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I'm stuck on a problem in the last chapter of Hamill's Student's Guide to Lagrangians and Hamiltonians. It asks why adding:

$$ \frac {\partial \mathscr{L}} {\partial t} + \frac {\partial \mathscr{L}} {\partial x} $$

to the Lagrangian density $\mathscr{L}$ won't change Lagrange's equations.

I've tried starting from Lagrange's equations themselves and from Hamilton's principle, but I can't get the new terms to cancel out. I also know that if the partial derivatives were of the variation, I could integrate and the variation would be zero at the boundary - but this hasn't really helped. Any help would be much appreciated!

References:

  1. P. Hamill, A Student’s Guide to Lagrangians and Hamiltonians, 2014; problem 7.1 p. 166.
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  • $\begingroup$ The linked post assumes that the fields go to zero at the boundary, but I don't think we can assume the Lagrangian density goes to zero at the boundary can we? $\endgroup$ – TKT Feb 18 at 19:57
  • $\begingroup$ Nevermind, of course the variation of the Lagrangian density has to be zero at the boundaries. Thanks for pointing me to that link! $\endgroup$ – TKT Feb 18 at 20:20
  • $\begingroup$ Related : Squaring the E&M (Maxwell) field strength tensor. $\endgroup$ – Frobenius Feb 19 at 0:34
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There are various issues with problem 7.1:

  1. Firstly, when reading chapter 7, it becomes clear that Hamill confusingly is denoting a total spacetime derivative $d/d x^{\mu}$ as $\partial/\partial x^{\mu}$, cf. e.g. this Phys.SE post.

  2. Secondly, for dimensional reasons, the function $F^{\mu}$ in the change in the Lagrangian density $$\Delta{\cal L}~=~\sum_{\mu=0}^3dF^{\mu}/d x^{\mu}\tag{*}$$ must be different than the Lagrangian density ${\cal L}$ itself.

  3. Thirdly, for given boundary conditions (BCs), the change (*) of the Lagrangian density may destroy the existence of the functional/variational derivative, cf. e.g. this Phys.SE post.

Altogether, this makes the corrected problem 7.1 effectively a duplicate of e.g. this Phys.SE post.

References:

  1. P. Hamill, A Student’s Guide to Lagrangians and Hamiltonians, 2014; problem 7.1 p. 166.
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