In Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations in one of two forms, either the Lagrange equations of the first kind, which treat constraints explicitly as extra equations, often using Lagrange multipliers; or the Lagrange equations of the second kind, which incorporate the constraints directly by judicious choice of generalized coordinates. In each case, a mathematical function called the Lagrangian is a function of the generalized coordinates, their time derivatives, and time, and contains the information about the dynamics of the system.
No new physics is introduced by Lagrangian mechanics; it is actually less general than Newtonian mechanics. Newton's laws can include non-conservative forces like friction, however they must include constraint forces explicitly and are best suited to Cartesian coordinates. Lagrangian mechanics is ideal for systems with conservative forces and for bypassing constraint forces, and some (not all) non-conservative forces, in any coordinate system. Generalized coordinates can be chosen by convenience, to exploit symmetries in the system or the geometry of the constraints, which considerably simplifies describing the dynamics of the system. Lagrangian mechanics also reveals conserved quantities and their symmetries in a direct way, although only as a special case of Noether's theorem. The theory connects with the principle of stationary action, although Lagrangian mechanics is less general because it is restricted to equilibrium problems. Also, Lagrangian mechanics can only be applied to systems with holonomic constraints, because the formulation does not work for nonholonomic constraints. Three examples 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 revert to Newtonian mechanics, or use other methods.
The Lagrangian formulation of mechanics is important not just for its broad applications, but also for its role in advancing deep understanding of physics. Although Lagrange only sought to describe classical mechanics, Hamilton's principle that can be used to derive the Lagrange equation was later recognized to be applicable to much of theoretical physics as well. In quantum mechanics, action and quantum-mechanical phase are related via Planck's constant, and the principle of stationary action can be understood in terms of constructive interference of wave functions. 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. The action principle, and the Lagrangian formalism, are tied closely to Noether's theorem, which connects physical conserved quantities to continuous symmetries of a physical system. Lagrangian mechanics and Noether's theorem together yield a natural formalism for first quantization by including commutators between certain terms of the Lagrangian equations of motion for a physical system.
Lagrangian mechanics is widely used to solve mechanical problems in physics and engineering when Newton's formulation of classical mechanics is not convenient. Lagrange's equations are also used in optimisation problems of dynamic systems. In mechanics, Lagrange's equations of the second kind are used much more than those of the first kind.
- 1 Introduction
- 2 Definition of the Lagrangian
- 3 Equations of motion
- 4 From Newtonian to Lagrangian mechanics
- 5 Properties of the Euler–Lagrange equation
- 6 Examples in specific coordinate systems
- 7 Non-relativistic examples
- 8 Extensions to include non-conservative forces
- 9 Applications or extensions of Lagrangian mechanics in other contexts
- 10 See also
- 11 Footnotes
- 12 Notes
- 13 References
- 14 Further reading
- 15 External links
The strength of Lagrangian mechanics is its ability to handle constrained mechanical systems. The following examples motivate the need for the concepts and terminology used to handle such systems.
For a bead sliding on a frictionless wire subject only to gravity in 2d space, the constraint on the bead can be stated in the form f(r) = 0, where the position of the bead can be written r = (x(s), y(s)), in which s is a parameter, the arc length s along the curve from some point on the wire. Only one coordinate is needed instead of two, because the position of the bead can be parameterized by one number, s, and the constraint equation connects the two coordinates x and y; either one is determined from the other. The constraint force is the reaction force the wire exerts on the bead to keep it on the wire, and the non-constraint applied force is gravity acting on the bead.
Suppose the wire changes its shape with time, by flexing. Then the constraint equation and position of the particle are respectively
which now both depend on time t due to the changing coordinates as the wire changes its shape. Notice time appears implicitly via the coordinates and explicitly in the constraint equations.
Another interesting 2d example is the chaotic double pendulum, again subject to gravity. The length of one pendulum is L1 and the length of the other is L2. Each pendulum bob has a constraint equation,
where the positions of the bobs are
and θ1 is the angle of pendulum 1 from some reference direction, likewise for pendulum 2. Each pendulum can be described by one coordinate since the constraint equation for each connects the two spatial coordinates.
