HOME

TheInfoList



OR:

In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the
stationary-action principle The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the ''action'' of a mechanical system, yields the equations of motion for that system. The principle states that ...
(also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer
Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaMécanique analytique ''Mécanique analytique'' (1788–89) is a two volume French treatise on analytical mechanics, written by Joseph-Louis Lagrange, and published 101 years following Isaac Newton's '' Philosophiæ Naturalis Principia Mathematica''. It consolidate ...
''. Lagrangian mechanics describes a mechanical system as a pair (M,L) consisting of a configuration space M and a smooth function L within that space called a ''Lagrangian''. By convention, L = T - V, where T and V are the
kinetic Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory of gases, Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to i ...
and potential 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 (a maximum, minimum, or saddle) 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

Suppose there exists a bead sliding around on a wire, or a swinging simple pendulum, etc. If one tracks each of the massive objects (bead, pendulum bob, etc.) as a particle, calculation of the motion of the particle using
Newtonian mechanics Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
would require solving for the time-varying constraint force required to keep the particle in the constrained motion (reaction force exerted by the wire on the bead, or tension in the pendulum rod). For the same problem using Lagrangian mechanics, one looks at the path the particle can take and chooses a convenient set of ''independent'' generalized coordinates that completely characterize the possible motion of the particle. This choice eliminates the need for the constraint force to enter into the resultant 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. For a system of ''N'' point particles with masses ''m''1, ''m''2, ..., ''mN'', 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 the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or s ...
, denoted r1, r2, ..., r''N''.
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
are often sufficient, so r1 = (''x''1, ''y''1, ''z''1), r2 = (''x''2, ''y''2, ''z''2) and so on. In three dimensional space, 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 the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
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 r = (''x'', ''y'', ''z''). The velocity of each particle is how fast the particle moves along its path of motion, and is the time derivative of its position, thus \mathbf_1 = \frac, \mathbf_2 = \frac,\ldots, \mathbf_N = \frac In Newtonian mechanics, the equations of motion are given by Newton's laws. The second law "net
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
equals mass times acceleration",\sum \mathbf = m\fracapplies to each particle. For an ''N'' particle system in 3 dimensions, there are 3''N'' second order ordinary differential equations in the positions of the particles to solve for.


