiplier In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function (mathematics), function subject to constraint (mathematics), equation constraints (i.e., subject to the conditio ...

''λ_i'' for each constraint equation ''f''_''i'', and

\frac \equiv \left(\frac, \frac, \frac\right), \quad
 \frac \equiv \left(\frac, \frac, \frac\right)

are each shorthands for a vector of

partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...

s with respect to the indicated variables (not a derivative with respect to the entire vector).Sometimes in this context the variational derivative denoted and defined as

\frac \equiv \frac - \frac\frac

is used. Throughout this article only partial and total derivatives are used. Each overdot is a shorthand for a

time derivative A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t. Notation A variety of notations are used to denote th ...

. This procedure does increase the number of equations to solve compared to Newton's laws, from 3''N'' to , because there are 3''N'' coupled second-order differential equations in the position coordinates and multipliers, plus ''C'' constraint equations. However, when solved alongside the position coordinates of the particles, the multipliers can yield information about the constraint forces. The coordinates do not need to be eliminated by solving the constraint equations. In the Lagrangian, the position coordinates and velocity components are all

independent variables A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical function ...

, and derivatives of the Lagrangian are taken with respect to these separately according to the usual

differentiation rules This article is a summary of differentiation rules, that is, rules for computing the derivative of a function (mathematics), function in calculus. Elementary rules of differentiation Unless otherwise stated, all functions are functions of real nu ...

(e.g. the partial derivative of ''L'' with respect to the ''z'' velocity component of particle 2, defined by , is just ; no awkward

chain rule In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h ...

s or total derivatives need to be used to relate the velocity component to the corresponding coordinate ''z''₂). In each constraint equation, one coordinate is redundant because it is determined from the other coordinates. The number of ''independent'' coordinates is therefore . We can transform each position vector to a common set of ''n''

generalized coordinates In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.p. 397 ...

, conveniently written as an ''n''-tuple , by expressing each position vector, and hence the position coordinates, as functions of the generalized coordinates and time:

\mathbf_k = \mathbf_k(\mathbf, t) = \big(x_k(\mathbf, t), y_k(\mathbf, t), z_k(\mathbf, t), t\big).

The vector q is a point in the configuration space of the system. The time derivatives of the generalized coordinates are called the generalized velocities, and for each particle the transformation of its velocity vector, the

total derivative In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with res ...

of its position with respect to time, is

\dot_j = \frac, \quad
 \mathbf_k = \sum_^n \frac\dot_j + \frac.

Given this v_''k'', the kinetic energy ''in generalized coordinates'' depends on the generalized velocities, generalized coordinates, and time if the position vectors depend explicitly on time due to time-varying constraints, so

T = T(\mathbf, \dot, t).

With these definitions, the Euler–Lagrange equations, or Lagrange's equations of the second kind are mathematical results from the

calculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

, which can also be used in mechanics. Substituting in the Lagrangian gives the

equations of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathem ...

of the system. The number of equations has decreased compared to Newtonian mechanics, from 3''N'' to coupled second-order differential equations in the generalized coordinates. These equations do not include constraint forces at all, only non-constraint forces need to be accounted for. Although the equations of motion include

partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...

s, the results of the partial derivatives are still

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...

s in the position coordinates of the particles. The total time derivative denoted d/d''t'' often involves

implicit differentiation In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit func ...

. Both equations are linear in the Lagrangian, but generally are nonlinear coupled equations in the coordinates.

From Newtonian to Lagrangian mechanics

Newton's laws

For simplicity, Newton's laws can be illustrated for one particle without much loss of generality (for a system of ''N'' particles, all of these equations apply to each particle in the system). The

equation of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathem ...

for a particle of constant mass ''m'' is

Newton's second law Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body re ...

of 1687, in modern vector notation

\mathbf = m \mathbf,

where a is its acceleration and F the resultant force acting ''on'' it. Where the mass is varying, the equation needs to be generalised to take the time derivative of the momentum. In three spatial dimensions, this is a system of three coupled second-order

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...

s to solve, since there are three components in this vector equation. The solution is the position vector r of the particle at time ''t'', subject to the initial conditions of r and v when Newton's laws are easy to use in Cartesian coordinates, but Cartesian coordinates are not always convenient, and for other coordinate systems the equations of motion can become complicated. In a set of

curvilinear coordinates In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is invertible, l ...

the law in

tensor index notation In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...

is the ''"Lagrangian form"''

F^a = m \left( \frac + \Gamma^a_ \frac\frac \right) = g^ \left(\frac \frac - \frac\right), \quad
 \dot^a \equiv \frac,

where ''F''^''a'' is the ''a''-th contravariant component of the resultant force acting on the particle, Γ^''a''_''bc'' are the

Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surface (topology), surfaces or other manifolds endowed with a metri ...

of the second kind,

T = \frac m g_ \frac \frac

is the kinetic energy of the particle, and ''g_bc'' the covariant components of the ''

metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...

'' of the curvilinear coordinate system. All the indices ''a'', ''b'', ''c'', each take the values 1, 2, 3. Curvilinear coordinates are not the same as generalized coordinates. It may seem like an overcomplication to cast Newton's law in this form, but there are advantages. The acceleration components in terms of the Christoffel symbols can be avoided by evaluating derivatives of the kinetic energy instead. If there is no resultant force acting on the particle, it does not accelerate, but moves with constant velocity in a straight line. Mathematically, the solutions of the differential equation are ''

geodesic In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...

s'', the curves of extremal length between two points in space (these may end up being minimal, that is the shortest paths, but not necessarily). In flat 3D real space the geodesics are simply straight lines. So for a free particle, Newton's second law coincides with the geodesic equation and states that free particles follow geodesics, the extremal trajectories it can move along. If the particle is subject to forces the particle accelerates due to forces acting on it and deviates away from the geodesics it would follow if free. With appropriate extensions of the quantities given here in flat 3D space to 4D

curved spacetime In physics, curved spacetime is the mathematical model in which, with Einstein's theory of general relativity, gravity naturally arises, as opposed to being described as a fundamental force in Isaac Newton, Newton's static Euclidean reference fra ...

