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17.6: Lorentz-Invariant Formulation of Lagrangian Mechanics

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Parametric Formulation

The Lagrangian and Hamiltonian formalisms in classical mechanics are based on the Newtonian concept of absolute time t which serves as the system evolution parameter in Hamilton’s Principle. This approach violates the Special Theory of Relativity. The extended Lagrangian and Hamiltonian formalism is a parametric approach, pioneered by Lanczos[La49], that introduces a system evolution parameter s that serves as the independent variable in the action integral, and all the space-time variables qi(s),t(s) are dependent on the evolution parameter s. This extended Lagrangian and Hamiltonian formalism renders it to a form that is compatible with the Special Theory of Relativity. The importance of the Lorentz-invariant extended formulation of Lagrangian and Hamiltonian mechanics has been recognized for decades.[La49, Go50, Sy60] Recently there has been a resurgence of interest in the extended Lagrangian and Hamiltonian formalism stimulated by the papers of Struckmeier[Str05, Str08] and this formalism has featured prominently in recent textbooks by Johns[Jo05] and Greiner[Gr10]. This parametric approach develops manifestly-covariant Lagrangian and Hamiltonian formalisms that treat equally all 2n+1 space-time canonical variables. It provides a plausible manifestly-covariant Lagrangian for the one-body system, but serious problems exist extending this to the N-body system when N>1. Generalizing the Lagrangian and Hamiltonian formalisms into the domain of the Special Theory of Relativity is of fundamental importance to physics, while the parametric approach gives insight into the philosophy underlying use of variational methods in classical mechanics.1

In conventional Lagrangian mechanics, the equations of motion for the n generalized coordinates are derived by minimizing the action integral, that is, Hamilton’s Principle.

δS(q,˙q,t)=δbaL(q(t),˙q(t),t)dt=0

where L(q(t),˙q(t),t) denotes the conventional Lagrangian. This approach implicitly assumes the Newtonian concept of absolute time t which is chosen to be the independent variable that characterizes the evolution parameter of the system. The actual path [q(t),˙q(t)] the system follows is defined by the extremum of the action integral S(q,˙q,t) which leads to the corresponding Euler-Lagrange equations. This assumption is contrary to the Theory of Relativity which requires that the space and time variables be treated equally, that is, the Lagrangian formalism must be covariant.

Extended Lagrangian

Lanczos[La49] proposed making the Lagrangian covariant by introducing a general evolution parameter s, and treating the time as a dependent variable t(s) on an equal footing with the configuration space variables qi(s). That is, the time becomes a dependent variable q0(s)=ct(s) similar to the spatial variables qμ(s) where 1μn. The dynamical system then is described as motion confined to a hypersurface within an extended space where the value of the extended Hamiltonian and the evolution parameter s constitute an additional pair of canonically conjugate variables in the extended space. That is, the canonical momentum p0, corresponding to q0=ct, is p0=Ec similar to the momentum-energy four vector, equation (17.5.21).

An extended Lagrangian L(q(s),dq(s)ds,t(s),dt(s)ds) can be defined which can be written compactly as L(qμ(s),dqμ(s)ds) where the index 0μn denotes the entire range of space-time variables.

This extended Lagrangian can be used in an extended action functional S(q,dqds,t,dtds) to give an extended version of Hamilton’s Principle2

δS(q,dqds,t,dtds)=δbaL(qμ(s),dqμ(s)ds)ds=0

The conventional action S, and extended action S, address alternate characterizations of the same underlying physical system, and thus the action principle implies that δS=δS=0 must hold simultaneously. That is,

δbaL(q,dqdt,t)dtdsds=δbaL(q,dqds,t,dtds)ds

As discussed in chapter 9.3, there is a continuous spectrum of equivalent gauge-invariant Lagrangians for which the Euler-Lagrange equations lead to identical equations of motion. Equation ??? is satisfied if the conventional and extended Lagrangians are related by

L(q,dqds,t,dtds)=L(q,dqdt,t)dtds+dΛ(q,t)ds

where Λ(q,t) is a continuous function of q and t that has continuous second derivatives. It is acceptable to assume that dΛ(q,t)ds=0, then the extended and conventional Lagrangians have a unique relation requiring no simultaneous transformation of the dynamical variables. That is, assume

L(q,dqds,t,dtds)=L(q,dqdt,t)dtds

Note that the time derivative of q can be expressed in terms of the s derivatives by

dqdt=dq/dsdt/ds

Thus, for a conventional Lagrangian with n variables, the corresponding extended Lagrangian is a function of n+1 variables while the conventional and extended Lagrangians are related using equations ???, and ???.