For a 3d example, a spherical pendulum with constant length l free to swing in any angular direction subject to gravity, the constraint on the pendulum bob can be stated in the form
where the position of the pendulum bob can be written
in which (θ, φ) are the spherical polar angles because the bob moves in the surface of a sphere. A logical choice of variables to describe the motion are the angles (θ, φ). Notice only two coordinates are needed instead of three, because the position of the bob can be parameterized by two numbers, and the constraint equation connects the three coordinates x, y, z so any one of them is determined from the other two.
For analyzing the small oscillations of multiple coupled simple harmonic oscillators, Lagrangian mechanics is especially natural, since the kinetic and potential energies of the system take a simple form despite the fact there are many particles, and the equations of motion can be derived immediately.
so overall, there are 3N coordinates to 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). If any or all of the particles are subject to a holonomic constraint, described by a constraint equation of the form f(r, t) = 0, then at any instant of time the position coordinates of those particles are linked together and not independent. If there are C constraints in the system, then each has a constraint equation,
and one coordinate can be eliminated from each constraint equation. The number of independent coordinates is therefore n = 3N − C. We can transform each position vector to a common set of n generalized coordinates, conveniently written as an n-tuple q = (q1, q2, ... qn), by expressing each position vector, and hence the position coordinates, as functions of the generalized coordinates and time,
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
(each overdot indicates a time derivative).
In the previous examples, if one tracks each of the massive objects as a particle (bead, pendulum bob, etc.), 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 in the pendulum rods). 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.
Definition of the Lagrangian
The central quantity of Lagrangian mechanics is the Lagrangian, a function which summarizes the dynamics of the entire system. Overall, the Lagrangian has no single expression for all physical systems. Any function which generates the correct equations of motion from the Euler–Lagrange equations, in agreement with physical laws, can be taken as a Lagrangian. However, it is 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 dot product. The kinetic energy is a function only of the velocities vk, not the positions rk nor time t, so T = T(v1, v2, ...). By contrast, the above expression for velocity shows the kinetic energy in generalized coordinates depends on the generalized velocities, coordinates, and time if the position vectors depend explicitly on time due to time-varying constraints, so T = T(q, dq/dt, t).
The potential energy V 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, 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, the velocities will appear also, V = V(r1, r2, ..., v1, v2, ...). If there is some external field changing with time, or external driving force, the potential will change with time, so most generally V = V(r1, r2, ..., v1, v2, ..., t).
If the potential or kinetic energy, or both, depend explicitly on time due to time-varying constraints or external influences, the Lagrangian L(q, dq/dt, t) is explicitly time-dependent. If neither the potential nor the kinetic energy depend on time, then the Lagrangian L(q, dq/dt) is explicitly independent of time. In either case, the Lagrangian will always have implicit time-dependence through the generalized coordinates.
In addition to the Lagrangian, it is sometimes possible to introduce another function to account for dissipative forces.
Equations of motion
In Newtonian mechanics, the equations are given by Newton's laws. The second law F = ma applies to each particle, so for an N particle system in 3d, there are 3N second order ordinary differential equations in the positions of the particles to solve for.
In Lagrangian mechanics, the Lagrangian function generates the equations of motion. Although the equations of motion include partial derivatives denoted by ∂/∂, they are still ordinary differential equations in the position coordinates of the particles. The total time derivative denoted d/dt may involve implicit differentiation.
There are two kinds, one which accounts for constraint forces by including the constraint equations, and another which excludes constraint forces and only accounts for non-constraint forces in terms of generalized coordinates. By substituting in the Lagrangian and solving the equations, subject to the initial values of the positions and velocities, will give the positions of the particles as functions of time, and one can see how the system evolves.
Lagrange's equations of the first kind
If a system of N particles in 3d is subject to C holonomic constraints, given by the equations f1, f2,..., fC, and the dynamics given by a Lagrangian L(r, dr/dt, t), Lagrange's equations of the first kind are
Lagrange's equations (First kind)
where k = 1, 2, ..., N labels the particles, there is a Lagrange multiplier λi for each constraint equation fi, and
are each shorthands for a vector of derivatives with respect to the indicated variables (not a derivative with respect to the entire vector).[nb 1]
This procedure does increase the number of equations to solve compared to Newton's laws, from 3N to 3N + C, because there are 3N 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.