, the above form of Newton's law also carries over to

Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...

's

general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...

, in which case free particles follow geodesics in curved spacetime that are no longer "straight lines" in the ordinary sense. However, we still need to know the total resultant force F acting on the particle, which in turn requires the resultant non-constraint force N plus the resultant constraint force C,

\mathbf = \mathbf + \mathbf.

The constraint forces can be complicated, since they generally depend on time. Also, if there are constraints, the curvilinear coordinates are not independent but related by one or more constraint equations. The constraint forces can either be eliminated from the equations of motion, so only the non-constraint forces remain, or included by including the constraint equations in the equations of motion.

D'Alembert's principle

A fundamental result in

analytical mechanics In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related formulations of classical mechanics. Analytical mechanics uses '' scalar'' properties of motion representing the sy ...

is

D'Alembert's principle D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental classical physics, classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond d' ...

, introduced in 1708 by Jacques Bernoulli to understand

static equilibrium In classical mechanics, a particle is in mechanical equilibrium if the net force on that particle is zero. By extension, a physical system made up of many parts is in mechanical equilibrium if the net force on each of its individual parts is ze ...

, and developed by

D'Alembert Jean-Baptiste le Rond d'Alembert ( ; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanics, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''E ...

in 1743 to solve dynamical problems. The principle asserts for ''N'' particles the virtual work, i.e. the work along a virtual displacement, ''δ''r_''k'', is zero:

\sum_^N ( \mathbf _k + \mathbf _k - m_k \mathbf_k )\cdot \delta \mathbf_k = 0.

The ''

virtual displacement In analytical mechanics, a branch of applied mathematics and physics, a virtual displacement (or infinitesimal variation) \delta \gamma shows how the mechanical system's trajectory can ''hypothetically'' (hence the term ''virtual'') deviate very ...

s'', ''δ''r_''k'', are by definition infinitesimal changes in the configuration of the system consistent with the constraint forces acting on the system ''at an instant of time'', i.e. in such a way that the constraint forces maintain the constrained motion. They are not the same as the actual displacements in the system, which are caused by the resultant constraint and non-constraint forces acting on the particle to accelerate and move it.Here the virtual displacements are assumed reversible, it is possible for some systems to have non-reversible virtual displacements that violate this principle, see Udwadia–Kalaba equation.

Virtual work In mechanics, virtual work arises in the application of the '' principle of least action'' to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different fo ...

is the work done along a virtual displacement for any force (constraint or non-constraint). Since the constraint forces act perpendicular to the motion of each particle in the system to maintain the constraints, the total virtual work by the constraint forces acting on the system is zero:In other words

\mathbf_k\cdot\delta\mathbf_k = 0

for particle ''k'' subject to a constraint force, however

C_ \delta x_k \neq 0,\quad C_ \delta y_k \neq 0,\quad C_ \delta z_k \neq 0

because of the constraint equations on the r_''k'' coordinates.

\sum_^N \mathbf _k \cdot \delta \mathbf_k = 0,

so that

\sum_^N (\mathbf _k - m_k \mathbf_k ) \cdot \delta \mathbf_k = 0.

Thus D'Alembert's principle allows us to concentrate on only the applied non-constraint forces, and exclude the constraint forces in the equations of motion. The form shown is also independent of the choice of coordinates. However, it cannot be readily used to set up the equations of motion in an arbitrary coordinate system since the displacements ''δ''r_''k'' might be connected by a constraint equation, which prevents us from setting the ''N'' individual summands to 0. We will therefore seek a system of mutually independent coordinates for which the total sum will be 0 if and only if the individual summands are 0. Setting each of the summands to 0 will eventually give us our separated equations of motion.

Equations of motion from D'Alembert's principle

If there are constraints on particle ''k'', then since the coordinates of the position are linked together by a constraint equation, so are those of the

virtual displacement In analytical mechanics, a branch of applied mathematics and physics, a virtual displacement (or infinitesimal variation) \delta \gamma shows how the mechanical system's trajectory can ''hypothetically'' (hence the term ''virtual'') deviate very ...

s . Since the generalized coordinates are independent, we can avoid the complications with the ''δ''r_''k'' by converting to virtual displacements in the generalized coordinates. These are related in the same form as a

total differential In calculus, the differential represents the principal part of the change in a function y = f(x) with respect to changes in the independent variable. The differential dy is defined by dy = f'(x)\,dx, where f'(x) is the derivative of with resp ...

,

\delta \mathbf_k = \sum_^n \frac   \delta q_j.

There is no partial time derivative with respect to time multiplied by a time increment, since this is a virtual displacement, one along the constraints in an ''instant'' of time. The first term in D'Alembert's principle above is the virtual work done by the non-constraint forces N_''k'' along the virtual displacements ''δ''r_''k'', and can without loss of generality be converted into the generalized analogues by the definition of generalized forces

Q_j = \sum_^N \mathbf _k \cdot \frac,

so that

\sum_^N \mathbf_k \cdot \delta \mathbf_k = \sum_^N  \mathbf _k \cdot \sum_^n \frac   \delta q_j = \sum_^n Q_j \delta q_j.

This is half of the conversion to generalized coordinates. It remains to convert the acceleration term into generalized coordinates, which is not immediately obvious. Recalling the Lagrange form of Newton's second law, the partial derivatives of the kinetic energy with respect to the generalized coordinates and velocities can be found to give the desired result:

\sum_^N m_k \mathbf_k \cdot \frac  = \frac\frac - \frac.

Now D'Alembert's principle is in the generalized coordinates as required,

\delta q_j = 0,

and since these virtual displacements ''δq''_''j'' are independent and nonzero, the coefficients can be equated to zero, resulting in Lagrange's equations or the generalized equations of motion,

Q_j = \frac\frac - \frac

These equations are equivalent to Newton's laws ''for the non-constraint forces''. The generalized forces in this equation are derived from the non-constraint forces only – the constraint forces have been excluded from D'Alembert's principle and do not need to be found. The generalized forces may be non-conservative, provided they satisfy D'Alembert's principle.

