Lagrangian mechanics
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In physics, Lagrangian mechanics is a formulation of classical mechanics 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 presentation to the Turin Academy of Science in 1760[1] culminating in his 1788 grand opus, Mécanique analytique.[2]
Lagrangian mechanics describes a mechanical system as a pair (M, L) consisting of a configuration space M and a smooth function within that space called a Lagrangian. For many systems, L = T − V, where T and V are the kinetic and potential energy of the system, respectively.[3]
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.[4]
Introduction
Suppose there exists a bead sliding around on a wire, or a swinging
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
Lagrangian
Instead of forces, Lagrangian mechanics uses the energies in the system. The central quantity of Lagrangian mechanics is the Lagrangian, a function which summarizes the dynamics of the entire system. Overall, the Lagrangian has units of energy, but no single expression for all physical systems. Any function which generates the correct equations of motion, in agreement with physical laws, can be taken as a Lagrangian. It is nevertheless possible to construct general expressions for large classes of applications. The non-relativistic Lagrangian for a system of particles in the absence of an electromagnetic field is given by[5]
Kinetic energy is the energy of the system's motion, and vk2 = vk · vk 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 vk, not the positions rk nor time t, so T = T(v1, v2, ...).
The
The above form of L does not hold in relativistic Lagrangian mechanics or in the presence of a magnetic field when using the typical expression for the potential energy, and must be replaced by a function consistent with special or general relativity. Also, for dissipative forces (e.g., friction), another function must be introduced alongside L.
One or more of the particles may each be subject to one or more
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[8]
where k = 1, 2, ..., N labels the particles, there is a Lagrange multiplier λi for each constraint equation fi, and
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 vz,2 = dz2/dt, is just ∂L/∂vz,2; no awkward chain rules or total derivatives need to be used to relate the velocity component to the corresponding coordinate z2).
In each constraint equation, one coordinate is redundant because it is determined from the other coordinates. 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 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
Given this vk, 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(q, , t).
With these definitions, the Euler–Lagrange equations, or Lagrange's equations of the second kind[9][10]
are mathematical results from the calculus of variations, which can also be used in mechanics. Substituting in the Lagrangian L(q, dq/dt, t), gives the equations of motion of the system. 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.
Although the equations of motion include
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
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
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
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
A fundamental result in
The virtual displacements, δrk, 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,[15] 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:[16][nb 3]
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.[17][18] 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 δrk 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 rk = (xk, yk, zk) are linked together by a constraint equation, so are those of 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 Nk along the virtual displacements δrk, and can without loss of generality be converted into the generalized analogues by the definition of
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:[6]
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 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.[22]
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[24][25]
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
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:
The time integral of the Lagrangian is another quantity called the action, defined as[26]
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
Historically, the idea of finding the shortest path a particle can follow subject to a force motivated the first applications of the
Hamilton's principle can be applied to
Lagrange multipliers and constraints
The Lagrangian L can be varied 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.[29] As always the end points are fixed δrk(t1) = δrk(t2) = 0 for all k. 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. Varying this new Lagrangian and integrating with respect to time gives
The introduced multipliers can be found so that the coefficients of δrk are zero, even though the rk are not independent. The equations of motion follow. From the preceding analysis, obtaining the solution to this integral is equivalent to the statement
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
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
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 q over a given time interval [tst, tfin]} and fixed end points Pst = q(tst) and Pfin = q(tfin), then two Lagrangians describing the same system can differ by the "total time derivative" of a function f(q, t):[30]
where means
Both Lagrangians L and L′ produce the same equations of motion[31][32] since the corresponding actions S and S′ are related via
with the last two components f(Pfin, tfin) and f(Pst, tst) independent of q.
Invariance under point transformations
Given a set of generalized coordinates q, if we change these variables to a new set of generalized coordinates Q according to a point transformation Q = Q(q, t) which is invertible as q = q(Q, t), the new Lagrangian L′ is a function of the new coordinates
and by the chain rule for partial differentiation, Lagrange's equations are invariant under this transformation;[33]
This may simplify the equations of motion.