Lagrange's equations of the second kind
Lagrange's equations (Second kind)
The number of equations has decreased compared to Newtonian mechanics, from 3N to n = 3N − 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.
From Newtonian to Lagrangian mechanics
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 particle of mass m is Newton's second law 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 equations to solve, since there are three components in this vector equation. The solutions are the position vectors r of the particles 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 ξ = (ξ1, ξ2, ξ3), the law in tensor index notation is the "Lagrangian form"
is the kinetic energy of the particle, and gbc 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, a scalar invariant which takes the same value in all frames of reference. 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 shortest length between two points in space. 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 free particles follow geodesics, the shortest 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.
The idea of finding the shortest path a particle can follow motivated the first applications of the calculus of variations to mechanical problems, such as the Brachistochrone problem solved by Jean Bernoulli in 1696, as well as Leibniz, 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 principles of mechanics, of Fermat, Maupertuis, Euler, Hamilton, and others. 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'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,
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.
A fundamental result in analytical mechanics is D'Alembert's principle, introduced in 1708 by Jacques Bernoulli to understand static equilibrium, and developed by D'Alembert in 1743 to solve dynamical problems. The principle asserts for N particles
The δrk are virtual displacements, by definition they are 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.[nb 2] 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;
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 is not readily usable to set up the equations of motion and solve for the motion of the system; therefore equations of motion in a set of independent coordinates are sought for.
Equations of motion from D'Alembert's principle
If there are constraints on particle k, then since the coordinates of the position rk = (xk, yk, zk) are linked together by a constraint equation, so are the those of the virtual displacements δrk = (δxk, δyk, δzk). Since the generalized coordinates are independent, we can avoid the complications with the δrk by converting to virtual displacements in the generalized coordinates. These are related in the same form as a total differential,
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 Nk along the virtual displacements δrk, and can without loss of generality be converted into the generalized analogues by the definition of generalized forces
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,
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 they 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
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. Lagrange's equations do not involve any potential, only forces; therefore they are more general than the Euler–Lagrange equations.
These equations also follow from the calculus of variations. The variation of the Lagrangian is
which has a similar form 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/∂(dqj/dt), in the process exchanging d(δqj)/dt for δqj, allowing the independent virtual displacements to be factorized from the derivatives of the Lagrangian,
Now, if the condition δqj(t1) = δqj(t2) = 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;
which is a functional; it takes in the Lagrangian function for all times between t1 and t2 and returns a scalar value. Its dimensions are the same as [ 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, 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. It is not widely stated that Hamilton's principle is a variational principle only with holonomic constraints, if we are dealing with nonholonomic systems then the variational principle should be replaced with one involving d'Alembert principle of virtual work.
Lagrange multipliers and constraints
We can vary L in the Cartesian rk coordinates, for N particles,
Hamilton's principle is still valid even if the coordinates L is expressed in are not independent, here rk, but the constraints are still assumed to be holonomic. What cannot be done is to simply equate the coefficients of δrk to zero because the δrk are not independent. Instead, the method of Lagrange multipliers can be used to include the constraints. Multiplying each constraint equation fi(rk, t) = 0 by a Lagrange multiplier λi 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 rk, so the multipliers are on equal footing with the position coordinates. With this new Lagrangian the Euler-Lagrange equations recover the constraint equations
and also obtain the equations of motion in terms of the rk, which are Lagrange's equations of the first kind
These equations also follow by varying the new Lagrangian and integrating over time;
where as always the end points are fixed, δrk(t1) = δrk(t2) = 0 for all k. The multipliers can be found so that the coefficients of δrk are zero, even though the rk are not independent. The equations of motion follow.
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
From the Lagrangian form of Newton's second law above, the derivatives of kinetic energy form the (negative of) resultant force acting on the particle k, and the derivatives of the potential form the non-constraint force acting on the particle k,
so it follows the constraint forces are given by
which relates the constraint equations to the constraint forces via the Lagrange multipliers.