Euler–Lagrange equations and Hamilton's principle

For a non-conservative force which depends on velocity, it ''may'' be possible to find a potential energy function ''V'' that depends on positions and velocities. If the generalized forces ''Q''_''i'' can be derived from a potential ''V'' such that

Q_j = \frac\frac - \frac,

equating to Lagrange's equations and defining the Lagrangian as obtains Lagrange's equations of the second kind or the Euler–Lagrange equations of motion

\frac - \frac\frac  = 0.

However, the Euler–Lagrange equations can only account for non-conservative forces ''if'' a potential can be found as shown. This may not always be possible for non-conservative forces, and Lagrange's equations do not involve any potential, only generalized forces; therefore they are more general than the Euler–Lagrange equations. The Euler–Lagrange equations also follow from the

calculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

. The ''variation'' of the Lagrangian is

\delta L = \sum_^n \left(\frac \delta q_j + \frac \delta \dot_j \right),\quad \delta \dot_j \equiv \delta\frac \equiv \frac,

which has a form similar to the

total differential In calculus, the differential represents the principal part of the change in a function y = f(x) with respect to changes in the independent variable. The differential dy is defined by dy = f'(x)\,dx, where f'(x) is the derivative of with resp ...

of ''L'', but the virtual displacements and their time derivatives replace differentials, and there is no time increment in accordance with the definition of the virtual displacements. An

integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivati ...

with respect to time can transfer the time derivative of ''δq''_''j'' to the ∂''L''/∂(d''q_j''/d''t''), in the process exchanging d(''δq''_''j'')/d''t'' for ''δq''_''j'', allowing the independent virtual displacements to be factorized from the derivatives of the Lagrangian,

^ + \int_^ \sum_^n \left(\frac - \frac\frac \right)\delta q_j \, \mathrmt. \end

Now, if the condition holds for all ''j'', the terms not integrated are zero. If in addition the entire time integral of ''δL'' is zero, then because the ''δq''_''j'' are independent, and the only way for a definite integral to be zero is if the integrand equals zero, each of the coefficients of ''δq''_''j'' must also be zero. Then we obtain the equations of motion. This can be summarized by Hamilton's principle:

\int_^\delta L \, \mathrmt = 0.

The time integral of the Lagrangian is another quantity called the action, defined as

S = \int_^ L\,\mathrmt,

which is a '' functional''; it takes in the Lagrangian function for all times between ''t''₁ and ''t''₂ and returns a scalar value. Its dimensions are the same as , ·, or ·. With this definition Hamilton's principle is

\delta S = 0.

Instead of thinking about particles accelerating in response to applied forces, one might think of them picking out the path with a stationary action, with the end points of the path in configuration space held fixed at the initial and final times. Hamilton's principle is one of several action principles. Historically, the idea of finding the shortest path a particle can follow subject to a force motivated the first applications of the

calculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

to mechanical problems, such as the

Brachistochrone problem In physics and mathematics, a brachistochrone curve (), or curve of fastest descent, is the one lying on the plane between a point ''A'' and a lower point ''B'', where ''B'' is not directly below ''A'', on which a bead slides frictionlessly under ...

solved by Jean Bernoulli in 1696, as well as

Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many ...

,

Daniel Bernoulli Daniel Bernoulli ( ; ; – 27 March 1782) was a Swiss people, Swiss-France, French mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applicati ...

, L'Hôpital around the same time, and Newton the following year. Newton himself was thinking along the lines of the variational calculus, but did not publish. These ideas in turn lead to the variational principles of mechanics, of

Fermat Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...

, Maupertuis,

Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

,

Hamilton Hamilton may refer to: * Alexander Hamilton (1755/1757–1804), first U.S. Secretary of the Treasury and one of the Founding Fathers of the United States * ''Hamilton'' (musical), a 2015 Broadway musical by Lin-Manuel Miranda ** ''Hamilton'' (al ...

, and others. Hamilton's principle can be applied to nonholonomic constraints if the constraint equations can be put into a certain form, a

linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...

of first order differentials in the coordinates. The resulting constraint equation can be rearranged into first order differential equation. This will not be given here.

Lagrange multipliers and constraints

The Lagrangian ''L'' can be varied in the Cartesian r_''k'' coordinates, for ''N'' particles,

\int_^ \sum_^N \left(\frac - \frac\frac \right)\cdot\delta \mathbf_k \, \mathrmt = 0.

Hamilton's principle is still valid even if the coordinates ''L'' is expressed in are not independent, here r_''k'', but the constraints are still assumed to be holonomic. As always the end points are fixed for all ''k''. What cannot be done is to simply equate the coefficients of ''δ''r_''k'' to zero because the ''δ''r_''k'' are not independent. Instead, the method of

Lagrange multiplier In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function (mathematics), function subject to constraint (mathematics), equation constraints (i.e., subject to the conditio ...

s can be used to include the constraints. Multiplying each constraint equation by a Lagrange multiplier ''λ''_''i'' for ''i'' = 1, 2, ..., ''C'', and adding the results to the original Lagrangian, gives the new Lagrangian

L' = L(\mathbf_1,\mathbf_2,\ldots,\dot_1,\dot_2,\ldots,t) + \sum_^C \lambda_i(t) f_i(\mathbf_k,t).

The Lagrange multipliers are arbitrary functions of time ''t'', but not functions of the coordinates r_''k'', so the multipliers are on equal footing with the position coordinates. Varying this new Lagrangian and integrating with respect to time gives

\int_^ \delta L' \mathrmt = \int_^ \sum_^N \left(\frac - \frac\frac + \sum_^C \lambda_i \frac\right)\cdot\delta \mathbf_k \, \mathrmt = 0.