Proof |
For a coordinate transformation , we have which implies that which implies that . It also follows that:
and similarly:
which imply that . The two derived relations can be employed in the proof.
Since the transformation from is invertible, it follows that the form of the Euler-Lagrange equation is invariant ie.:
|
Cyclic coordinates and conserved momenta
An important property of the Lagrangian is that conserved quantities can easily be read off from it. The generalized momentum "canonically conjugate to" the coordinate qi is defined by
If the Lagrangian L does not depend on some coordinate qi, it follows immediately from the Euler–Lagrange equations that
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
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. However complicated the motion of the system is, all the coordinates and velocities will vary in such a way that these momenta are conserved.
Energy
Given a Lagrangian the Hamiltonian of the corresponding mechanical system is, by definition,
This quantity will be equivalent to energy if the generalized coordinates are natural coordinates, ie. they have no explicit time dependance when expressing position vector: . From:
- where is a symmetric matrix that is defined for the derivation.
Invariance under coordinate transformations
At every time instant t, the energy is invariant under configuration space coordinate changes q → Q, i.e. (using natural coordinates)
Besides this result, the proof below shows that, under such change of coordinates, the derivatives change as coefficients of a linear form.
Proof |
For a coordinate transformation Q = F(q), we have where is the tangent map of the vector space to the vector space and is the Jacobian. In the coordinates and the previous formula for has the form After differentiation involving the product rule, where In vector notation, On the other hand, It was mentioned earlier that Lagrangians do not depend on the choice of configuration space coordinates, i.e. One implication of this is that and This demonstrates that, for each and is a well-defined linear form whose coefficients are contravariant 1-tensors. Applying both sides of the equation to and using the above formula for yields The invariance of the energy follows. |
Conservation
In Lagrangian mechanics, the system is closed if and only if its Lagrangian does not explicitly depend on time. The
More precisely, let q = q(t) be an extremal. (In other words, q satisfies the Euler–Lagrange equations). Taking the total time-derivative of L along this extremal and using the EL equations leads to
If the Lagrangian L does not explicitly depend on time, then ∂L/∂t = 0, then H does not vary with time evolution of particle, indeed, an integral of motion, meaning that
Hence, if the chosen coordinates were natural coordinates, the energy is conserved.
Kinetic and potential energies
Under all these circumstances,[34] the constant
is the total energy of the system. The kinetic and potential energies still change as the system evolves, but the motion of the system will be such that their sum, the total energy, is constant. This is a valuable simplification, since the energy E is a constant of integration that counts as an arbitrary constant for the problem, and it may be possible to integrate the velocities from this energy relation to solve for the coordinates.
Mechanical similarity
If the potential energy is a homogeneous function of the coordinates and independent of time,[35] and all position vectors are scaled by the same nonzero constant α, rk′ = αrk, so that
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
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:[30]
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 straightforward – the overall Lagrangian is the sum of the separate Lagrangians for each subsystem. If there are interactions, then interaction Lagrangians may be added.
Consequences of singular Lagrangians
From the Euler-Lagrange equations, it follows that:
Where the matrix is defined as . If the matrix is non-singular, the above equations can be solved to represent as a function of . If the matrix is non-invertible, it would not be possible to represent all 's as a function of but also, the Hamiltonian equations of motions will not take the standard form.[36]
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,
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
The equations of motion for the particle are found by applying the Euler–Lagrange equation, for the x coordinate
with derivatives
hence
and similarly for the y and z coordinates. Collecting the equations in vector form we find
which is
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
So, in spherical coordinates, 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 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 -direction. Let be the coordinate along the line of the support, and let us denote the position of the pendulum by the angle from the vertical. The coordinates and velocity components of the pendulum bob are
The generalized coordinates can be taken to be and . The kinetic energy of the system is then
and the potential energy is
giving the Lagrangian
Since x is absent from the Lagrangian, it is a cyclic coordinate. The conserved momentum is
and the Lagrange equation for the support coordinate is
The Lagrange equation for the angle θ is
and simplifying
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
Two-body central force problem
Two bodies of masses m1 and m2 with position vectors r1 and r2 are in orbit about each other due to an attractive
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 splits 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.