The 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 can be defined by L = T - V where T = \frac \sum_^N m_k v^2_k is the total kinetic energy of the system, equalling the
sum Sum most commonly means the total of two or more numbers added together; see addition. Sum can also refer to: Mathematics * Sum (category theory), the generic concept of summation in mathematics * Sum, the result of summation, the additio ...
Σ of the kinetic energies of the particles, and ''V'' is the
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
of the system. Kinetic energy is the energy of the system's motion, and ''vk''2 = v''k'' · v''k'' is the magnitude squared of velocity, equivalent to the dot product of the velocity with itself. The kinetic energy is a function only of the velocities v''k'', not the positions r''k'' nor time ''t'', so ''T'' = ''T''(v1, v2, ...). The
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
of the system reflects the energy of interaction between the particles, i.e. how much energy any one particle will have due to all the others and other external influences. For conservative forces (e.g. Newtonian gravity), it is a function of the position vectors of the particles only, so ''V'' = ''V''(r1, r2, ...). For those non-conservative forces which can be derived from an appropriate potential (e.g. electromagnetic potential), the velocities will appear also, ''V'' = ''V''(r1, r2, ..., v1, v2, ...). If there is some external field or external driving force changing with time, the potential will change with time, so most generally ''V'' = ''V''(r1, r2, ..., v1, v2, ..., ''t''). The above form of ''L'' does not hold in relativistic Lagrangian mechanics, and must be replaced by a function consistent with special or general relativity. Also, for dissipative forces another function must be introduced alongside ''L''. 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 ''f''(r, ''t'') = 0. If the number of constraints in the system is ''C'', then each constraint has an equation, ''f''1(r, ''t'') = 0, ''f''2(r, ''t'') = 0, ..., ''fC''(r, ''t'') = 0, each of which could apply to any of the particles. If particle ''k'' is subject to constraint ''i'', then ''fi''(r''k'', ''t'') = 0. 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 A nonholonomic system in physics and mathematics is a physical system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differential constraints and non-linear constraints, s ...
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 nonintegrable, when the constraints have inequalities, or with complicated non-conservative forces like friction. Nonholonomic constraints require special treatment, and one may have to revert to Newtonian mechanics, or use other methods. If ''T'' or ''V'' or both depend explicitly on time due to time-varying constraints or external influences, the Lagrangian ''L''(r1, r2, ... v1, v2, ... ''t'') is ''explicitly time-dependent''. If neither the potential nor the kinetic energy depend on time, then the Lagrangian ''L''(r1, r2, ... v1, v2, ...) is ''explicitly independent of time''. In either case, the Lagrangian will always have 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 ''λi'' for each constraint equation ''fi'', 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). Part ...
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. This procedure does increase the number of equations to solve compared to Newton's laws, from 3''N'' to 3''N'' + ''C'', 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 variable Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...
s, and derivatives of the Lagrangian are taken with respect to these separately according to the usual
differentiation rules This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus. Elementary rules of differentiation Unless otherwise stated, all functions are functions of real numbers (R) that return real ...
(e.g. the partial derivative of ''L'' with respect to the ''z''-velocity component of particle 2, defined by v_ = dz_2 / dt , is just \partial L / \partial v_ ; no awkward chain rules 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 ''n'' = 3''N'' − ''C''. We can transform each position vector to a common set of ''n'' generalized coordinates, conveniently written as an ''n''-tuple q = (''q''1, ''q''2, ... ''qn''), by expressing each position vector, and hence the position coordinates, as functions of the generalized coordinates and time, \mathbf_k = \mathbf_k(\mathbf, t) = (x_k(\mathbf, t), y_k(\mathbf, t), z_k(\mathbf, t), t)\,. 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 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 functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
, which can also be used in mechanics. Substituting in the Lagrangian ''L''(q, dq/d''t'', ''t''), gives the equations of motion of the system. The number of equations has decreased compared to Newtonian mechanics, from 3''N'' to ''n'' = 3''N'' − ''C'' 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). Part ...
s, the results of the partial derivatives are still ordinary differential equations in the position coordinates of the particles. The total time derivative denoted d/d''t'' often involves implicit differentiation. Both equations are linear in the Lagrangian, but will generally be 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 for a particle of mass ''m'' is Newton's second law of 1687, in modern vector notation \mathbf = m \mathbf \,, where a is its acceleration and F the resultant force acting ''on'' it. In three spatial dimensions, this is a system of three coupled second order ordinary differential equations 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 condition In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For ...
s of r and v when ''t'' = 0. 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 ξ = (''ξ''1, ''ξ''2, ''ξ''3), the law in tensor index notation 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 ''Fa'' is the ''a''th contravariant component of the resultant force acting on the particle, Γ''abc'' are the Christoffel symbols of the second kind, T = \frac m g_ \frac \frac is the kinetic energy of the particle, and ''gbc'' 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, F = 0, 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 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 connection. ...