The introduced multipliers can be found so that the coefficients of ''δ''r_''k'' are zero, even though the r_''k'' are not independent. The equations of motion follow. From the preceding analysis, obtaining the solution to this integral is equivalent to the statement

\frac - \frac\frac = 0 \quad \Rightarrow \quad \frac - \frac\frac + \sum_^C \lambda_i \frac = 0,

which are Lagrange's equations of the first kind. Also, the ''λ''_''i'' Euler-Lagrange equations for the new Lagrangian return the constraint equations

\frac - \frac\frac  = 0 \quad \Rightarrow \quad f_i(\mathbf_k,t) = 0.

For the case of a conservative force given by the gradient of some potential energy ''V'', a function of the r_k coordinates only, substituting the Lagrangian gives

\underbrace_ + \underbrace_ + \sum_^C \lambda_i \frac = 0,

and identifying the derivatives of kinetic energy as the (negative of the) resultant force, and the derivatives of the potential equaling the non-constraint force, it follows the constraint forces are

\mathbf_k = \sum_^C \lambda_i \frac,

thus giving the constraint forces explicitly in terms of the constraint equations and the Lagrange multipliers.

Properties of the Lagrangian

Non-uniqueness

The Lagrangian of a given system is not unique. A Lagrangian ''L'' can be multiplied by a nonzero constant ''a'' and shifted by an arbitrary constant ''b'', and the new Lagrangian will describe the same motion as ''L''. If one restricts as above to trajectories q over a given time interval } and fixed end points and , then two Lagrangians describing the same system can differ by the "total time derivative" of a function :

L'(\mathbf,\dot\mathbf,t) = L(\mathbf,\dot\mathbf,t) + \frac,

where

\frac

means

\frac+\sum_i \frac _i.

Both Lagrangians ''L'' and ''L''′ produce the same equations of motion since the corresponding actions ''S'' and ''S''′ are related via

+ f(P_\text,t_\text) - f(P_\text,t_\text), \end

with the last two components and independent of q.

Invariance under point transformations

Given a set of generalized coordinates q, if we change these variables to a new set of generalized coordinates Q according to a point transformation which is invertible as , the new Lagrangian ''L''′ is a function of the new coordinates and similarly for the constraints

\beginL'(\mathbf, \dot,t) &= L(\mathbf(\mathbf,t), \dot(\mathbf,\dot,t), t ),\\
\phi_j'(\mathbf,t)&=\phi_j(\mathbf(\mathbf,t),t)
\end

and by the

chain rule In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h ...

for partial differentiation, Lagrange's equations are invariant under this transformation;

\frac\frac = \frac+\sum_j\lambda_j\frac.

Cyclic coordinates and conserved momenta

An important property of the Lagrangian is that conserved quantities can easily be read off from it. The ''generalized momentum'' "canonically conjugate to" the coordinate ''q_i'' is defined by

p_i =\frac.

If the Lagrangian ''L'' does ''not'' depend on some coordinate ''q_i'', it follows immediately from the Euler–Lagrange equations that

\dot_i = \frac\frac = \frac = 0

and integrating shows the corresponding generalized momentum equals a constant, a conserved quantity. This is a special case of

Noether's theorem Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems (see Noether's second theorem) published by the mat ...

. Such coordinates are called "cyclic" or "ignorable". For example, a system may have a Lagrangian

L(r,\theta,\dot,\dot,\dot,\dot,\dot,t),

where ''r'' and ''z'' are lengths along straight lines, ''s'' is an arc length along some curve, and ''θ'' and ''φ'' are angles. Notice ''z'', ''s'', and ''φ'' are all absent in the Lagrangian even though their velocities are not. Then the momenta

p_z =\frac,\quad p_s =\frac,\quad p_\phi =\frac,

are all conserved quantities. The units and nature of each generalized momentum will depend on the corresponding coordinate; in this case ''p''_''z'' is a translational momentum in the ''z'' direction, ''p''_''s'' is also a translational momentum along the curve ''s'' is measured, and ''p''_''φ'' is an angular momentum in the plane the angle ''φ'' is measured in. However complicated the motion of the system is, all the coordinates and velocities will vary in such a way that these momenta are conserved.

Energy

Given a Lagrangian

L,

the

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

of the corresponding mechanical system is, by definition,

H = \biggl(\sum^n_ _i\frac\biggr) - L.

This quantity will be equivalent to energy if the generalized coordinates are natural coordinates, i.e., they have no explicit time dependence when expressing position vector:

\mathbf= \mathbf(q_1, \cdots, q_n)

. From:

T = \fracv^2 = \frac \sum_ \left(\frac \dot_i       \right)\cdot \left(\frac \dot_j\right) = \frac \sum_ a_ \dot_i\dot_j

\sum^n_ \dot_k \frac = \sum^n_ \dot_k \frac = \frac\left( 2 \sum_ a_ \dot_i\dot_j\right) = 2T

H = \left(\sum^n_ _i\frac\right) - L = 2T - (T - V) = T + V = E

where

a_=\frac\cdot \frac

is a symmetric matrix that is defined for the derivation.

Invariance under coordinate transformations

At every time instant ''t'', the energy is invariant under configuration space coordinate changes , i.e. (using natural coordinates)

E(\mathbf,\dot\mathbf,t) = E(\mathbf,\dot\mathbf,t).

Besides this result, the proof below shows that, under such change of coordinates, the derivatives

\partial L/\partial _i

change as coefficients of a linear form.

Conservation

In Lagrangian mechanics, the system is closed if and only if its Lagrangian

L

does not explicitly depend on time. The energy conservation law states that the energy

E

of a closed system is an integral of motion. More precisely, let be an ''extremal''. (In other words, satisfies the Euler–Lagrange equations). Taking the total time-derivative of ''L'' along this extremal and using the EL equations leads to

\begin
\frac &= \dot \frac+\ddot \frac + \frac \\
- \frac &= \frac\left(\frac\right)\dot+\ddot \frac - \dot \\
- \frac &= \frac\left( \frac \mathbf \dot q  - L \right)= \frac
\end

If the Lagrangian ''L'' does not explicitly depend on time, then , then ''H'' does not vary with time evolution of particle, indeed, an integral of motion, meaning that

H(\mathbf(t), \dot\mathbf(t), t) = \text.

Hence, if the chosen coordinates were natural coordinates, the energy is conserved.