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|,
so θ is a cyclic coordinate with the corresponding conserved (angular) momentum
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,[39]
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 −dV/dr and a second outward force, called in this context the (Lagrangian) centrifugal force (see centrifugal force#Other uses of the term):
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:[40]
"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[41] 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.[42] 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.[43]
Extensions to include non-conservative forces
Dissipative 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.[44][45][46][47]
In a more general formulation, the forces could be both conservative and
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[48]
and
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. Not only can the fields form non conservative potentials, these potentials can also be velocity dependent.
The Lagrangian for a
The Lagrangian of a massive charged test particle in an electromagnetic field
is called minimal coupling. This is a good example of when the common rule of thumb that the Lagrangian is the kinetic energy minus the potential energy is incorrect. Combined with Euler–Lagrange equation, it produces the Lorentz force law
Under
where f(r,t) is any scalar function of space and time, the aforementioned Lagrangian transforms like:
which still produces the same Lorentz force law.
Note that the
This relation is also used in the
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
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 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/dt, t) obtains the generalized momenta Lagrangian L′(p, dp/dt, 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
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
Quantum mechanics
In
In 1948,
Classical field theory
In Lagrangian mechanics, the generalized coordinates form a discrete set of variables that define the configuration of a system. In
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
where d3r is a 3D
Noether's theorem
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.
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
- Fundamental lemma of the calculus of variations
- Functional derivative
- Generalized coordinates
- Hamiltonian mechanics
- Hamiltonian optics
- Inverse problem for Lagrangian mechanics, 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
- Lagrangian system
- Non-autonomous mechanics
- Plateau's problem
- Restricted three-body problem
Footnotes
- variational derivativedenoted and defined asis used. Throughout this article only partial and total derivatives are used.
- Udwadia–Kalaba equation.
- ^ In other words
for particle k subject to a constraint force, howeverbecause of the constraint equations on the rk coordinates.
- ^ The Lagrangian also can be written explicitly for a rotating frame. See Padmanabhan, 2000.
Notes
- ^ Fraser, Craig. "J. L. Lagrange's Early Contributions to the Principles and Methods of Mechanics". Archive for History of Exact Sciences, vol. 28, no. 3, 1983, pp. 197–241. JSTOR, http://www.jstor.org/stable/41133689. Accessed 3 Nov. 2023.
- ^ Hand & Finch 1998, p. 23
- ^ Hand & Finch 1998, pp. 18–20
- ^ Hand & Finch 1998, pp. 46, 51
- ^ Torby 1984, p. 270
- ^ a b c d Torby 1984, p. 269
- ^ Hand & Finch 1998, p. 36–40
- ^ Hand & Finch 1998, p. 60–61
- ^ Hand & Finch 1998, p. 19
- ^ Penrose 2007
- ^ Kay 1988, p. 156
- ^ Synge & Schild 1949, p. 150–152
- ^ Foster & Nightingale 1995, p. 89
- ^ Hand & Finch 1998, p. 4
- ^ Goldstein 1980, pp. 16–18
- ^ Hand & Finch 1998, p. 15
- ^ Hand & Finch 1998, p. 15
- ^ Fetter & Walecka 1980, p. 53
- ^ Kibble & Berkshire 2004, p. 234
- ^ Fetter & Walecka 1980, p. 56
- ^ Hand & Finch 1998, p. 17
- ^ Hand & Finch 1998, p. 15–17
- ISBN 978-0-679-77631-4.
