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Variational Principle for Electrodynamics of Moving Particles
Jerzy Kijowski Centrum Fizyki Teoretycznej PAN
Aleja Lotników 32/46, 02-668 Warsaw, Poland and
Dariusz Chruściński Institute of Physics, Nicholas Copernicus University
ul. Grudzia̧dzka 5/7, 87-100 Toruń, Poland
Consistent relativistic theory of the classical Maxwell field interacting with clas- sical, charged, point–like particles, proposed in , is now derived from a variational principle. For this purpose a new electrodynamical Lagrangian based on fluxes is constructed. As a result, we obtain the action principle where 1) field degrees of freedom and particle degrees of freedom are kept at the same footing, 2) contrary to the standard formulation, no infinities arise, 3) energy (Hamiltonian) is obtained from the Lagrangian via the Legendre transformation, without any need of “adding a complete divergence”.
PACS: 03.50.De; 04.20.Fy; 41.10.-j; 41.70.+t
1 Introduction 2
2 Electrodynamics of moving particles: statement of results 5
3 Co-moving description of a relativistic field theory 7
4 Point particles and extended particles. Renormalization 12
5 Equations of motion from the variational principle 16
6 Variational principle based on fluxes 18
7 The Lagrangian in the co-moving frame 25
8 Particle in an external potential 27
A Hamiltonian structure for a 2-nd order Lagrangian theory 28
B Maxwell equations in the co-moving frame 29
C Boundary momenta 32
D Proof of the conservation laws 35
Recently, one of us (J.K) proposed a consistent relativistic theory of the classical Maxwell field interacting with classical, charged, point-like particles (cf. ). For this purpose an “already renormalized” formula for the total four-momentum of a system composed of both the moving particles and the surrounding electromagnetic field was proposed. It was proved, that the conservation of the total four-momentum defined by this formula is equivalent to a certain boundary condition for the behaviour of the Maxwell field in the vicinity of the particle trajectories (in  this condition is called the fundamental equation).
Field equations of such a theory are, therefore, precisely the linear, inhomogeneous Maxwell equations for the electromagnetic field surrounding the point-like sources. The new element introduced in , which completes this standard theory, is the above boundary condition, with the particle trajectories playing the role of the moving boundary. Together with this condition, the theory (called electrodynamics of moving particles) becomes causal and complete: initial data for both the field and the particles uniquely imply the evolution of the system. This means e. g. that the particle trajectories may also be calculated uniquely from the initial data.
It was proved that the limit of this theory for e → 0 and m → 0 with their ratio being fixed, coincides with the Maxwell–Lorentz theory of test particles moving on the background described by the free field. However, for any finite value of e, the acceleration of the particle can not be equal to the Lorentz force, the latter being always ill defined, because of the field singularities implied by Maxwell equations.
Physically, the “already renormalized” formula for the total four-momentum (formula (33) in the present paper) was suggested by a suitable approximation procedure applied to an extended-particle model. In such a model we suppose that the particle is a stable, soliton-like solution of a hypothetical fundamental theory of interacting electromagnetic
and matter fields. Assuming, that for weak electromagnetic fields and vanishing matter fields (i. e. outside of the particles) the theory coincides with linear Maxwell electrodynam- ics, a formula was found, which gives in a good approximation the total four-momentum of a system composed of both the moving particles and the surrounding electromagnetic field. The formula uses only the “mechanical” information about the particle (position, velocity, mass m and the electric charge e) and the free electromagnetic field outside of the particle. It turns out, that the formula is meaningful also in the case of point particles. Hence, it can be taken as a starting point for a mathematically self-consistent theory of point-like particles interacting with the linear Maxwell field.
In the present paper we give the variational formulation of the above theory. The corresponding, highly nontrivial canonical (Hamiltonian) structure, will be given in the next paper.