s'', the curves of extremal length between two points in space (these may end up being minimal so the shortest paths, but that is not necessary). 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, F ≠ 0, 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, the above form of Newton's law also carries over to
Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
's general relativity, 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 will 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 alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the ...
is D'Alembert's principle, introduced in 1708 by
Jacques Bernoulli Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Leib ...
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 zero ...
, and developed by D'Alembert in 1743 to solve dynamical problems. The principle asserts for ''N'' particles the virtual work, i.e. the work along a virtual displacement, δr''k'', is zero: \sum_^N ( \mathbf _k + \mathbf _k - m_k \mathbf_k )\cdot \delta \mathbf_k = 0\,. The '' virtual displacements'', δr''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 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 δr''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 r''k'' = (''xk'', ''yk'', ''zk'') are linked together by a constraint equation, so are those of the virtual displacements ''δ''r''k'' = (''δxk'', ''δyk'', ''δzk''). 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, \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, \sum_^n \left Q_j - \left(\frac\frac - \frac \right) \right\delta q_j = 0 \,, and since these virtual displacements ''δqj'' 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 ''Qi'' can be derived from a potential ''V'' such that Q_j = \frac\frac - \frac \,, equating to Lagrange's equations and defining the Lagrangian as ''L'' = ''T'' − ''V'' 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 functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. 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 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 with respect to time can transfer the time derivative of ''δqj'' to the ∂''L''/∂(d''qj''/d''t''), in the process exchanging d(''δqj'')/d''t'' for ''δqj'', allowing the independent virtual displacements to be factorized from the derivatives of the Lagrangian, \int_^ \delta L \, \mathrmt =\int_^ \sum_^n \left(\frac\delta q_j +\frac \left(\frac\delta q_j\right) - \frac\frac\delta q_j \right) \, \mathrmt \, = \sum_^n\left frac\delta q_j\right^ + \int_^ \sum_^n \left(\frac - \frac\frac \right)\delta q_j \, \mathrmt \,. Now, if the condition ''δqj''(''t''1) = ''δqj''(''t''2) = 0 holds for all ''j'', the terms not integrated are zero. If in addition the entire time integral of ''δL'' is zero, then because the ''δqj'' are independent, and the only way for a definite integral to be zero is if the integrand equals zero, each of the coefficients of ''δqj'' 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 Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...
''; 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 angular momentum ">angular_momentum.html" ;"title="angular momentum">angular momentum [energy]·[time], or [length]·[momentum]. With this definition Hamilton's principle is \delta S = 0\,. Thus, 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 sometimes referred to as the '' principle of least action'', however the action functional need only be ''stationary'', not necessarily a maximum or a minimum value. Any variation of the functional gives an increase in the functional integral of the action. 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 functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
to mechanical problems, such as the Brachistochrone problem solved by
Jean Bernoulli Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating L ...
in 1696, as well as Leibniz,
Daniel Bernoulli Daniel Bernoulli FRS (; – 27 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applications of mathematics to mechan ...
, L'Hôpital around the same time, and
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * ''Newton'' (film), a 2017 Indian film * 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, Maupertuis,
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
, Hamilton, and others. Hamilton's principle can be applied to
nonholonomic constraints A nonholonomic system in physics and mathematics is a physical system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differential constraints and non-linear constraints, s ...
if the constraint equations can be put into a certain form, a linear combination 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 ''δ''r''k''(''t''1) = ''δ''r''k''(''t''2) = 0 for all ''k''. What cannot be done is to simply equate the coefficients of δr''k'' to zero because the δr''k'' are not independent. Instead, the method of Lagrange multipliers can be used to include the constraints. Multiplying each constraint equation ''fi''(r''k'', ''t'') = 0 by a Lagrange multiplier ''λi'' 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 ''λi'' 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 rk coordinates only, substituting the Lagrangian ''L'' = ''T'' − ''V'' 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 ''L' = aL'' + ''b'' will describe the same motion as ''L''. If one restricts as above to trajectories \mathbf over a given time interval _\text,t_\text/math> and fixed end points P_\text = \mathbf(t_\text) and P_\text = \mathbf(t_\text), then two Lagrangians describing the same system can differ by the "total time derivative" of a function f(\mathbf,t): :L'(\mathbf,\dot\mathbf,t) = L(\mathbf,\dot\mathbf,t) + \frac, where \textstyle \frac means \textstyle \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 \begin S' mathbf= \int\limits^_ L'(\mathbf(t),\dot\mathbf(t),t)\,dt = \int\limits^_ L(\mathbf(t),\dot\mathbf(t),t)\,dt + \int^_ \frac\, dt \\ = S mathbf+ f(P_\text,t_\text) - f(P_\text,t_\text), \end with the last two components f(P_\text,t_\text) and f(P_\text,t_\text) independent of \mathbf q.