Kinetic and potential energies

Under all these circumstances, the constant

E = T + V

is the total energy of the system. The kinetic and potential energies still change as the system evolves, but the motion of the system will be such that their sum, the total energy, is constant. This is a valuable simplification, since the energy ''E'' is a constant of integration that counts as an arbitrary constant for the problem, and it may be possible to integrate the velocities from this energy relation to solve for the coordinates.

Mechanical similarity

If the potential energy is a

homogeneous function In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar (mathematics), scalar, then the function's value is multiplied by some p ...

of the coordinates and independent of time, and all position vectors are scaled by the same nonzero constant ''α'', , so that

V(\alpha\mathbf_1,\alpha\mathbf_2,\ldots, \alpha\mathbf_N)=\alpha^N V(\mathbf_1,\mathbf_2,\ldots, \mathbf_N)

and time is scaled by a factor ''β'', ''t''′ = ''βt'', then the velocities v_''k'' are scaled by a factor of ''α''/''β'' and the kinetic energy ''T'' by (''α''/''β'')². The entire Lagrangian has been scaled by the same factor if

\frac=\alpha^N \quad\Rightarrow\quad \beta = \alpha^.

Since the lengths and times have been scaled, the trajectories of the particles in the system follow geometrically similar paths differing in size. The length ''l'' traversed in time ''t'' in the original trajectory corresponds to a new length ''l''′ traversed in time ''t''′ in the new trajectory, given by the ratios

\frac= \left(\frac\right)^.

Interacting particles

For a given system, if two subsystems ''A'' and ''B'' are non-interacting, the Lagrangian ''L'' of the overall system is the sum of the Lagrangians ''L''_''A'' and ''L''_''B'' for the subsystems:

L = L_A + L_B.

If they do interact this is not possible. In some situations, it may be possible to separate the Lagrangian of the system ''L'' into the sum of non-interacting Lagrangians, plus another Lagrangian ''L''_''AB'' containing information about the interaction,

L = L_A + L_B + L_.

This may be physically motivated by taking the non-interacting Lagrangians to be kinetic energies only, while the interaction Lagrangian is the system's total potential energy. Also, in the limiting case of negligible interaction, ''L''_''AB'' tends to zero reducing to the non-interacting case above. The extension to more than two non-interacting subsystems is straightforward – the overall Lagrangian is the sum of the separate Lagrangians for each subsystem. If there are interactions, then interaction Lagrangians may be added.

Consequences of singular Lagrangians

From the Euler-Lagrange equations, it follows that:

\begin
&\frac\frac-\frac =0\\
&++\frac-=0\\
&\sum_W_(q,\dot,t)\ddot_=-\frac-\sum_\dot_,\\
\end

where the matrix is defined as

W_=\frac

. If the matrix

W

is non-singular, the above equations can be solved to represent

\ddot q

as a function of

(\dot q, q, t)

. If the matrix is non-invertible, it would not be possible to represent all

\ddot q

's as a function of

(\dot q, q, t)

but also, the Hamiltonian equations of motions will not take the standard form.

Examples

The following examples apply Lagrange's equations of the second kind to mechanical problems.

Conservative force

A particle of mass ''m'' moves under the influence of a

conservative force In physics, a conservative force is a force with the property that the total work done by the force in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the total work don ...

derived from the

gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...

∇ of a

scalar potential In mathematical physics, scalar potential describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one p ...

,

\mathbf = -\boldsymbol V(\mathbf).

If there are more particles, in accordance with the above results, the total kinetic energy is a sum over all the particle kinetic energies, and the potential is a function of all the coordinates.

Cartesian coordinates

The Lagrangian of the particle can be written

L(x,y,z, \dot, \dot,\dot) = \frac m (\dot^2 + \dot^2 + \dot^2) - V(x,y,z).

The equations of motion for the particle are found by applying the Euler–Lagrange equation, for the ''x'' coordinate

\frac \left( \frac \right) = \frac,

with derivatives

\frac = - \frac,\quad \frac  = m \dot,\quad \frac \left( \frac \right) = m \ddot,

hence

m \ddot = - \frac,

and similarly for the ''y'' and ''z'' coordinates. Collecting the equations in vector form we find

m\ddot=-\boldsymbol V

which is

Newton's second law of motion Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body re ...

for a particle subject to a conservative force.

Polar coordinates in 2D and 3D

Using the spherical coordinates as commonly used in physics (ISO 80000-2:2019 convention), where ''r'' is the radial distance to origin, ''θ'' is polar angle (also known as colatitude, zenith angle, normal angle, or inclination angle), and ''φ'' is the azimuthal angle, the Lagrangian for a central potential is

L = \frac(\dot^2+r^2\dot^2 +r^2\sin^2\theta \, \dot^2)-V(r).

So, in spherical coordinates, the Euler–Lagrange equations are

m\ddot-mr(\dot^2+\sin^2\theta \, \dot^2)+\frac =0,

\frac(mr^2\dot) -mr^2\sin\theta\cos\theta \, \dot^2=0,

\frac(mr^2\sin^2\theta \, \dot)=0.

The ''φ'' coordinate is cyclic since it does not appear in the Lagrangian, so the conserved momentum in the system is the angular momentum

p_\varphi = \frac = mr^2\sin^2\theta \dot,

in which ''r'', ''θ'' and ''dφ''/''dt'' can all vary with time, but only in such a way that ''p''_''φ'' is constant. The Lagrangian in two-dimensional polar coordinates is recovered by fixing ''θ'' to the constant value ''π''/2.

