Work and energy for particles in electromagnetic field
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ELEMENTARY PARTICLES AND FIELDS Theory
Work and Energy for Particles in Electromagnetic Field∗ S. G. Babajanyan** Yerevan Physics Institute, Alikhanian Brothers str. 2, Yerevan, 375036 Armenia Received May 18, 2016
Abstract—Defining the energy and work for particles interacting with electromagnetic field (EMF) is an open problem, because—due to the gauge-freedom—there exist various non-equivalent possibilities. It is argued that a consistent definition can be provided via the Lorenz gauge. To this end, I work out a system of two electromagnetically coupled classical particles. One of them is much heavier and models the source of work. The definition of energy in the Lorenz gauge is causal and consistent, because it leads to an approximate conservation law due to which the work done by the heavy particle (source of work) can be defined either via the kinetic energy of the heavy particle, or via the full time-dependent energy (kinetic + potential in the Lorenz gauge) of the light particle. DOI: 10.1134/S1063778817040032
1. INTRODUCTION Work is the basic quantity for statistical mechanics, e.g. because it appears in the formulations of the first and second laws of thermodynamics [1]. The virtue of this quantity is that it is well-defined both in and out of equilibrium (e.g. in contrast to entropy) and thus serves as the basis for non-equilibrium statistical mechanics. Consider a system of non-relativistic particles interacting via Hamiltonian H(p, q; f (t)), where (p, q) denotes all (canonical) momenta and coordinates of the particles, and where f (t) is an external field acting on the system. (If there are thermal baths, they are assumed to be included into the system.) Then the work done by the external field in the time interval [t1 , t2 ] is defined as [1]: H(p(t2 ), q(t2 ); f (t2 )) − H(p(t1 ), q(t1 ); f (t1 )), (1) i.e. it equals the energy change. The alternative definition of work goes via force, t2
df dt ∂f H(p(t), q(t); f (t)), dt
t1
and leads to (1) due to the Hamilton equations of motion. According to (1) no work is done if f is time independent. Definition (1) is well known and applies widely [1]. It generalizes to the statistical situation, ∗ **
The text was submitted by the author in English. E-mail: [email protected]
where the description goes not via trajectories, but probability densities or via density matrices (as in the quantum situation) [2]. Consider the Hamiltonian of a particle in electromagnetic field (EMF) [3] H(π, x; t) = p2 + m2 + eφ(x, t), π = p + eA(x, t), (2) i where √ A = (φ, A) is the 4-potential of EMF, p = mv/ 1 − v 2 is the kinetic momentum, π is the canonical momentum, x, m, and e are the (canonical) coordinate, mass, and charge, respectively. We employ the Gaussian units and c = 1. For a nonrelativistic particle, p2 + m2 in (2) is replaced by p2 /2m.
When attempting to define the work via Eqs. (1), (2) we meet the following problem [4–10]. Equation (2) is not invariant under a gauge transformation (defined via a function χ): φ(x, t) → φ(x, t) + ∂χ(x, t)/∂t, A(x, t) → A(x, t) −
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