Abstract
The stationary action principle and the general form of the Euler–Lagrange equations. The notion of symmetry in classical field theory. Noether’s conserved currents.
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Notes
- 1.
L is regarded as a function of q, \(\dot{q}\) and t.
- 2.
This assumption has been made in the derivation of the Euler–Lagrange equation (2.6).
- 3.
It may happen that some of the relations (2.9) reduce to trivial identities like \(0=0\).
- 4.
- 5.
\(\partial \Omega \) is called surface in the space-time because its dimension, equal to 3, differs from the dimension of the space-time by 1.
- 6.
One should not confuse a symmetry of a model with a symmetry of a concrete physical state. For example, a model which is invariant under rotations can predict the existence of physical states which are not invariant under rotations.
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Arodź, H., Hadasz, L. (2017). The Euler–Lagrange Equations and Noether’s Theorem. In: Lectures on Classical and Quantum Theory of Fields. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-55619-2_2
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DOI: https://doi.org/10.1007/978-3-319-55619-2_2
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