Properties of the Euler–Lagrange equation
In some cases, the Lagrangian has properties which can provide information about the system without solving the equations of motion. These follow from Lagrange's equations of the second kind.
The Lagrangian of a given system is not unique. A Lagrangian L can be multiplied by a nonzero constant a, an arbitrary constant b can be added, and the new Lagrangian aL + b will describe exactly the same motion as L. A less obvious result is that two Lagrangians describing the same system can differ by the total derivative (not partial) of some function f(q, t) with respect to time;
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
Cyclic coordinates and conservation laws
An important property of the Lagrangian is that conserved quantities can easily be read off from it. The generalized momentum "canonically conjugate to" the coordinate qi is defined by
If the Lagrangian L does not depend on some coordinate qi, then it follows from the Euler–Lagrange equations that the corresponding generalized momentum will be a conserved quantity, because its time derivative is zero so the momentum must be a constant of the motion;
This is a special case of Noether's theorem. Such coordinates are called "cyclic" or "ignorable".
For example, a system may have a Lagrangian
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
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. Whatever the complicated the motion of the system is; all the coordinates and velocities will vary in such a way that these momenta are conserved.
Taking the total derivative of the Lagrangian L = T − V with respect to time leads to the general result
If the entire Lagrangian is explicitly independent of time, it follows the partial time derivative of the Lagrangian is zero, ∂L/∂t = 0, so the quantity under the total time derivative in brackets
must be a constant for all times during the motion of the system, and it also follows 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
Under all these circumstances, the constant
is the total conserved 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.
Under the same conditions, the Hamiltonian equals the total energy of the system and is conserved.
and time is scaled by a factor β, t′ = βt, then the velocities vk 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
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
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:
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,
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 straightforwards - the overall Lagrangian is the sum of the separate Lagrangians for each subsystem. If there are interactions, then interaction Lagranians may be added.
Examples in specific coordinate systems
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.
The Lagrangian of the particle can be written
The equations of motion for the particle are found by applying the Euler–Lagrange equation, for the x coordinate
and similarly for the y and z coordinates. For the x coordinate
and similarly for the y and z coordinates. Collecting the equations in vector form we find
which is Newton's second law of motion for a particle subject to a conservative force.
Polar coordinates in 2d and 3d
The Lagrangian for the above problem in spherical coordinates is
so the Euler–Lagrange equations are
The φ coordinate is cyclic since it does not appear in the Lagrangian, so the conserved momentum in the system is the angular momentum
in which r, θ and dφ/dt can all vary with time, but only in such a way that pφ is constant.
The following examples apply Lagrange's equations of the second kind to mechanical problems.
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 θ from the vertical.
The kinetic energy can then be shown to be
and the potential energy of the system is
The Lagrangian is therefore
Since x is absent from the Lagrangian, it is a cyclic coordinate. The conserved momentum is
The Lagrange equation for the support coordinate x is therefore
The Lagrange equation for the angle θ is
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, should give the equations of motion for a pendulum that is at rest in some inertial frame, while 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
The basic problem is that of two bodies of masses m1 and m2 with position vectors r1 and r2 are in orbit about each other due to an attractive central force V. We may naïvely write down the Lagrangian in terms of the position coordinates as they are, but it is an established procedure to convert the two-body problem into a one-body problem as follows. Introduce the Jacobi coordinates; the separation of the bodies r = r2 − r1 and the location of the center of mass R = (m1r1 + m2r2)/(m1 + m2). The Lagrangian is then[nb 3]
where M = m1 + m2 is the total mass, μ = m1m2/(m1 + m2) is the reduced mass, and V the potential of the radial force, which depends only on the magnitude of the separation |r| = |r2 − r1|. The Lagrangian is divided into a center-of-mass term Lcm and a relative motion term Lrel. The Euler–Lagrange equation for R is simply
which states the center of mass moves in a straight line at constant velocity. The since the relative motion only depends on the magnitude of the separation, it is ideal to use polar coordinates (r, θ) and take r = |r|,
which does not depend upon θ, therefore θ is an ignorable coordinate. The conserved momentum corresponding to θ is
which will be abbreviated ℓ. The radial coordinate r and angular velocity dθ/dt can vary with time, but only in such a way that ℓ is constant. The Lagrange equation for r is
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 dθ/dt from this radial equation,
which is the equation of motion for a one-dimensional problem in which a particle of mass μ is subjected to the inward central force −dU/dr and a second outward force, called in this context the centrifugal force
Of course, if one remains entirely within the one-dimensional formulation, ℓ 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 (r, θ) 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 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.
