# Extended Hamilton-Lagrange formalism and its application

to Feynman’s path integral for relativistic quantum physics

###### Abstract

We present a consistent and comprehensive treatise on the foundations of the extended Hamilton-Lagrange formalism—where the dynamical system is parameterized along a general system evolution parameter , and the time is treated as a dependent variable on equal footing with all other configuration space variables . In the action principle, the conventional classical action is then replaced by the generalized action , with and denoting the conventional and the extended Lagrangian, respectively. It is shown that a unique correlation of and exists if we refrain from performing simultaneously a transformation of the dynamical variables. With the appropriate correlation of and in place, the extension of the formalism preserves its canonical form.

In the extended formalism, the dynamical system is described as a constrained motion within an extended space. We show that the value of the constraint and the parameter constitutes an additional pair of canonically conjugate variables. In the corresponding quantum system, we thus encounter an additional uncertainty relation.

As a consequence of the formal similarity of conventional and extended Hamilton-Lagrange formalisms, Feynman’s non-relativistic path integral approach can be converted on a general level into a form appropriate for relativistic quantum physics. In the emerging parameterized quantum description, the additional uncertainty relation serves as the means to incorporate the constraint and hence to finally eliminate the parameterization.

We derive the extended Lagrangian of a classical relativistic point particle in an external electromagnetic field and show that the generalized path integral approach yields the Klein-Gordon equation as the corresponding quantum description. We furthermore derive the space-time propagator for a free relativistic particle from its extended Lagrangian . These results can be regarded as the proof of principle of the relativistic generalization of Feynman’s path integral approach to quantum physics.

PACS numbers: 04.20.Fy, 03.65.-w, 03.65.Pm

## 1 Introduction

Even more than hundred years after the emerging of Einstein’s special theory of relativity, the presentation of classical dynamics in terms of the Lagrangian and the Hamiltonian formalism is still usually based in literature on the Newtonian absolute time as the system evolution parameter[1, 2, 3, 4, 5, 6, 7]. The idea how the Hamilton-Lagrange formalism is to be generalized in order to be compatible with relativity is obvious and well-established. It consists of introducing a system evolution parameter, , as the new independent variable, and of subsequently treating the time as a dependent variable of , in parallel to all configuration space variables . This idea has been pursued in numerous publications, only a few of them being cited here. To mention only one, Duru and Kleinert[8] succeeded in solving the quantum mechanical Coulomb problem in terms of a parameterized Feynman path integral[9] that is based on a particular, not explicitly time-dependent extended Lagrangian.

Despite this unambiguity in the foundations and the huge pile of publications on the matter—dating back to P. Dirac[10] and C. Lanczos[11]—there is up to date no consensus in literature how this extension of the Hamilton-Lagrange formalism is actually to be devised. The reason is that on the basis of the action principle, there is apparently some freedom in defining how the conventional Hamiltonian relates to the extended Hamiltonian . On one hand, we often find in literature approaches based on the trivial extended Hamiltonian[12, 13, 14] that simply reduces the extended formalism to the conventional one by identifying the parameter with the time . On the other hand, we encounter quite general ad-hoc definitions of extended Hamiltonians that contain additional functions of the dynamical variables whose physical meaning and compatibility with the action principle is unclear[8, 15].

Thus, the key issue for casting the extended Hamilton-Lagrange formalism into its canonical form is to clarify how the Lagrangians and , as well as the Hamiltonians and should be correlated. To this end, we must separate the task of relating with , and with , from the task of performing a transformation of the dynamical variables. With these different matters clearly distinguished, we show that consistent and unique representations of both the extended Lagrangian as well as the extended Hamiltonian exist. With our relation of and in place, we find the subsequent extended set of canonical equations to perfectly coincide in its form with the conventional one, which means that no additional functions are involved. This is also true for the theory of extended canonical transformations. The connection of the extended with the conventional Hamiltonian description is established by the trivial extended Hamiltonian , whose canonical equations coincide with those of a conventional Hamiltonian . Correspondingly, the trivial extended generating function generates exactly the subgroup of conventional canonical transformations within the group of extended canonical transformations. This subgroup consists of exactly those canonical mappings that leave the time variable unchanged.