Pendulum on a movable support

Consider a pendulum of mass ''m'' and length ''ℓ'', which is attached to a support with mass ''M'', which can move along a line in the

x

-direction. Let

x

be the coordinate along the line of the support, and let us denote the position of the pendulum by the angle

\theta

from the vertical. The coordinates and velocity components of the pendulum bob are

\begin
& x_\mathrm = x + \ell \sin \theta & \quad \Rightarrow \quad \dot_\mathrm = \dot + \ell \dot \cos \theta \\
& y_\mathrm = - \ell \cos\theta & \quad \Rightarrow \quad \dot_\mathrm = \ell \dot \sin \theta.
\end

The generalized coordinates can be taken to be

x

and

\theta

. The kinetic energy of the system is then

T = \frac M \dot^2 + \frac m \left( \dot_\mathrm^2 + \dot_\mathrm^2 \right)

and the potential energy is

V = m g y_\mathrm

giving the Lagrangian

+ m g \ell \cos \theta \\ &=& \frac \left( M + m \right) \dot x^2 + m \dot x \ell \dot \theta \cos \theta + \frac m \ell^2 \dot \theta ^2 + m g \ell \cos \theta. \end

Since ''x'' is absent from the Lagrangian, it is a cyclic coordinate. The conserved momentum is

p_x = \frac = (M + m) \dot x + m \ell \dot\theta \cos\theta,

and the Lagrange equation for the support coordinate

x

is

(M + m) \ddot x + m \ell \ddot\theta\cos\theta-m \ell \dot\theta ^2 \sin\theta = 0.

The Lagrange equation for the angle ''θ'' is

+ m \ell (\dot x \dot \theta + g) \sin\theta = 0;

and simplifying

\ddot\theta + \frac \cos\theta + \frac \sin\theta = 0.

These equations may look quite complicated, but finding them with Newton's laws would have required carefully identifying all forces, which would have been much more laborious and prone to errors. By considering limit cases, the correctness of this system can be verified: For example,

\ddot x \to 0

should give the equations of motion for a

simple pendulum A pendulum is a device made of a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate i ...

that is at rest in some

inertial frame In classical physics and special relativity, an inertial frame of reference (also called an inertial space or a Galilean reference frame) is a frame of reference in which objects exhibit inertia: they remain at rest or in uniform motion relative ...

, while

\ddot\theta \to 0

should give the equations for a pendulum in a constantly accelerating system, etc. Furthermore, it is trivial to obtain the results numerically, given suitable starting conditions and a chosen time step, by stepping through the results iteratively.

Two-body central force problem

Two bodies of masses and with position vectors and are in orbit about each other due to an attractive central potential . We may write down the Lagrangian in terms of the position coordinates as they are, but it is an established procedure to convert the two-body problem into a one-body problem as follows. Introduce the Jacobi coordinates; the separation of the bodies and the location of the

center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the barycenter or balance point) is the unique point at any given time where the weight function, weighted relative position (vector), position of the d ...

. The Lagrangian is thenThe Lagrangian also can be written explicitly for a rotating frame. See Padmanabhan, 2000.

L = \underbrace_ + \underbrace_

where is the total mass, is the

reduced mass In physics, reduced mass is a measure of the effective inertial mass of a system with two or more particles when the particles are interacting with each other. Reduced mass allows the two-body problem to be solved as if it were a one-body probl ...

, and the potential of the radial force, which depends only on the

magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...

of the separation . The Lagrangian splits into a ''center-of-mass'' term and a ''relative motion'' term . The Euler–Lagrange equation for is simply

M\ddot = 0,

which states the center of mass moves in a straight line at constant velocity. Since the relative motion only depends on the magnitude of the separation, it is ideal to use polar coordinates and take ,

L_\text = \frac \mu \left(\dot^2 +r^2 \dot^2 \right) - V(r),

so is a cyclic coordinate with the corresponding conserved (angular) momentum

p_\theta = \frac  = \mu r^2 \dot \theta = \ell.

The radial coordinate and angular velocity can vary with time, but only in such a way that is constant. The Lagrange equation for is

\mu r \dot \theta ^2 -\frac  = \mu \ddot.

This equation is identical to the radial equation obtained using Newton's laws in a ''co-rotating'' reference frame, that is, a frame rotating with the reduced mass so it appears stationary. Eliminating the angular velocity from this radial equation,

\mu \ddot r = -\frac + \frac.

which is the equation of motion for a one-dimensional problem in which a particle of mass is subjected to the inward central force and a second outward force, called in this context the ''(Lagrangian) centrifugal force'' (see centrifugal force#Other uses of the term):

F_ = \mu r \dot \theta ^2 = \frac .

Of course, if one remains entirely within the one-dimensional formulation, enters only as some imposed parameter of the external outward force, and its interpretation as angular momentum depends upon the more general two-dimensional problem from which the one-dimensional problem originated. If one arrives at this equation using Newtonian mechanics in a co-rotating frame, the interpretation is evident as the centrifugal force in that frame due to the rotation of the frame itself. If one arrives at this equation directly by using the generalized coordinates and simply following the Lagrangian formulation without thinking about frames at all, the interpretation is that the centrifugal force is an outgrowth of ''using polar coordinates''. As Hildebrand says: "Since such quantities are not true physical forces, they are often called ''inertia forces''. Their presence or absence depends, not upon the particular problem at hand, but ''upon the coordinate system chosen''." In particular, if Cartesian coordinates are chosen, the centrifugal force disappears, and the formulation involves only the central force itself, which provides the

centripetal force Centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is the force that makes a body follow a curved trajectory, path. The direction of the centripetal force is always orthogonality, orthogonal to the motion of the bod ...

for a curved motion. This viewpoint, that fictitious forces originate in the choice of coordinates, often is expressed by users of the Lagrangian method. This view arises naturally in the Lagrangian approach, because the frame of reference is (possibly unconsciously) selected by the choice of coordinates. For example, see for a comparison of Lagrangians in an inertial and in a noninertial frame of reference. See also the discussion of "total" and "updated" Lagrangian formulations in. Unfortunately, this usage of "inertial force" conflicts with the Newtonian idea of an inertial force. In the Newtonian view, an inertial force originates in the acceleration of the frame of observation (the fact that it is not an

inertial frame of reference In classical physics and special relativity, an inertial frame of reference (also called an inertial space or a Galilean reference frame) is a frame of reference in which objects exhibit inertia: they remain at rest or in uniform motion relative ...

