Abstract
In this chapter we apply the variational principle to rederive the dynamics of classical electrodynamics, together with its conservation laws. This physical system represents a field theory, described by the fields \(A_\mu (x)\), interacting with a system of charged particles, described by the world lines \(y^\mu _r(\lambda _r)\). Before considering the coupled system, we establish the form of the action of a free relativistic particle.
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Notes
- 1.
Alternatively, the action (4.6) could be considered as a functional of the Lagrangian coordinates \(\mathbf {y}(t)\), in place of the \(y^\mu (\lambda )\), in which case, choosing the parameter \(\lambda =y^0/c= t\), it would become
$$\begin{aligned} I[\mathbf {y}]=-mc^2\int \sqrt{1-v^2/c^2}\, dt.\qquad {(4.7)} \end{aligned}$$The action (4.7) yields the three equations of motion \(d\mathbf {p}/dt=0\), where \(\mathbf {p}\) is the relativistic momentum (2.11), which, although not manifestly covariant, are physically equivalent to the four equations (4.1).
- 2.
The world line \(\gamma _r\) intersects the hypersurfaces \(\varSigma _a\) and \(\varSigma _b\) at most once, because the first is time-like and the latter are space-like.
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Lechner, K. (2018). Variational Methods in Electrodynamics. In: Classical Electrodynamics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-91809-9_4
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DOI: https://doi.org/10.1007/978-3-319-91809-9_4
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