![Langrange portrait](https://upload.wikimedia.org/wikipedia/commons/1/19/Lagrange_portrait.jpg)
In
physics
Physics is the natural science that studies matter, its 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 which r ...
, Lagrangian mechanics is a formulation of
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
founded on the
stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer
Joseph-Louis Lagrange in his 1788 work, ''
Mécanique analytique''.
Lagrangian mechanics describes a mechanical system as a pair
consisting of a
configuration space and a smooth function
within that space called a ''Lagrangian''. By convention,
where
and
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
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
![Bead on wire constraint](https://upload.wikimedia.org/wikipedia/commons/8/8e/Bead_on_wire_constraint.svg)
Suppose there exists a bead sliding around on a wire, or a swinging
simple pendulum
A pendulum is 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 it back toward th ...
, 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 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
Tension may refer to:
Science
* Psychological stress
* Tension (physics), a force related to the stretching of an object (the opposite of compression)
* Tension (geology), a stress which stretches rocks in two opposite directions
* Voltage or el ...
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 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. 39 ...
s 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
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 up ...
. For a system of ''N'' point particles with
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
es ''m''
1, ''m''
2, ..., ''m
N'', each particle has a
position vector, denoted r
1, r
2, ..., r
''N''.
Cartesian coordinates are often sufficient, so r
1 = (''x''
1, ''y''
1, ''z''
1), r
2 = (''x''
2, ''y''
2, ''z''
2) and so on. In
three dimensional space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
, each position vector requires three
coordinates 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
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
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
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.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
are given by
Newton's laws
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 motio ...
. The second law "net
force equals mass times
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
",
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 whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s in the positions of the particles to solve for.
The Lagrangian
Instead of forces, Lagrangian mechanics uses the
energies
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat ...
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
where
is the total
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
of the system, equalling the
sum Σ of the kinetic energies of the particles,
and ''V'' is the
potential energy of the system.
Kinetic energy is the energy of the system's motion, and ''v
k''
2 = v
''k'' · v
''k'' is the magnitude squared of velocity, equivalent to the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
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''(v
1, v
2, ...).
The
potential energy 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 force
In physics, a conservative force is a force with the property that the total work done 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 done (the sum ...
s (e.g.
Newtonian gravity
Newton's law of universal gravitation is usually stated as 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 square of the distan ...
), it is a function of the position vectors of the particles only, so ''V'' = ''V''(r
1, r
2, ...). For those non-conservative forces which can be derived from an appropriate potential (e.g.
electromagnetic potential
An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. ...
), the velocities will appear also, ''V'' = ''V''(r
1, r
2, ..., v
1, v
2, ...). If there is some external field or external driving force changing with time, the potential will change with time, so most generally ''V'' = ''V''(r
1, r
2, ..., v
1, v
2, ..., ''t'').
The above form of ''L'' does not hold in
relativistic Lagrangian mechanics
In theoretical physics, relativistic Lagrangian mechanics is Lagrangian mechanics applied in the context of special relativity and general relativity.
Lagrangian formulation in special relativity
Lagrangian mechanics can be formulated in specia ...
, 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
In classical mechanics, holonomic constraints are relations between the position variables (and possibly time) that can be expressed in the following form:
:f(u_1, u_2, u_3,\ldots, u_n, t) = 0
where \ are the ''n'' generalized coordinates that d ...
; 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, ..., ''f
C''(r, ''t'') = 0, each of which could apply to any of the particles. If particle ''k'' is subject to constraint ''i'', then ''f
i''(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 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''(r
1, r
2, ... v
1, v
2, ... ''t'') is ''explicitly time-dependent''. If neither the potential nor the kinetic energy depend on time, then the Lagrangian ''L''(r
1, r
2, ... v
1, v
2, ...) 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
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied e ...
''λ
i'' for each constraint equation ''f
i'', and
are each shorthands for a vector of
partial derivatives with respect to the indicated variables (not a derivative with respect to the entire vector).
[Sometimes in this context the ]variational 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 w ...
denoted and defined as
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 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 variables, and derivatives of the Lagrangian are taken with respect to these separately according to the usual
differentiation rules (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 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(x)=f(g(x)) for every , ...
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 ''n'' = 3''N'' − ''C''. 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. 39 ...
