Abstract
In this chapter we shall start with one of the oldest and most useful branches of mathematical physics, the calculus of variations. After giving the fundamentals and some examples, we shall investigate the consequences of symmetries associated with variational problems. The chapter then ends with Noether’s theorem, which connects such symmetries with their associated conservation laws. All vector spaces of relevance in this chapter will be assumed to be real.
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Notes
- 1.
Do not confuse this functional with the linear functional of Chap. 2.
- 2.
The reader need not be concerned about lack of consistency in the location of indices (upper vs. lower), because we are dealing with indexed objects, such as F i , which are not tensors!
- 3.
In order to avoid confusion in applying formula (33.20), we use x (instead of t) as the independent variable and u (instead of x) as the dependent variable.
- 4.
See [Math 70, pp. 331–341] for a discussion of Lagrange multipliers and their use in variational techniques, especially those used in approximating solutions of the Schrödinger equation.
- 5.
We have multiplied J i by a negative sign to conform to physicists’ convention.
- 6.
The reader notes that the superscript α, which labeled components of the independent variable u, is now the label of the irreducible representation. The components of the dependent variable (now denoted by ϕ) are labeled by j.
- 7.
Only in three dimensions can one label rotations with a single index. This is because each coordinate plane has a unique direction (by the use of the right-hand rule) perpendicular to it that can be identified as the direction of rotation.
References
Mathews, J., Walker, R.: Mathematical Methods of Physics, 2nd edn. Benjamin, Elmsford (1970)
Olver, P.: Application of Lie Groups to Differential Equations. Springer, Berlin (1986)
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Hassani, S. (2013). Calculus of Variations, Symmetries, and Conservation Laws. In: Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-01195-0_33
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DOI: https://doi.org/10.1007/978-3-319-01195-0_33
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