Non-relativistic test particles in fields
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.
Non-relativistic test particle in a Newtonian gravitational field
For a particle with mass m in a Newtonian gravitation potential
since the force is conservative, one can follow the same procedure in the Cartesian coordinates example to find
where the mass cancels algebraically, and physically it should because the acceleration of all massive objects due to gravity is independent of the mass.
Non-relativistic test particle in an electromagnetic field
The case of a charged particle with electrical charge q interacting with an electromagnetic field is more complicated. The electric scalar potential ϕ(r(t), t) and magnetic vector potential A(r(t), t) are defined from the electric field
and magnetic field
Notice all the fields depend on the position r of the particle at time t, and depend explicitly on time as well as implicitly via the position.
The Lagrangian of a massive charged test particle in an electromagnetic field is
which produces the Lorentz force law
An interesting point in this example is the generalized momentum conjugate to r is the ordinary momentum plus a contribution from the A field,
Extensions to include non-conservative forces
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:
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
Applications or extensions of Lagrangian mechanics in other contexts
Relation to other formulations of classical mechanics
The Hamiltonian, denoted by H, is obtained by performing a Legendre transformation on the Lagrangian, which introduces new variables, canonically conjugate to the original variables. This doubles the number of variables, but makes differential equations first order. The Hamiltonian is the basis for an alternative formulation of classical mechanics known as Hamiltonian mechanics. It is a particularly ubiquitous quantity in quantum mechanics (see Hamiltonian (quantum mechanics)). In the classical view, time is an independent variable and qi (and dqi/dt) are dependent variables as is often seen in phase space explanations of systems.
Routhian mechanics is a hybrid formulation of Lagrangian and Hamiltonian mechanics, which is not often used in practice but an efficient formulation for cyclic coordinates.
Lagrangian mechanics can be formulated in special relativity and general relativity. The equations of motion take the same form, but the Lagrangian is not simply the kinetic minus the potential energy.
Applications in quantum mechanics
In 1948, Feynman discovered the path integral formulation extending the principle of least action to quantum mechanics for electrons and photons. 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. The continuum analogue for defining a field are field variables, say ϕ(r, t), which represents some density function varying with position and time.
defined in terms of the field and its space and time derivatives at a location r and time t. The Lagrangian is then the integral of the Lagrangian density over 3d space (see volume integral):
where d3r is a 3d differential volume element, must be used instead. The Lagrangian is a function of time since the Lagrangian density has implicit space dependence via the fields, and may explicit spatial dependence, but these are removed in the integral, leaving only time in as the variable for the Lagrangian.
Uses in engineering
Circa 1963[when?] Lagrangians were a general part of the engineering curriculum, but a quarter of a century later, even with the ascendency of dynamical systems, they were dropped as requirements for some engineering programs, and are generally considered to be the domain of theoretical dynamics. Circa 2003[when?] this changed dramatically, and Lagrangians are not only a required part of many ME and EE graduate-level curricula, but also find applications in finance, economics, and biology, mainly as the basis of the formulation of various path integral schemes to facilitate the solution of parabolic partial differential equations via random walks.
Circa 2013,[when?] Lagrangians find their way into hundreds of direct engineering solutions, including robotics, turbulent flow analysis (Lagrangian and Eulerian specification of the flow field), signal processing, microscopic component contact and nanotechnology (superlinear convergent augmented Lagrangians), gyroscopic forcing and dissipation, semi-infinite supercomputing (which also involve Lagrange multipliers in the subfield of semi-infinite programming), chemical engineering (specific heat linear Lagrangian interpolation in reaction planning), civil engineering (dynamic analysis of traffic flows), optics engineering and design (Lagrangian and Hamiltonian optics) aerospace (Lagrangian interpolation), force stepping integrators, and even airbag deployment (coupled Eulerian-Lagrangians as well as SELM—the stochastic Eulerian Lagrangian method).