On grounds of the formal similarity of conventional and extended Hamilton-Lagrange formalisms, it is possible to formally convert non-relativistic approaches that are based on conventional Lagrangians into relativistic approaches in terms of extended Lagrangians. This idea is worked out exemplarily for Feynman’s path integral approach to quantum physics[16].

The paper is organized as follows. We start in Sect. 2.1 with the Lagrangian description and derive from the extended form of the action integral the extended Lagrangian , together with its relation to the conventional Lagrangian . It is shown that this relation reduces to the factor . More “general” correlations are shown to correspond to an additional transformation of the dynamical variables. The extended set of Euler-Lagrange equations then follows trivially from the dependencies of the extended Lagrangian. It is shown that the extended Lagrangian description of dynamics consists in a constrained motion in an extended space, namely in the tangent bundle over the space-time configuration manifold .

To provide a simple example, we derive in Sect. 3.1 the extended Lagrangian for a free relativistic point particle. This Lorentz-invariant Lagrangian has the remarkable feature to be quadratic in the velocities. This contrasts to the conventional Lorentz-invariant Lagrangian that describes the identical dynamics. For this system, the constraint depicts the constant square of the four-velocity vector.

We show in Sect. 3.2 that the extended Lagrangian of a relativistic particle in an external electromagnetic field agrees in its form with the corresponding non-relativistic conventional Lagrangian . The difference between both is that the derivatives in the extended Lagrangian are being defined with respect to the particle’s proper time, which are converted into derivatives with respect to the Newtonian absolute time in the non-relativistic limit.

In Sect. 2.2, we switch to the extended Hamiltonian description. As the extended Hamiltonian springs up from the extended Lagrangian by means of a Legendre transformation, both functions have equally the total information content on the dynamical system in question. The Hamiltonian counterparts of the Lagrangian description, namely, the extended set of canonical equations, the constraint function, and the correlation of the extended Hamiltonian to the conventional Hamiltonian are presented. On this basis, the theory of extended canonical transformations and the extended version of the Hamilton-Jacobi equation are worked out as straightforward generalizations of the conventional theory. As a mapping of the time is incorporated in an extended canonical transformation, not only the transformed coordinates emerging from the Hamilton-Jacobi equation are constants, as usual, but also the transformed time . The extended Hamilton-Jacobi equation may thus be interpreted as defining the mapping of the entire dynamical system into its state at a fixed instant of time, i.e., for instance, into its initial state. In the extended formulation, the Hamilton-Jacobi equation thus reappears in a new perspective.

We furthermore show that the value of the extended Hamiltonian and the system evolution parameter yield an additional pair of canonically conjugate variables. For the corresponding quantum system, we thus encounter an additional uncertainty relation. Based on both the extended Lagrangian and the additional uncertainty relation, we present in Sect. 2.5 the path integral formalism in a form appropriate for relativistic quantum systems. An extension of Feynman’s approach was worked out earlier[8] for a particular system. Nevertheless, the most general form of the extended path integral formalism that applies for any extended Lagrangian is presented here for the first time. By consistently treating space and time variables on equal footing, the generalized path integral formalism is shown to apply as well for Lagrangians that explicitly depend on time. In particular, the transition of a wave function is presented here as a space-time integral over a space-time propagator. In this context, we address the physical meaning of the additional integration over . The uncertainty relation is exhibited as the quantum physics’ means to incorporate the constraint in order to finally eliminate the parameterization.

On grounds of a generalized understanding of the action principle, Feynman showed that the Schrödinger equation emerges as the non-relativistic quantum description of a dynamical system if the corresponding classical system is described by the non-relativistic Lagrangian of a point particle in an external potential. Parallel to this beautiful approach, we derive in Section 3.6 the Klein-Gordon equation as the relativistic quantum description of a system, whose classical counterpart is described by the extended Lagrangian of a relativistic point particle in an external electromagnetic field. The reason for this to work is twofold. Since the extended Lagrangian agrees in its form with the conventional non-relativistic Lagrangian , the generalized path integral formalism can be worked out similarly to the non-relativistic case. Furthermore, as we proceed in our derivation an infinitesimal proper time step only and consider the limit , the constraint disappears by virtue of the uncertainty relation.