), not in the choice of coordinate system. To keep matters clear, it is safest to refer to the Lagrangian inertial forces as ''generalized'' inertial forces, to distinguish them from the Newtonian vector inertial forces. That is, one should avoid following Hildebrand when he says (p. 155) "we deal ''always'' with ''generalized'' forces, velocities accelerations, and momenta. For brevity, the adjective "generalized" will be omitted frequently." It is known that the Lagrangian of a system is not unique. Within the Lagrangian formalism the Newtonian fictitious forces can be identified by the existence of alternative Lagrangians in which the fictitious forces disappear, sometimes found by exploiting the symmetry of the system.

Extensions to include non-conservative forces

Dissipative forces

Dissipation In thermodynamics, dissipation is the result of an irreversible process that affects a thermodynamic system. In a dissipative process, energy ( internal, bulk flow kinetic, or system potential) transforms from an initial form to a final form, wh ...

(i.e. non-conservative systems) can also be treated with an effective Lagrangian formulated by a certain doubling of the degrees of freedom. In a more general formulation, the forces could be both conservative and

viscous Viscosity is a measure of a fluid's rate-dependent resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for example, syrup h ...

. If an appropriate transformation can be found from the F_''i'',

Rayleigh Rayleigh may refer to: Science *Rayleigh scattering *Rayleigh–Jeans law *Rayleigh waves *Rayleigh (unit), a unit of photon flux named after the 4th Baron Rayleigh *Rayl, rayl or Rayleigh, two units of specific acoustic impedance and characte ...

suggests using a dissipation function, ''D'', of the following form:

D = \frac  \sum_^m \sum_^m C_ \dot_j \dot_k,

where ''C''_''jk'' are constants that are related to the damping coefficients in the physical system, though not necessarily equal to them. If ''D'' is defined this way, then

Q_j = - \frac  - \frac

and

\frac \left ( \frac  \right ) - \frac  + \frac  = 0.

Electromagnetism

A test particle is a particle whose

mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...

and charge are assumed to be so small that its effect on external system is insignificant. It is often a hypothetical simplified point particle with no properties other than mass and charge. Real particles like

electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...

s and

up quark The up quark or u quark (symbol: u) is the lightest of all quarks, a type of elementary particle, and a significant constituent of matter. It, along with the down quark, forms the neutrons (one up quark, two down quarks) and protons (two up quark ...

s are more complex and have additional terms in their Lagrangians. Not only can the fields form non conservative potentials, these potentials can also be velocity dependent. The Lagrangian for a

charged particle In physics, a charged particle is a particle with an electric charge. For example, some elementary particles, like the electron or quarks are charged. Some composite particles like protons are charged particles. An ion, such as a molecule or atom ...

with

electrical charge Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...

, interacting with an

electromagnetic field An electromagnetic field (also EM field) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...

, is the prototypical example of a velocity-dependent potential. The electric

scalar potential In mathematical physics, scalar potential describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one p ...

and

magnetic vector potential In classical electromagnetism, magnetic vector potential (often denoted A) is the vector quantity defined so that its curl is equal to the magnetic field, B: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the ma ...

are defined from the

electric field An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...

and

magnetic field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...

as follows:

\mathbf = - \boldsymbol\phi - \frac, \quad \mathbf = \boldsymbol \times \mathbf.

The Lagrangian of a massive charged test particle in an electromagnetic field

L = \tfracm \dot^2 + q\, \dot \cdot \mathbf - q \phi,

is called minimal coupling. This is a good example of when the common

rule of thumb In English language, English, the phrase ''rule of thumb'' refers to an approximate method for doing something, based on practical experience rather than theory. This usage of the phrase can be traced back to the 17th century and has been associat ...

that the Lagrangian is the kinetic energy minus the potential energy is incorrect. Combined with Euler–Lagrange equation, it produces the

Lorentz force In electromagnetism, the Lorentz force is the force exerted on a charged particle by electric and magnetic fields. It determines how charged particles move in electromagnetic environments and underlies many physical phenomena, from the operation ...

law

m \ddot = q \mathbf + q \dot \times \mathbf

Under

gauge transformation In the physics of gauge theory, gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant Degrees of freedom (physics and chemistry), degrees of freedom in field (physics), field variab ...

:

\mathbf \rightarrow \mathbf+\boldsymbol f, \quad \phi \rightarrow \phi-\dot f,

where is any scalar function of space and time, the aforementioned Lagrangian transforms like:

L \rightarrow L+q\left(\dot\cdot\boldsymbol+\frac\right)f=L+q\frac,

which still produces the same Lorentz force law. Note that the canonical momentum (conjugate to position ) is the kinetic momentum plus a contribution from the field (known as the potential momentum):

\mathbf = \frac = m \dot + q \mathbf.

This relation is also used in the minimal coupling prescription in

quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

and

quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...

. From this expression, we can see that the canonical momentum is not gauge invariant, and therefore not a measurable physical quantity; However, if is cyclic (i.e. Lagrangian is independent of position ), which happens if the and fields are uniform, then this canonical momentum given here is the conserved momentum, while the measurable physical kinetic momentum is not.

Other contexts and formulations

The ideas in Lagrangian mechanics have numerous applications in other areas of physics, and can adopt generalized results from the calculus of variations.

Alternative formulations of classical mechanics

A closely related formulation of classical mechanics is

Hamiltonian mechanics In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gener ...

. The Hamiltonian is defined by

H = \sum_^n \dot_i\frac - L

and can be obtained by performing a

Legendre transformation In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, is an involutive transformation on real-valued functions that are convex on a rea ...

on the Lagrangian, which introduces new variables canonically conjugate to the original variables. For example, given a set of generalized coordinates, the variables canonically conjugate are the generalized momenta. This doubles the number of variables, but makes differential equations first order. The Hamiltonian is a particularly ubiquitous quantity in

quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

(see

Hamiltonian (quantum mechanics) In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's ''energy spectrum'' or its set of ''energy eigenvalu ...

).