Invariance under point transformations

Given a set of generalized coordinates q, if we change these variables to a new set of generalized coordinates s according to a point transformation q = q(s, ''t''), the new Lagrangian ''L''′ is a function of the new coordinates :L(\mathbf(\mathbf,t), \dot(\mathbf,\dot,t), t ) = L'(\mathbf, \dot,t) \,, and by the chain rule for partial differentiation, Lagrange's equations are invariant under this transformation; :\frac\frac = \frac \,. This may simplify the equations of motion.


Cyclic coordinates and conserved momenta

An important property of the Lagrangian is that
conserved quantities In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables, the value of which remains constant (mathematics), constant along each trajectory of the system. Not all systems have conserved quantities, and c ...
can easily be read off from it. The ''generalized momentum'' "canonically conjugate to" the coordinate ''qi'' is defined by :p_i =\frac. If the Lagrangian ''L'' does ''not'' depend on some coordinate ''qi'', 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. 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 ''pz'' is a translational momentum in the ''z'' direction, ''ps'' is also a translational momentum along the curve ''s'' is measured, and ''pφ'' 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


Definition

Given a Lagrangian L, the ''energy'' of the corresponding mechanical system is, by definition, :E = \biggl(\sum^n_ _i\frac\biggr) - L.


Invariance under coordinate transformations

At every time instant t, the energy is invariant under configuration space coordinate changes \mathbf \to \mathbf, i.e. :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 Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
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 In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a ''mathematical'' constraint, the natural consequence of the equations of motion, rather than ...
. More precisely, let \mathbf=\mathbf(t) be an ''extremal''. (In other words, \mathbf satisfies the Euler-Lagrange equations). Taking the total time-derivative of L along this extremal and using the EL equations leads to : - \frac\biggl, _ = \frac\left _\right If the Lagrangian L does not explicitly depend on time, then \partial L/\partial t = 0, so E is, indeed, an integral of motion, meaning that :E(\mathbf(t), \dot\mathbf(t), t) = \text. Hence, the energy is conserved.


Kinetic and potential energies

It also follows that the kinetic energy is a homogenous function of degree 2 in the generalized velocities. If in addition the potential ''V'' is only a function of coordinates and independent of velocities, it follows by direct calculation, or use of Euler's theorem for homogenous functions, that :\sum_^n \dot_i\frac = \sum_^n \dot_i\frac = 2T \,. 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. In the case the velocity or kinetic energy or both depends on time, then the energy is ''not'' conserved.


Mechanical similarity

If the potential energy is a
homogeneous function In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''deg ...
of the coordinates and independent of time, and all position vectors are scaled by the same nonzero constant ''α'', r''k''′ = ''α''r''k'', 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 ''LA'' and ''LB'' 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 ''LAB'' 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, ''LAB'' 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.


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 derived from the gradient ∇ of a scalar potential, :\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 basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
for a particle subject to a conservative force.


Polar coordinates in 2D and 3D

Using the spherical coordinates (r, ''θ'', ''φ'') 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 ''pφ'' 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 : \begin L &=& T - V \\ &=& \frac M \dot^2 + \frac m \left \left( \dot x + \ell \dot\theta \cos \theta \right)^2 + \left( \ell \dot\theta \sin \theta \right)^2 \right+ 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 \theta is :\frac\left m( \dot x \ell \cos\theta + \ell^2 \dot\theta ) \right+ 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 that is at rest in some
inertial frame In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
, 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 In the theory of many-particle systems, Jacobi coordinates often are used to simplify the mathematical formulation. These coordinates are particularly common in treating polyatomic molecules and chemical reactions, and in celestial mechanics ...
; 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 balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
. 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, and the potential of the radial force, which depends only on the magnitude 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 (\dot^2 +r^2 \dot^2 ) - 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 centrifugal force :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 A centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous c ...
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), 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.