- Fundamental lemma of the calculus of variations
- Canonical coordinates
- Functional derivative
- Generalized coordinates
- Hamiltonian mechanics
- Hamiltonian optics
- Lagrangian analysis (applications of Lagrangian mechanics)
- Lagrangian point
- Lagrangian system
- Non-autonomous mechanics
- Restricted three-body problem
- Plateau's problem
- Sometimes in this context the variational derivative denoted and defined as
- 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.
- The Lagrangian also can be written explicitly for a rotating frame. See Padmanabhan, 2000.
- Dvorak & Freistetter 2005, p. 24
- Haken 2006, p. 61
- Lanczos 1986, p. 43
- Menzel & Zatzkis 1960, p. 160
- Goldstien, Poole & Safko 2001, p. 35
- Hand & Finch 2008, p. 36–40
- Torby1984, p.270
- Torby 1984, p. 269
- Hand & Finch 2008, p. 60–61
- Hand & Finch 2008, p. 19
- Penrose 2007
- Schuam 1988, p. 156
- Synge & Schild 1949, p. 150–152
- Hand & Finch 2008, p. 44–45
- Hand & Finch 2008, p. 44–45
- Foster & Nightingale 1995, p. 89
- Hand & Finch 2008, p. 4
- Goldstein 1980, p. 16–18
- Hand 2008, p. 15
- Hand & Finch 2008, p. 15
- Fetter & Walecka 1980, p. 53
- Torby 1984, p. 264
- Torby 1984, p. 269
- Kibble & Berkshire 2004, p. 234
- Fetter & Walecka 1980, p. 56
- Hand & Finch 2008, p. 17
- Hand & Finch 2008, p. 15–17
- R. Penrose (2007). The Road to Reality. Vintage books. p. 474. ISBN 0-679-77631-1.
- Goldstien 1980, p. 23
- Kibble & Berkshire 2004, p. 234–235
- Hand & Finch 2008, p. 51
- Fetter Walecka, pp. 68–70
- Landau & Lifshitz 1976, p. 4
- Goldstien, Poole & Safko 2002, p. 21
- Landau & Lifshitz 1976, p. 4
- Goldstien 1980, p. 21
- Landau & Lifshitz 1976, p. 14
- Landau & Lifshitz 1976, p. 22
- Taylor 2005, p. 297
- Padmanabhan 2000, p. 48
- Hand & Finch 1998, pp. 140–141
- Hildebrand 1992, p. 156
- Zak, Zbilut & Meyers 1997, pp. 202
- Shabana 2008, pp. 118–119
- Gannon 2006, p. 267
- Kosyakov 2007
- Galley 2013
- Hadar, Shahar & Kol 2014
- Birnholtz, Hadar & Kol 2013
- Torby 1984, p. 271
- Gans 2013
- Landau, L. D.; Lifshitz, E. M.. Mechanics (3rd ed.). Butterworth Heinemann. p. 134. ISBN 9780750628969.
- Landau, Lev; Lifshitz, Evgeny (1975). The Classical Theory of Fields. Elsevier Ltd. ISBN 978-0-7506-2768-9.
- Hand, L. N.; Finch, J. D. Analytical Mechanics (2nd ed.). Cambridge University Press. p. 23. ISBN 9780521575720.
- Louis N. Hand, Janet D. Finch (1998). Analytical mechanics. Cambridge University Press. pp. 140–141. ISBN 0-521-57572-9.
- Kibble, T. W. B.; Berkshire, F. H. (2004). Classical Mechanics (5th ed.). Imperial College Press. p. 236. ISBN 9781860944352.
- Goldstein, Herbert (1980). Classical Mechanics (2nd ed.). San Francisco, CA: Addison Wesley. pp. 352–353. ISBN 0201029189.
- Goldstein, Herbert; Poole, Charles P., Jr.; Safko, John L. (2002). Classical Mechanics (3rd ed.). San Francisco, CA: Addison Wesley. pp. 347–349. ISBN 0-201-65702-3.