We finally derive in Sect. 3.7 the space-time propagator for the wave function of a free particle with spin zero from the extended Lagrangian of a free relativistic point particle. The constraint function, as the companion of the classical extended description, is taken into account in the quantum description by integrating over all possible parameterizations of the system’s variables. This integration is now explained in terms of the uncertainty relation. We regard these results as the ultimate confirmation of the relativistic generalization of Feynman’s path integral formalism.

## 2 Extended Hamilton-Lagrange formalism

### 2.1 Extended set of Euler-Lagrange equations

The conventional formulation of the principle of least action is based on the action functional , defined by

(1) |

with denoting the system’s conventional Lagrangian, and the vector of configuration space variables as a function of time. In this formulation, the independent variable time plays the role of the Newtonian absolute time. The clearest reformulation of the least action principle for relativistic physics is accomplished by treating the time —like the vector of configuration space variables—as a dependent variable of a newly introduced independent variable, [11, 17, 18, 19]. The action functional then writes in terms of an extended Lagrangian

(2) |

As the action functional (2) has the form of (1), the subsequent Euler-Lagrange equations that determine the particular path on which the value of the functional takes on an extremum, adopt the customary form,

(3) |

Here, the index spans the entire range of extended configuration space variables. In particular, the Euler-Lagrange equation for writes

The equations of motion for both and are thus determined by the extended Lagrangian . The solution of the Euler-Lagrange equations that equivalently emerges from the corresponding conventional Lagrangian may then be constructed by eliminating the evolution parameter .

As the actions, and , are supposed to be alternative characterizations of the same underlying physical system, the action principles and must hold simultaneously. This means that

which, in turn, is assured if both integrands differ at most by the -derivative of an arbitrary differentiable function

Functions define a particular class of point transformations of the dynamical variables, namely those ones that preserve the form of the Euler-Lagrange equations. Such a transformation can be applied at any time in the discussion of a given Lagrangian system and should be distinguished from correlating and . We may thus restrict ourselves without loss of generality to those correlations of and , where . In other words, we correlate and without performing simultaneously a transformation of the dynamical variables. We will discuss this issue in the more general context of extended canonical transformations in Sect. 2.3. The extended Lagrangian is then related to the conventional Lagrangian, , by

(4) |

The derivatives of from Eq. (4) with respect to its arguments can now be expressed in terms of the conventional Lagrangian as

(5) | |||||

(6) | |||||

(7) | |||||

(8) |

Equations (7) and (8) yield for the following sum over the extended range of dynamical variables

The extended Lagrangian thus satisfies the constraint

(9) |

The correlation (4) and the pertaining condition (9) allows two interpretations, depending on which Lagrangian is primarily given, and which one is derived. If the conventional Lagrangian is the given function to describe the dynamical system in question and is derived from according to Eq. (4), then is a homogeneous form of first order in the variables . This may be seen by replacing all derivatives with , in Eq. (4). Consequently, Euler’s theorem on homogeneous functions states that Eq. (9) constitutes an identity for [11]. The Euler-Lagrange equation involving then also yields an identity, hence, we do not obtain a substantial equation of motion for . In this case, the parameterizations of time is left undetermined—which reflects the fact that a conventional Lagrangian does not provide any information on a parameterization of time.

In the opposite case, if an extended Lagrangian is the primary function to describe our system, then Eq. (9) furnishes a constraint function for the system. Furthermore, the Euler-Lagrange equation involving then yields a non-trivial equation of motion for . The conventional Lagrangian may then be deduced from (4) by means of the constraint function (9).

To summarize, by switching from the conventional variational principle (1) to the extended representation (2), we have introduced an extended Lagrangian that additionly depends on . Due to the emerging constraint function (9), the actual number of degrees of freedom is unchanged. Geometrically, the system’s motion now takes place on a hypersurface, defined by Eq. (9), within the tangent bundle over the space-time configuration manifold . This contrasts with the conventional, unconstrained Lagrangian description on the time-dependent tangent bundle .