Electromagnetism

A test particle is a particle whose mass 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 electrons and up quarks are more complex and have additional terms in their Lagrangians. The Lagrangian for a charged particle with electrical charge , interacting with an
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
, is the prototypical example of a velocity-dependent potential. The electric scalar potential and
magnetic vector potential In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic v ...
are defined from the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
and
magnetic field A magnetic field is a vector 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 to its own velocity and to ...
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. Combined with Euler–Lagrange equation, it produces the
Lorentz force In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
law :m \ddot = q \mathbf + q \dot \times \mathbf Under gauge transformation: :\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 and
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
. 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.


Extensions to include non-conservative forces

Dissipation (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. If an appropriate transformation can be found from the Fi, Rayleigh suggests using a dissipation function, ''D'', of the following form: :D = \frac \sum_^m \sum_^m C_ \dot_j \dot_k \, where ''Cjk'' 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\,.


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. The Hamiltonian is defined by : H = \sum_^n \dot_i\frac - L \, and can be obtained by performing a Legendre transformation 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 (see Hamiltonian (quantum mechanics)).
Routhian mechanics alt= 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 ...
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 ''L''(q, dq/d''t'', ''t'') obtains the generalized momenta Lagrangian ''L''′(p, dp/d''t'', ''t'') 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 In applied mathematics, the Ostrogradsky instability is a feature of some solutions of theories having equations of motion with more than two time derivatives (higher-derivative theories). It is suggested by a theorem of Mikhail Ostrogradsky in cla ...


Optics

Lagrangian mechanics can be applied to geometrical optics, 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 and general relativity. 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 In general relativity, a manifestly covariant equation is one in which all expressions are tensors. The operations of addition, tensor multiplication, tensor contraction, raising and lowering indices, and covariant differentiation may appear in t ...
way, it may be possible if a particular frame of reference is singled out.


Quantum mechanics

In quantum mechanics, action and quantum-mechanical phase are related via Planck's constant, and the
principle of stationary action The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the ''action'' of a mechanical system, yields the equations of motion for that system. The principle states that ...
can be understood in terms of constructive interference of wave functions. In 1948, Feynman discovered the path integral formulation extending the principle of least action to quantum mechanics for electrons 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, so they alway ...
. 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 in optics.


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 physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum ...
, the physical system is not a set of discrete particles, but rather a continuous field ''ϕ''(r, ''t'') defined over a region of 3D space. Associated with the field is a Lagrangian density :\mathcal(\phi, \nabla \phi, \partial\phi/\partial t , \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 of the Lagrangian density over 3D space : L(t) = \int \mathcal \, \mathrm^3 \mathbf where d3r is a 3D differential volume element. 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, which connects physical
conserved quantities In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables, the value of which remains constant (mathematics), constant along each trajectory of the system. Not all systems have conserved quantities, and c ...
to continuous symmetries 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 or general relativity.


See also

*
Canonical coordinates In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of ...
*
Fundamental lemma of the calculus of variations In mathematics, specifically in the calculus of variations, a variation of a function can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum (functional derivative equal ze ...
*
Functional derivative In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on ...
* Generalized coordinates * Hamiltonian mechanics * Hamiltonian optics *
Inverse problem for Lagrangian mechanics In mathematics, the inverse problem for Lagrangian mechanics is the problem of determining whether a given system of ordinary differential equations can arise as the Euler–Lagrange equations for some Lagrangian function. There has been a gr ...
, the general topic of finding a Lagrangian for a system given the equations of motion. * Lagrangian and Eulerian specification of the flow field *
Lagrangian point In celestial mechanics, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies. Mathematically, this involves the solution of th ...
*
Lagrangian system In mathematics, a Lagrangian system is a pair , consisting of a smooth fiber bundle and a Lagrangian density , which yields the Euler–Lagrange differential operator acting on sections of . In classical mechanics, many dynamical systems are Lagr ...
* Non-autonomous mechanics * Plateau's problem *
Restricted three-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...


Footnotes


Notes


References

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


Further reading

* Gupta, Kiran Chandra, ''Classical mechanics of particles and rigid bodies'' (Wiley, 1988). * * Goldstein, Herbert, et al. '' Classical Mechanics''. 3rd ed., Pearson, 2002.


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 Dynamical systems Mathematical physics