Abstract
Hamilton’s variational principle and the Lagrangian mechanics that rests on it are exceedingly successful in their application to mechanical systems with a finite number of degrees of freedom. Hamilton’s principle characterizes the physically realizable orbits, among the set of all possible orbits, as being the critical elements of the action integral. The Lagrangian function, although not an observable on its own, is not only useful in deriving the equations of motion but is also an important tool for identifying symmetries of the theory and constructing the corresponding conserved quantities, via Noether’s theorem.
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Notes
- 1.
Although spin is a quantum property and, hence, is described by self-adjoint operators \({\hat{S}}=\{\hat{S}_{i}\}\), \(i=1,2,3\), of quantum mechanics, its expectation values \(\langle{\hat{S}}\rangle\) in quantum states are classical observables.
- 2.
More on this example is found in [QP], Sect. 7.1.
- 3.
In quantum field theory, it is useful to provide the (quantized) scalar field \(\phi\) with the dimension (length) \({}^{{-1}}\). As one verifies, the density \(\mathcal{L}\) then has dimension \(E/L^{3}\), i.e. energy/volume.
- 4.
The precise statement is this: If an initial state which has a well-defined behaviour under space reflection went over into another state which exhibits such a momentum-angular momentum correlation, by the effect of electromagnetic interaction, the interaction would contain both parity-even and parity-odd terms.
- 5.
We follow the generally accepted convention of denoting the Fourier components by a “tilde”.
- 6.
Compared to (3.65b), the other sign was chosen on the right-hand side. This choice, which entails a sign change of the corresponding Green functions, does not limit the generality of the method.
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© 2012 Springer-Verlag Berlin Heidelberg
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Scheck, F. (2012). Maxwell Theory as a Classical FieldTheory. In: Classical Field Theory. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27985-0_3
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DOI: https://doi.org/10.1007/978-3-642-27985-0_3
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