Routhian mechanics In classical mechanics, Routh's procedure or Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the Routhian is the function which replaces both the La ...

is a hybrid formulation of Lagrangian and Hamiltonian mechanics, which is not often used in practice but an efficient formulation for cyclic coordinates.

Momentum space formulation

The Euler–Lagrange equations can also be formulated in terms of the generalized momenta rather than generalized coordinates. Performing a Legendre transformation on the generalized coordinate Lagrangian obtains the generalized momenta Lagrangian in terms of the original Lagrangian, as well the EL equations in terms of the generalized momenta. Both Lagrangians contain the same information, and either can be used to solve for the motion of the system. In practice generalized coordinates are more convenient to use and interpret than generalized momenta.

Higher derivatives of generalized coordinates

There is no mathematical reason to restrict the derivatives of generalized coordinates to first order only. It is possible to derive modified EL equations for a Lagrangian containing higher order derivatives, see Euler–Lagrange equation for details. However, from the physical point-of-view there is an obstacle to include time derivatives higher than the first order, which is implied by Ostrogradsky's construction of a canonical formalism for nondegenerate higher derivative Lagrangians, see Ostrogradsky instability

Optics

Lagrangian mechanics can be applied to

geometrical optics Geometrical optics, or ray optics, is a model of optics that describes light Wave propagation, propagation in terms of ''ray (optics), rays''. The ray in geometrical optics is an abstract object, abstraction useful for approximating the paths along ...

, by applying variational principles to rays of light in a medium, and solving the EL equations gives the equations of the paths the light rays follow.

Relativistic formulation

Lagrangian mechanics can be formulated in

special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...

and

general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...

. Some features of Lagrangian mechanics are retained in the relativistic theories but difficulties quickly appear in other respects. In particular, the EL equations take the same form, and the connection between cyclic coordinates and conserved momenta still applies, however the Lagrangian must be modified and is not simply the kinetic minus the potential energy of a particle. Also, it is not straightforward to handle multiparticle systems in a manifestly covariant way, it may be possible if a particular frame of reference is singled out.

Quantum mechanics

In

quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

, action and quantum-mechanical phase are related via the

Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...

, and the principle of stationary action can be understood in terms of

constructive interference In physics, interference is a phenomenon in which two coherence (physics), coherent waves are combined by adding their intensities or displacements with due consideration for their phase (waves), phase difference. The resultant wave may have ...

of

wave function In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...

s. In 1948, Feynman discovered the

path integral formulation The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or ...

extending the

principle of least action Action principles lie at the heart of fundamental physics, from classical mechanics through quantum mechanics, particle physics, and general relativity. Action principles start with an energy function called a Lagrangian describing the physical sy ...

to

quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...

for

electrons The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...

and

photons A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that ...

. In this formulation, particles travel every possible path between the initial and final states; the probability of a specific final state is obtained by summing over all possible trajectories leading to it. In the classical regime, the path integral formulation cleanly reproduces Hamilton's principle, and

Fermat's principle Fermat's principle, also known as the principle of least time, is the link between geometrical optics, ray optics and physical optics, wave optics. Fermat's principle states that the path taken by a Ray (optics), ray between two given ...

in

optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...

.

Classical field theory

In Lagrangian mechanics, the generalized coordinates form a discrete set of variables that define the configuration of a system. In

classical field theory A classical field theory is a physical theory that predicts how one or more fields in physics interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called qua ...

, the physical system is not a set of discrete particles, but rather a continuous field defined over a region of 3D space. Associated with the field is a Lagrangian density

\mathcal(\phi, \nabla \phi, \dot\phi, \mathbf,t)

defined in terms of the field and its space and time derivatives at a location r and time ''t''. Analogous to the particle case, for non-relativistic applications the Lagrangian density is also the kinetic energy density of the field, minus its potential energy density (this is not true in general, and the Lagrangian density has to be "reverse engineered"). The Lagrangian is then the

volume integral In mathematics (particularly multivariable calculus), a volume integral (∭) is an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applica ...

of the Lagrangian density over 3D space

L(t) = \int  \mathcal \, \mathrm^3 \mathbf

where d³r is a 3D differential

volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form \ma ...

. The Lagrangian is a function of time since the Lagrangian density has implicit space dependence via the fields, and may have explicit spatial dependence, but these are removed in the integral, leaving only time in as the variable for the Lagrangian.

Noether's theorem

The action principle, and the Lagrangian formalism, are tied closely to

Noether's theorem Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems (see Noether's second theorem) published by the mat ...

, which connects physical conserved quantities to continuous

symmetries Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

of a physical system. If the Lagrangian is invariant under a symmetry, then the resulting equations of motion are also invariant under that symmetry. This characteristic is very helpful in showing that theories are consistent with either

special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...

or

general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...

.

Footnotes

Notes

References

* * * * * * * * * * * * * ''The Principle of Least Action'', R. Feynman * * * * * * * * * * * * * * * * * * * *

External links

*
Principle of least action interactive
Excellent interactive explanation/webpage
Joseph Louis de Lagrange - Œuvres complètes
(Gallica-Math)
Constrained motion and generalized coordinates
page 4 {{Authority control

Introduction

Lagrangian

From Newtonian to Lagrangian mechanics

Newton's laws

D'Alembert's principle

Equations of motion from D'Alembert's principle

Euler–Lagrange equations and Hamilton's principle

Lagrange multipliers and constraints

Properties of the Lagrangian

Non-uniqueness

Invariance under point transformations

Cyclic coordinates and conserved momenta

Energy

Invariance under coordinate transformations

Conservation

Kinetic and potential energies

Mechanical similarity

Interacting particles

Consequences of singular Lagrangians

Examples

Conservative force

Cartesian coordinates

Polar coordinates in 2D and 3D

Pendulum on a movable support

Two-body central force problem

Extensions to include non-conservative forces

Dissipative forces

Electromagnetism

Other contexts and formulations

Alternative formulations of classical mechanics

Momentum space formulation

Higher derivatives of generalized coordinates

Optics

Relativistic formulation

Quantum mechanics

Classical field theory

Noether's theorem

See also

Footnotes

Notes

References

Further reading

External links