### 2.2 Extended set of canonical equations

The Lagrangian formulation of particle dynamics can equivalently be expressed as a Hamiltonian description. The complete information on the given dynamical system is then contained in a Hamiltonian , which carries the same information content as the corresponding Lagrangian . It is defined by the Legendre transformation

(10) |

with the covariant momentum vector components being defined by

Correspondingly, the extended Hamiltonian is defined as the extended Legendre transform of the extended Lagrangian as

(11) |

We know from Eq. (7) that for the momentum variable is equally obtained from the extended Lagrangian ,

(12) |

This fact ensures the Legendre transformations (10) and (11) to be compatible. For the index , i.e., for we must take some care as the derivative of with respect to evaluates to

The momentum coordinate that is conjugate to must therefore be defined as

(13) |

with representing the instantaneous value of the Hamiltonian at , but not the Hamilton function itself. This distinction is essential as the canonical coordinate must be defined—like all other canonical coordinates—as a function of the independent variable only. The reason is that the with depict the coordinates pertaining to the base vectors that span the (symplectic) extended phase space. We may express this fact by means of the comprehensible notation

(14) |

The constraint function from Eq. (9) translates in the extended Hamiltonian description simply into

(15) |

This means that the extended Hamiltonian directly defines the hypersurface on which the classical motion of the system takes place. The hypersurface lies within the cotangent bundle over the same extended configuration manifold as in the case of the Lagrangian description. Inserting Eqs. (12) and (14) into the extended set of Euler-Lagrange equations (3) yields the extended set of canonical equations,

(16) |

The right-hand sides of these equations follow directly from the Legendre transformation (11) since the Lagrangian does not depend on the momenta and has, up to the sign, the same space-time dependence as the Hamiltonian . The extended set is characterized by the additional pair of canonical equations for the index , which reads in terms of and

(17) |

By virtue of the Legendre transformations (10) and (11), the correlation from Eq. (4) of extended and conventional Lagrangians is finally converted into

(18) |

as only the term for the index does not cancel after inserting Eqs. (10) and (11) into (4).

The conventional Hamiltonian is defined as the particular function whose value coincides with the extended phase-space variable . In accordance with Eqs. (13) and (15), we thus determine for any given extended Hamiltonian by solving for . Then, emerges as the right-hand side of the equation .

In the converse case, if merely a conventional Hamiltonian is given, and is set up according to Eq. (18), then the canonical equation for yields an identity, hence allows arbitrary parameterizations of time. This is not astonishing as a conventional Hamiltonian generally does not provide the information for an equation of motion for .

Corresponding to Eq. (13), we may introduce the variable as the value of the extended Hamiltonian . We can formally define to be in addition a function of ,

(19) |

By virtue of the extended set of canonical equations (16), we find that is a constant of motion if and only if does not explicitly depend on ,

In this case, can be regarded as a cyclic variable, with the pertaining constant of motion, and hence its conjugate. Thus, in the same way as constitutes a pair of canonically conjugate variables, so does the pair , i.e., the value of the extended Hamiltonian and the parameterization of the system’s variables in terms of . In the context of a corresponding quantum description, this additional pair of canonically conjugate variables gives rise to the additional uncertainty relation

(20) |

Thus, in a quantum system whose classical limit is described by an extended Hamiltonian , we cannot simultaneously measure exactly a deviation from the constraint condition from Eqs. (15), (19) and the actual value of the system parameter . For the particular extended Hamiltonian of a relativistic particle in an external electromagnetic field, to be discussed in Sect. 3.3, the constraint reflects the relativistic energy-momentum correlation, whereas the parameter represents the particle’s proper time. For this particular system, the uncertainty relation (20) thus states the we cannot have simultaneous knowledge on a deviation from the relativistic energy-momentum correlation (53) and the particle’s proper time. The extended Lagrangian and the uncertainty relation (20) constitute together the cornerstones for deriving the relativistic generalization of Feynman’s path integral approach to non-relativistic quantum physics, to be presented in Sect. 2.5.

To end this section, we remark that the extended Hamiltonian most frequently found in literature is given by (cf, for instance, Refs. [11, 12, 14, 13, 15, 20])

(21) |

According to Eqs. (17), the canonical equation for is obtained as

Up to arbitrary shifts of the origin of our time scale, we thus identify with . As all other partial derivatives of coincide with those of , so do the respective canonical equations. The system description in terms of from Eq. (21) is thus identical to the conventional description and does not provide any additional information. The extended Hamiltonian (21) thus constitutes the trivial extended Hamiltonian.