- Lanczos, Cornelius (1986). "II §5 Auxiliary conditions: the Lagrangian λ-method". The variational principles of mechanics (Reprint of University of Toronto 1970 4th ed.). Courier Dover. p. 43. ISBN 0-486-65067-7.
- Fetter, A. L.; Walecka, J. D. (1980). Theoretical Mechanics of Particles and Continua. Dover. pp. 53–57. ISBN 978-0-486-43261-8.
- The Principle of Least Action, R. Feynman
- Dvorak, R.; Freistetter, Florian (2005). "§ 3.2 Lagrange equations of the first kind". Chaos and stability in planetary systems. Birkhäuser. p. 24. ISBN 3-540-28208-4.
- Henry Zatzkis (1960). "§1.4 Lagrange equations of the second kind". In DH Menzel. Fundamental formulas of physics 1 (2nd ed.). Courier Dover. p. 160. ISBN 0-486-60595-7.
- Francis Begnaud Hildebrand (1992). Methods of applied mathematics (Reprint of Prentice-Hall 1965 2nd ed.). Courier Dover. p. 156. ISBN 0-486-67002-3.
- Michail Zak, Joseph P. Zbilut, Ronald E. Meyers (1997). From instability to intelligence. Springer. p. 202. ISBN 3-540-63055-4.
- Ahmed A. Shabana (2008). Computational continuum mechanics. Cambridge University Press. pp. 118–119. ISBN 0-521-88569-8.
- John Robert Taylor (2005). Classical mechanics. University Science Books. p. 297. ISBN 1-891389-22-X.
- Padmanabhan, Thanu (2000). "§2.3.2 Motion in a rotating frame". Theoretical Astrophysics: Astrophysical processes (3rd ed.). Cambridge University Press. p. 48. ISBN 0-521-56632-0.
- Kosyakov, B. P. (2007). Introduction to the classical theory of particles and fields. Berlin, Germany: Springer. doi:10.1007/978-3-540-40934-2.
- Galley, Chad R. (2013). "Classical Mechanics of Nonconservative Systems". Physical Review Letters 110 (17): 174301. arXiv:1210.2745. Bibcode:2013PhRvL.110q4301G. doi:10.1103/PhysRevLett.110.174301. PMID 23679733.
- Birnholtz, Ofek; Hadar, Shahar; Kol, Barak (2014). "Radiation reaction at the level of the action". International Journal of Modern Physics A 29 (24): 1450132. arXiv:1402.2610. Bibcode:2014IJMPA..2950132B. doi:10.1142/S0217751X14501322.
- Birnholtz, Ofek; Hadar, Shahar; Kol, Barak (2013). "Theory of post-Newtonian radiation and reaction". Physical Review D 88 (10): 104037. arXiv:1305.6930. Bibcode:2013PhRvD..88j4037B. doi:10.1103/PhysRevD.88.104037.
- Roger F Gans (2013). Engineering Dynamics: From the Lagrangian to Simulation. New York: Springer. ISBN 978-1-4614-3929-5.
- Terry Gannon (2006). Moonshine beyond the monster: the bridge connecting algebra, modular forms and physics. Cambridge University Press. p. 267. ISBN 0-521-83531-3.
- Torby, Bruce (1984). "Energy Methods". Advanced Dynamics for Engineers. HRW Series in Mechanical Engineering. United States of America: CBS College Publishing. ISBN 0-03-063366-4.
- Foster, J; Nightingale, J.D. (1995). A Short Course in General Relativity (2nd ed.). Springer. ISBN 0-03-063366-4.
- M. P. Hobson, G. P. Efstathiou, A. N. Lasenby (2006). General Relativity: An Introduction for Physicists. Cambridge University Press. p. 79–80. ISBN 9780521829519.
- Gupta, Kiran Chandra, Classical mechanics of particles and rigid bodies (Wiley, 1988).
- Cassel, Kevin W.: Variational Methods with Applications in Science and Engineering, Cambridge University Press, 2013.
- Tong, David, Classical Dynamics Cambridge lecture notes
- Principle of least action interactive Excellent interactive explanation/webpage
- Joseph Louis de Lagrange - Œuvres complètes (Gallica-Math)