The standard variational principle used in electrodynamics cannot be extended to the theory containing also point-like particles interacting with the electromagnetic field. Such a principle is based on the following Lagrangian, written usually in textbooks (see e. g. , ):
Ltotal = LMaxwell + Lparticle + Lint , (1)
LMaxwell = −1 4
√−gfµνfµν , (2)
Lparticle := −mδζ , (3)
and the interaction term given by
Lint := eAµu µδζ . (4)
Here by δζ we denote the Dirac delta distribution localized on the particle trajectory ζ. The above Lagrangian may be used to derive the trajectories of the test particles, when the field is given a priori. In a different context, it may also be used to derive Maxwell equations, if the particle trajectories are given a priori. Simultaneous variation with respect to both fields and particles leads, however, to a contradiction, since the Lorentz force will be always ill defined due to Maxwell equations.
But already in the context of the inhomogeneous Maxwell theory with given point- like sources, the variational principle based on Lagrangian (1) is of very limited use, since the interaction term Lint becomes infinite. As a consequence, the hamiltonian of such a theory will always be ill defined, although the theory displays a perfectly causal behaviour.
The main result of the present paper consists in removing this difficulty. Applying an appropriate Legendre transformation to the field Lagrangian LMaxwell, we obtain a new, quasi-local variational principle for the Maxwell field. Already in the context of the inhomogeneous Maxwell theory with given (i. e. non-dynamical) point-like sources, our Lagrangian produces no infinities and enables us to describe the dynamics of the field influenced by moving particles as an infinite-dimensional Hamiltonian system. It turns
out, that adding the particle Lagrangian (3) and varying it with respect to both fields and particles is now possible and does not lead to any contradiction. As a result, we obtain precisely the “electrodynamics of moving particles” proposed in .
It is not unusual that the same physical theory is described by different variational principles. We derive our new variational principle, transforming the standard Maxwell Lagrangian by an appropriately chosen Legendre transformation. Each of these two vari- ational principles is related to a specific way of controlling the boundary data of the field. Changing the variational principle means changing the physical quantities, which are kept fixed at the boundary during the variation. In the standard approach, the variation is performed with values of the electromagnetic potentials Aµ being kept on the boundary. In our approach, we keep at the boundary the value of the electric and the magnetic fluxes.
Passing from the lagrangian description to the hamiltonian one, different Lagrangians lead to different field Hamiltonians, describing the field dynamics with different boundary conditions. It is worthwhile to notice that the Hamiltonian obtained from our Lagrangian is equal to the field energy: 1
2 (D2 + B2). Other Hamiltonians, related to other boundary
conditions, which may be obtained from other Lagrangians (e. g. the standard one (2)), are not positive and even not bounded from below (see ).
The relation between different Hamiltonians, corresponding to different boundary con- ditions, is similar to the relation between the internal energy and the free (Helmholtz) energy in thermodynamics. The first one describes the evolution of the thermodynamic system, when insulated adiabatically from any external influence, whereas the latter de- scribes the (completely different) evolution of the same system, when put into a thermal bath. From this point of view, controlling the electric and the magnetic fluxes on the boundary of a 3-dimensional volume V means insulating it adiabatically from any exter- nal influence, whereas the standard control of potentials still leaves the possibility of the energy exchange between the exterior and the interior of V .
The paper is organized as follows. Section 2 contains the main results of the theory proposed in . In Section 3 we develop a new technique, which enables us to describe at the same
footing the field and the particle degrees of freedom. For this purpose we formulate any relativistic, hyperbolic field theory with respect to a non-inertial reference frame defined as a rest-frame for an arbitrarily moving observer. Such a formulation will be used as a starting point for our renormalization procedure.
In Section 4 we show how to extend the above approach to the case of electrodynamical field interacting with point particles. This enables us to derive in subsequent Sections the Electrodynamics of Moving Particles from a variational principle.
Finally, in Section 8 we present the lagrangian formulation for the particle interacting not only with