### 2.3 Extended canonical transformations

The conventional theory of canonical transformations is built upon the conventional action integral from Eq. (1). In this theory, the Newtonian absolute time plays the role of the common independent variable of both original and destination system. Similarly to the conventional theory, we may build the extended theory of canonical equations on the basis of the extended action integral from Eq. (2). With the time and the configuration space variables treated on equal footing, we are enabled to correlate two Hamiltonian systems, and , with different time scales, and , hence to canonically map the system’s time and its conjugate quantity in addition to the mapping of generalized coordinates and momenta . The system evolution parameter is then the common independent variable of both systems, and . A general mapping of all dependent variables may be formally expressed as

(22) |

Completely parallel to the conventional theory, the subgroup of general transformations (22) that preserve the action principle of the system is referred to as “canonical”. The action integral (2) may be expressed equivalently in terms of an extended Hamiltonian by means of the Legendre transformation (11). We thus get the following condition for a transformation (22) to be canonical

(23) |

As we are operating with functionals, the condition (23) holds if the integrands differ at most by the derivative of an arbitrary differentiable function

(24) |

We restrict ourselves to functions of the old and the new extended configuration space variables, hence to a function of those variables, whose derivatives are contained in Eq. (24). Calculating the -derivative of ,

(25) |

we then get unique transformation rules by comparing the coefficients of Eq. (25) with those of (24)

(26) |

is referred to as the extended generating function of the—now generalized—canonical transformation. The extended Hamiltonian has the important property to be conserved under these transformations. Corresponding to the extended set of canonical equations, the additional transformation rule is given for the index . This transformation rule may be expressed equivalently in terms of , , and , as

(27) |

with , correspondingly to Eq. (13), the value of the transformed Hamiltonian

(28) |

The addressed transformed Hamiltonian is finally obtained from the general correlation of conventional and extended Hamiltonians from Eq. (18), and the transformation rule for the extended Hamiltonian from Eq. (26)

Eliminating the evolution parameter , we arrive at the following two equivalent transformation rules for the conventional Hamiltonians under extended canonical transformations

(29) |

The transformation rules (29) are generalizations of the rule for conventional canonical transformations as now cases with are included. We will see at the end of this section that the rules (29) merge for the particular case into the corresponding rules of conventional canonical transformation theory.

By means of the Legendre transformation

(30) |

we may express the extended generating function of a generalized canonical transformation equivalently as a function of the original extended configuration space variables and the extended set of transformed canonical momenta . As, by definition, the functions and agree in their dependence on the , so do the corresponding transformation rules

This means that all do not take part in the transformation defined by (30). Hence, for the Legendre transformation, we may regard the functional dependence of the generating functions to be reduced to and . The new transformation rule pertaining to thus follows from the -dependence of

The new set of transformation rules, which is, of course, equivalent to the previous set from Eq. (26), is thus

(31) |

Expressed in terms of the variables , , , , and , , , the new set of coordinate transformation rules takes on the more elaborate form

(32) |

Similarly to the conventional theory of canonical transformations, there are two more possibilities to define a generating function of an extended canonical transformation. By means of the Legendre transformation

we find in the same manner as above the transformation rules

(33) |

Finally, applying the Legendre transformation, defined by

the following equivalent version of transformation rules emerges

Calculating the second derivatives of the generating functions, we conclude that the following correlations for the derivatives of the general mapping from Eq. (22) must hold for the entire set of extended phase-space variables,

Exactly if these conditions are fulfilled for all , then the extended coordinate transformation (22) is canonical and preserves the form of the extended set of canonical equations (16). Otherwise, we are dealing with a general, non-canonical coordinate transformation that does not preserve the form of the canonical equations.

The connection of the extended canonical transformation theory with the conventional one is furnished by the particular extended generating function

(34) |

with denoting a conventional generating function. According to Eqs. (32), the coordinate transformation rules following from (34) are

Together with the general transformation rule (29) for conventional Hamiltonians, we find the well-known rule for Hamiltonians under conventional canonical transformations,

Canonical transformations that are defined by extended generating functions of the form of Eq. (34) leave the time variable unchanged and thus define the subgroup of conventional canonical transformations within the general group of extended canonical transformations. Corresponding to the trivial extended Hamiltonian from Eq. (21), we may refer to (34) as the trivial extended generating function.