The derivatives of the relation between the extended and conventional Lagrangians lead to

Lqμ=Lqμdtds

Lt=Ltdtds

L(dqμds)=L(dqμdt)

L(dtds)=Lnμ=1L(dqμdt)dqμdt

where 1μn since the μ=0 time derivatives are written explicitly in equations ???, ???.

Equations ??????, summed over the extended range 0μn of time and spatial dynamical variables, imply

nμ=0L(dqμds)(dqμds)=Ldtdsnμ=1L(dqμdt)dqμdtdtds+ni=1L(dqμdt)dqμds=L

Equation ??? can be written in the form

Lnμ=0L(dqμds)dqμds={=0 if L is not homogeneous in dqμds0 if L is homogeneous in dqμds

If the extended Lagrangian L(q,dqds,t,dtds) is homogeneous to first order in the n+1 variables dqμds, then Euler’s theorem on homogeneous functions trivially implies the relation given in Equation ???. Struckmeier[Str08] identified a subtle but important point that if L is not homogeneous in dqμds, then Equation ??? is not an identity but is an implicit equation that is always satisfied as the system evolves according to the solution of the extended Euler-Lagrange equations. Then Equation ??? is satisfied without it being a homogeneous form in the n+1 velocities dqμds. This introduces a new class of non-homogeneous Lagrangians. The relativistic free particle, discussed in example 17.6.1, is a case of a non-homogeneous extended Lagrangian.

Extended generalized momenta

The generalized momentum is defined by

pμ=L(qμt)

Assume that the definitions of the extended Lagrangian L, and the extended Hamiltonian H, are related by a Legendre transformation, and are based on variational principles, analogous to the relation that exists between the conventional Lagrangian L and Hamiltonian H. The Legendre transformation requires defining the extended generalized (canonical) momentum-energy four vector P(s)=(E(s)c,p(s)). The momentum components of the momentum-energy four vector P(s)=(E(s)c,p(s)) are given by the 1μn components using Equation ???.

pμ(s)=L(dqμds)=L(dqμdt)

The μ=0 component of the momentum-energy four vector can be derived by recognizing that the right-hand side of Equation ??? is equal to H(pμ,qμ,t). That is, the corresponding generalized momentum p0, that is conjugate to q0=ct, is given by

p0=L(dq0ds)=1c(L(dtds))=1c(Lnμ=1L(dqμdt)dqμdt)=H(pμ,qμ,t)c

Extended Lagrange equations of motion

By direct analogy with the non-relativistic action integral ???, the extremum for the relativistic action integral S(q,dqds,t,dtds) is obtained using the Euler-Lagrange equations derived from Equation ??? where the independent variable is s. This implies that for 0μn

dds(L(dqμds))Lqμ=QEXμ=mk=1dtdsλkgkqμ+QEXCμdtds

where the extended generalized force QEXμ shown on the right-hand side of Equation ???, accounts for all forces not included in the potential energy term in the Lagrangian. The extended generalized force QEXμ can be factored into two terms as discussed in chapter 6, equation (6.5.12). The Lagrange multiplier term includes 1km holonomic constraint forces where the m holonomic constraints, which do no work, are expressed in terms of the m algebraic equations of holonomic constraint gk. The QEXCμ term includes the remaining constraint forces and generalized forces that are not included in the Lagrange multiplier term or the potential energy term of the Lagrangian.

For the case where μ=0, since q0=ct, then Equation ??? reduces to

dds(L(dtds))Lt=mk=1dtdsλkgktnν=1QEXCνdqνds

These Euler-Lagrange equations of motion ???, ??? determine the 1μn generalized coordinates qμ(s), plus q0=ct(s) in terms of the independent variable s.

If the holonomic equations of constraint are time independent, that is gkt=0 and if QEXC0=0, then the μ=0 term of the Euler-Lagrange equations simplifies to

dds(L(dtds))Lt=0

One interpretation is to select L to be primary. Then L is derived from L using Equation ??? and L must satisfy the identity given by Equation ??? while the Euler-Lagrange equations containing dtds yield an identity which implies that L does not provide an equation of motion in terms of t(s). Conversely, if L is chosen to be primary, then L is no longer a homogeneous function and Equation ??? serves as a constraint on the motion that can be used to deduce L, while dtds yields a non-trivial equation of motion in terms of t(s). In both cases the occurrence of a constraint surface results from the fact that the extended space has 2n+2 variables to describe 2n+1 degrees of freedom, that is, one more degree of freedom than required for the actual system.