, conveniently written as an ''n''-tuple q = (''q''
1, ''q''
2, ... ''q
n''), by expressing each position vector, and hence the position coordinates, as
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
s of the generalized coordinates and time,
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
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
.
With these definitions, the
Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
s, or Lagrange's equations of the second kind
are mathematical results from the
calculus of variations, which can also be used in mechanics. Substituting in the Lagrangian ''L''(q, dq/d''t'', ''t''), 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.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
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 derivatives, the results of the partial derivatives are still
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
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 will generally be nonlinear coupled equations in the coordinates.
From Newtonian to Lagrangian mechanics
Newton's laws
![GodfreyKneller-IsaacNewton-1689](https://upload.wikimedia.org/wikipedia/commons/3/39/GodfreyKneller-IsaacNewton-1689.jpg)
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.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
for a particle of mass ''m'' is
Newton's second law
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 moti ...
of 1687, in modern vector notation
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 equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
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 ''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
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 locally inve ...
ξ = (''ξ''
1, ''ξ''
2, ''ξ''
3), the law in
tensor index notation
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be c ...
is the ''"Lagrangian form"''
where ''F
a'' is the ''a''th
contravariant component of the resultant force acting on the particle, Γ''
abc'' 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 surfaces or other manifolds endowed with a metric, allowing distanc ...
of the second kind,
is the kinetic energy of the particle, and ''g
bc'' the
covariant components of the ''
metric tensor'' 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 ''
geodesics'', 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
Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Cu ...
, the above form of Newton's law also carries over to
Einstein's
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, 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,
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
![Alembert](https://upload.wikimedia.org/wikipedia/commons/d/df/Alembert.jpg)
A fundamental result in
analytical mechanics is
D'Alembert's principle
D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond d'Alembert. D'Alembert ...
, introduced in 1708 by
Jacques Bernoulli to understand
static equilibrium, and developed by
D'Alembert
Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the '' Encyclopé ...
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:
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'', 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
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 for ...
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
for particle ''k'' subject to a constraint force, however
because of the constraint equations on the r''k'' coordinates.]
so that
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'' = (''x
k'', ''y
k'', ''z
k'') 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 ''δ''r
''k'' = (''δx
k'', ''δy
k'', ''δz
k''). 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 ...
,
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 force Generalized forces find use in Lagrangian mechanics, where they play a role conjugate to generalized coordinates. They are obtained from the applied forces, Fi, i=1,..., n, acting on a system that has its configuration defined in terms of generali ...
s
so that
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:
Now D'Alembert's principle is in the generalized coordinates as required,
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,
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
![Least action principle](https://upload.wikimedia.org/wikipedia/commons/1/1c/Least_action_principle.svg)
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
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
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 ''variation'' of the Lagrangian is
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 ...
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 antiderivative. ...
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,
Now, if the condition ''δq
j''(''t''
1) = ''δq
j''(''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 ''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
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function, ...
:
The time integral of the Lagrangian is another quantity called the
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
, defined as
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
angular_momentum_.html" ;"title="angular_momentum.html" ;"title="angular momentum">angular momentum ">angular_momentum.html" ;"title="angular momentum">angular momentum [energy]·[time], or [length]·[momentum]. With this definition Hamilton's principle is
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
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 tha ...
'', 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 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 (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
,
Daniel Bernoulli,
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 principle
In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those funct ...
s of mechanics, of
Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 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 ...
,
Maupertuis
Pierre Louis Moreau de Maupertuis (; ; 1698 – 27 July 1759) was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Prussian Academy of Science, at the ...
,
Euler,
Hamilton Hamilton may refer to:
People
* Hamilton (name), a common British surname and occasional given name, usually of Scottish origin, including a list of persons with the surname
** The Duke of Hamilton, the premier peer of Scotland
** Lord Hamilt ...
, 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 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,
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 multiplier
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied e ...
s can be used to include the constraints. Multiplying each constraint equation ''f
i''(r
''k'', ''t'') = 0 by a Lagrange multiplier ''λ
i'' for ''i'' = 1, 2, ..., ''C'', and adding the results to the original Lagrangian, gives the new Lagrangian
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
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
which are Lagrange's equations of the first kind. Also, the ''λ
i'' Euler-Lagrange equations for the new Lagrangian return the constraint equations
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 ''L'' = ''T'' − ''V'' gives
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
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
over a given time interval