### 2.4 Extended Hamilton-Jacobi equation

In the context of the extended canonical transformation theory, we may derive an extended version of the Hamilton-Jacobi equation. We are looking for a generating function of an extended canonical transformation that maps a given extended Hamiltonian into a transformed extended Hamiltonian that vanishes identically, , in the sense that all partial derivatives of vanish. Then, according to the extended set of canonical equations (16), the derivatives of all canonical variables with respect to the system’s evolution parameter must vanish as well

(35) |

This means that all transformed canonical variables must be constants of motion. Writing the variables for the index separately, we thus have

Thus, corresponding to the conventional Hamilton-Jacobi formalism, the vectors of the transformed canonical variables, and , are constant. Yet, in the extended formalism, the transformed time is also a constant. The particular generating function that defines transformation rules for the extended set of canonical variables such that Eqs. (35) hold for the transformed variables thus defines a mapping of the entire system into its state at a fixed instant of time, hence—up to trivial shifts in the origin of the time scale—into its initial state at

We may refer to this particular generating function as the extended Hamiltonian action function . According to the transformation rule for extended Hamiltonians from Eq. (26), we obtain the transformed extended Hamiltonian simply by expressing the original extended Hamiltonian in terms of the transformed variables. This means for the conventional Hamiltonian according to Eq. (18) in conjunction with the transformation rules from Eqs. (32),

As we have in general, we finally get the generalized form of the Hamilton-Jacobi equation,

(36) |

Equation (36) has exactly the form of the conventional Hamilton-Jacobi equation. Yet, it is actually a generalization as the extended action function represents an extended generating function of type , as defined by Eq. (30). This means that is also a function of the (constant) transformed energy .

Summarizing, the extended Hamilton-Jacobi equation may be interpreted as defining the mapping of all canonical coordinates , , , and of the actual system into constants , , , and . In other words, it defines the mapping of the entire dynamical system from its actual state at time into its state at a fixed instant of time, , which could be the initial conditions.

### 2.5 Generalized path integral with extended Lagrangians

In Feynman’s path integral approach to quantum mechanics, the space and time evolution of a wave function is formulated in terms of a transition amplitude density , also referred to as a kernel, or, a propagator. Its space-time generalization writes

(37) |

Obviously, this propagator has the dimension of a space-time density. The justification for integrating over all times is that in relativistic quantum physics we must allow the laboratory time to take any value—negative and even positive ones—if we regard from the viewpoint of a particle with its proper time . We thus additionally integrate over all histories of the particle. The integration over all futures can then be interpreted as integration over all histories of the anti-particle, whose proper time scale runs backwards in terms of the particle’s proper time scale[21].

The kernel for a parameterized action is given by the multiple path integral[22]

(38) |

Herein, the integrals are to be taken over all paths that go from at to at . The action functional stands for the -integral over the extended Lagrangian , as defined by Eq. (2).

In classical dynamics, the parameterization of space and time variables can be eliminated by means of the constraint function (9). For the corresponding quantum description, the uncertainty principle from Eq. (20) applies. It tells us that an accurate fulfillment of the constraint is related to a complete uncertainty about the parameterization of the system’s variables in terms of . Therefore, in the context of the path integral approach, the constraint is incorporated by integrating the parameterized kernel over all possible parameterizations of coordinates and time . The transition amplitude density is thus given by

(39) |

This means that all parameterized kernels contribute with equal weight to the total transition amplitude . The normalization factor is determined by the requirement that the integration (37) should preserve the norm of the wave function . As an example, we calculate in Sect. 3.7 the explicit form of the space-time propagator for the wave function of a relativistic free particle from the extended Lagrangian of the pertaining classical system.

For an infinitesimal step , we may approximate the action functional from Eq. (2) by

For , the kernel from Eq. (38) that yields the transition amplitude density for a particle along this infinitesimal interval is accordingly given by

As we proceed an infinitesimal step only, and then take the limit , the integration (39) over all possible parameterizations of this step must be omitted. For, conversely to the situation discussed beforehand, a small is related to a large uncertainty with respect to satisfying the constraint, so that in the limit the constraint ceases to exist.

The yet to be determined normalization factor represents the integration measure for one step of the multiple path integral (38). Clearly, this measure must depend on the step size . The transition of a given wave function at the particle’s proper time to the wave function that is separated by an infinitesimal proper time interval can now be formulated according to Eq. (37) as

(40) |

Note that we integrate here over the entire space-time. To serve as test for this approach, we derive in Sect. 3.6 the Klein-Gordon equation on the basis of the extended Lagrangian for a relativistic particle in an external electromagnetic field.