Example 17.6.1: Lagrangian for a relativistic free particle

The standard Lagrangian L=TU is not Lorentz invariant. The extended Lagrangian L(q,dqds,t,dtds) introduces the independent variable s which treats both the space variables q(s) and time variable q0=ct(s) equally. This can be achieved by defining the non-standard Lagrangian

L(q,dqds,t,dtds)=12mc2[1c2(dqds)2(dtds)21]

The constant third term in the bracket is included to ensure that the extended Lagrangian converges to the standard Lagrangian in the limit dtds1.

Note that the extended Lagrangian α is not homogeneous to first order in the velocities dqds as is required. Equation ??? must be used to ensure that Equation α is homogeneous. That is, it must satisfy the constraint relation

(dtds)21c2(dqds)21=0

Inserting β into the extended Lagrangian α yields that the square bracket in Equation α must equal 2. Thus

|L|=12mc2[2]=mc2

The constraint Equation β implies that

dsdt=11c2(dqdt)2=1γ

Using Equation δ gives that the relativistic Lagrangian is

L=Lγ=mc2γ=mc21β2

Equation ϵ is the conventional relativistic Lagrangian derived by assuming that the system evolution parameter s is transformed to be along the world line ds, where the invariant length ds replaces the proper time interval

ds=cdτ=cdtγ

The definition of the generalized (canonical) momentum

pi=Ld˙qi=γm˙qi

leads to the relativistic expression for momentum given in equation (17.4.6).

The relativistic Lagrangian is an important example of a non-standard Lagrangian. Equation α does not equal the difference between the kinetic and potential energies, that is, the relativistic expression for kinetic energy is given by (17.4.13) to be

T=(γ1)mc2

The non-standard relativistic Lagrangian ϵ can be used with the Euler-Lagrange equations to derive the second-order equations of motion for both relativistic and non-relativistic problems within the Special Theory of Relativity.

Example 17.6.2: Relativistic particle in an external elctromagnetic field

A charged particle moving at relativistic speed in an external electromagnetic field provides an example of the use of the relativistic Lagrangian.

In the discussion of classical mechanics it was shown that the velocity-dependent Lorentz force can be absorbed into the scalar electric potential Φ plus the vector magnetic potential A. That is, the potential energy is given by equation (17.3.4) to be U=q(ΦAv). Including this in the Lagrangian, ???, gives

L=mc2γU=mc21β2qΦ+qAv

The three spatial partial derivatives can be written in vector notation as

Lr=qΦ+qc(vA)

and the generalized momentum is given by

p=Ldv=γmv+qA

which is identical to the non-relativistic answer given by equation 7.6. That is, it includes the momentum of the electromagnetic field plus the classical linear momentum of the moving particle.

The total time derivative of the generalized momentum is

dpdt=ddt(Ldv)=ddt(γmv)+qdAdt

where the last term is given by the chain rule

dAdt=At+(v)A

Using equations a, b, c in the Euler-Lagrange equation gives

ddt(Ldv)=Lr

ddt(γmv)+qdAdt=qΦ+q(vA)

Collecting terms and using the well-known vector-product identity, plus the definition B=×A, gives

ddt(γmv)=[qΦqAt]+q[(vA)(v)A]=q[ΦAt]+q[v××A]F=q[E+v×B]

If we adopt the definition that the relativistic canonical momentum is p=γmv then the left hand side is the relativistic force while the right-hand side is the well-known Lorentz force of electromagnetism. Thus the extended Lagrangian formulation correctly reproduces the well-known Lorentz force for a charged particle moving in an electromagnetic field.


1Chapters 17.6 and 17.7 reproduce the Struckmeier presentation.[Str08]

2These formula involve total and partial derivatives with respect to both time, t and parameter s. For clarity, the derivatives are written out in full because Lanczos[La49] and Johns[Jo05] use the opposite convention for the dot and prime superscripts as abbreviations for the differentials with respect to t and s. The blackboard bold format is used to designate the extended versions of the action S, Lagrangian L and Hamiltonian H.


This page titled 17.6: Lorentz-Invariant Formulation of Lagrangian Mechanics is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Douglas Cline via source content that was edited to the style and standards of the LibreTexts platform.

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