- ^ Goldstein 1980, p. 23
- ^ Kibble & Berkshire 2004, p. 234–235
- ^ Hand & Finch 1998, p. 51
- ^ a b Hand & Finch 1998, p. 44–45
- ^ Goldstein 1980
- ^ Fetter & Walecka 1980, pp. 68–70
- ^ a b Landau & Lifshitz 1976, p. 4
- ^ Goldstein, Poole & Safko 2002, p. 21
- ^ Landau & Lifshitz 1976, p. 4
- ^ Goldstein 1980, p. 21
- ^ Landau & Lifshitz 1976, p. 14
- ^ Landau & Lifshitz 1976, p. 22
- ISBN 978-981-4299-64-0.
- ^ 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
- ^ Birnholtz, Hadar & Kol 2014
- ^ Birnholtz, Hadar & Kol 2013
- ^ a b Torby 1984, p. 271
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- Lagrange, J. L. (1815). Mécanique analytique. Vol. 2.
- Penrose, Roger (2007). The Road to Reality. Vintage books. ISBN 978-0-679-77631-4.
- ISBN 9780750628969.
- ISBN 978-0-7506-2768-9.
- Hand, L. N.; Finch, J. D. (1998). Analytical Mechanics (2nd ed.). Cambridge University Press. ISBN 9780521575720.
- Saletan, E. J.; José, J. V. (1998). Classical Dynamics: A Contemporary Approach. Cambridge University Press. ISBN 9780521636360.
- Kibble, T. W. B.; Berkshire, F. H. (2004). Classical Mechanics (5th ed.). Imperial College Press. p. 236. ISBN 9781860944352.
- ISBN 0201029189.
- 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.
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- 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.
- Haken, H (2006). Information and self-organization (3rd ed.). Springer. p. 61. ISBN 3-540-33021-6.
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- Hildebrand, Francis Begnaud (1992). Methods of applied mathematics (Reprint of Prentice-Hall 1965 2nd ed.). Courier Dover. p. 156. ISBN 0-486-67002-3.
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- 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.
- Doughty, Noel A. (1990). Lagrangian Interaction. Addison-Wesley Publishers Ltd. ISBN 0-201-41625-5.
- Kosyakov, B. P. (2007). Introduction to the classical theory of particles and fields. Berlin, Germany: Springer. ISBN 978-3-540-40933-5.
- Galley, Chad R. (2013). "Classical Mechanics of Nonconservative Systems". Physical Review Letters. 110 (17): 174301. S2CID 14591873.
- Birnholtz, Ofek; Hadar, Shahar; Kol, Barak (2014). "Radiation reaction at the level of the action". International Journal of Modern Physics A. 29 (24): 1450132–1450190. S2CID 118541484.
- Birnholtz, Ofek; Hadar, Shahar; Kol, Barak (2013). "Theory of post-Newtonian radiation and reaction". Physical Review D. 88 (10): 104037. S2CID 119170985.
- Roger F Gans (2013). Engineering Dynamics: From the Lagrangian to Simulation. New York: Springer. ISBN 978-1-4614-3929-5.
- Gannon, Terry (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.
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- M. P. Hobson; G. P. Efstathiou; A. N. Lasenby (2006). General Relativity: An Introduction for Physicists. Cambridge University Press. pp. 79–80. ISBN 9780521829519.
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Further reading
- Gupta, Kiran Chandra, Classical mechanics of particles and rigid bodies (Wiley, 1988).
- Cassel, Kevin (2013). Variational methods with applications in science and engineering. Cambridge: Cambridge University Press. ISBN 978-1-107-02258-4.
- Classical Mechanics. 3rd ed., Pearson, 2002.
External links
- David Tong. "Cambridge Lecture Notes on Classical Dynamics". DAMTP. Retrieved 2017-06-08.
- Principle of least action interactive Excellent interactive explanation/webpage
- Joseph Louis de Lagrange - Œuvres complètes (Gallica-Math)
- Constrained motion and generalized coordinates, page 4