## 3 Examples of extended Hamilton-Lagrange systems

### 3.1 Extended Lagrangian for a relativistic free particle

As only expressions of the form are preserved under the Lorentz group, the conventional Lagrangian for a free point particle of mass , given by

(41) |

is obviously not Lorentz-invariant. Yet, in the extended description, a corresponding Lorentz-invariant Lagrangian can be constructed by introducing as the new independent variable, and by treating the space and time variables, and equally. This is achieved by adding the corresponding derivative of the time variable ,

(42) |

The constant third term has been defined accordingly to ensure that converges to in the limit . Of course, the dynamics following from (41) and (42) are different—which reflects the modification our dynamics encounters if we switch from a non-relativistic to a relativistic description. With the Lagrangian (42), we obtain from Eq. (9) the constraint

(43) |

As usual for constrained Lagrangian systems, we must not insert back the constraint function into the Lagrangian prior to setting up the Euler-Lagrange equations. Physically, the constraint (43) reflects the fact that the square of the four-velocity vector is constant. It equals if the sign convention of the Minkowski metric is defined as . We thus find that in the case of the Lagrangian (42) the system evolution parameter is physically nothing else than the particle’s proper time. In contrast to the non-relativistic description, the constant rest energy term in the extended Lagrangian (42) is essential. The constraint can alternatively be expressed as

which yields the usual relativistic scale factor, . The conventional Lagrangian that describes the same dynamics as the extended Lagrangian from Eq. (42) is derived according to Eq. (4)

(44) | |||||

We thus encounter the well-known conventional Lagrangian of a relativistic free particle. In contrast to the equivalent extended Lagrangian from Eq. (42), the Lagrangian (44) is not quadratic in the derivatives of the dependent variables, . The loss of the quadratic form originates from the projection of the constrained description on the tangent bundle to the unconstrained description on . The quadratic form is recovered in the non-relativistic limit by expanding the square root, which yields the Lagrangian from Eq. (41).

### 3.2 Extended Lagrangian for a relativistic particle in an external electromagnetic field

The extended Lagrangian of a point particle of mass and charge in an external electromagnetic field that is described by the potentials is given by

(45) |

The associated constraint function coincides with that for the free-particle Lagrangian from Eq. (43) as all terms linear in the velocities drop out calculating the difference in Eq. (9). Similar to the free particle case from Eq. (44), the extended Lagrangian (45) may be projected into to yield the well-known conventional relativistic Lagrangian

(46) |

Again, the quadratic form of the velocity terms is lost owing to the projection.

For small velocity , the quadratic form is regained as the square root in (46) may be expanded to yield the conventional non-relativistic Lagrangian for a point particle in an external electromagnetic field,

(47) |

Significantly, this Lagrangian can be derived directly, hence without the detour over the projected Lagrangian (46), from the extended Lagrangian (45) by letting .

It is instructive to review the Lagrangian (45) and its non-relativistic limit (47) in covariant notation. Denoting by the components of the contravariant four-vector of space-time variables , the corresponding covariant vector is then for the metric used here. With Einstein’s summation convention and the notation , the extended Lagrangian (45) then writes

(48) |

Correspondingly, the non-relativistic Lagrangian (47) has the equivalent representation

(49) |

Note that , which yields the second half of the rest energy term, so that (49) indeed agrees with (47). Comparing the Lagrangian(49) with the extended Lagrangian from Eq. (48)—and correspondingly the Lagrangians (45) and (47)—we notice that the transition to the non-relativistic description is made by identifying the proper time with the laboratory time . The remarkable formal similarity of the Lorentz-invariant extended Lagrangian (48) with the non-invariant conventional Lagrangian (49) suggests that approaches based on non-relativistic Lagrangians may be transposed to a relativistic description by (i) introducing the proper time as the new system evolution parameter, (ii) treating the time as an additional dependent variable on equal footing with the configuration space variables —commonly referred to as the “principle of homogeneity in space-time”—and (iii) by replacing the conventional non-relativistic Lagrangian with the corresponding Lorentz-invariant extended Lagrangian