# Questions tagged [variational-calculus]

Variational calculus is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find extrema of functionals: mappings from a set of functions to the real numbers. The archetype application in physics is Lagrangian mechanics, seeking extrema of action functionals.

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### Where exactly does the integral definition of the gradient come from? [migrated]

In the book "Essential mathematical methods for physicists" from Weber and Arfken, they define the integral form of the gradient,divergence and curl, althougth they give sections before an ...
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### Help deriving Maxwell's equations from the Lagrangian [duplicate]

Starting with the Lagrangian density $$\mathcal{L} = -\frac{1}{2}(\partial_\mu \mathcal{A}_\nu)(\partial^\mu \mathcal{A}^\nu)+\frac{1}{2}(\partial_\mu \mathcal{A}^\mu)^2,$$ I don't understand how to ...
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### If the solution of a field vanishes on-shell does it mean anything particular?

Let us consider an action $S=S(a,b,c)$ which is a functional of the fields $a,\, b,\,$ and $c$. The solution of the field $c$ is given by the expression $f(a,b)$. On taking into account the relations ...
1 vote
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### Feynman's Derivation for Principle of Least Action

In the Feynman Lectures on Physics (Addison–Wesley, Reading, MA, 1964), Vol. II, Chap. 19, Feynman demonstrates how the principle of stationary action for one particle implies Newton's second law (or ...
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### Variation of action for Jackiw-Teitelboim (JT) gravity in order to get equations of motion

Im having a bit of trouble understanding the variation of action for JT gravity (for example in this article https://arxiv.org/abs/1606.01857 the equation 3.8). I dont really get how do they get from ...
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### How did Landau & Lifshitz (Mechanics) get Equation 2.5?

I understood everything in Landau & Lifshitz's mechanics book until Equation 2.4,but I'm not sure what he means when he says "effecting the variation" and gets Equation 2.5. Can anyone ...
1 vote
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### If two different gauge transformations of an action commute, does it imply anything?

If two different gauge transformations of a Lagrangian commute with each other, does it imply anything?
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### How to distinguish a trivial gauge transformation from a non-trivial one?

Two days ago I posted a post that discusses a very generic gauge transformation. I repeat it here. Suppose we have an action $S=S(a,b,c)$ which is a functional of the fields $a,\, b,\,$ and $c$. We ...
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### Question about Trivial Gauge Transformation

Suppose we have an action $S=S(a,b,c)$ which is a functional of the fields $a,\, b,\,$ and $c$. We denote the variation of $S$ wrt to a given field, say $a$, i.e. $\frac{\delta S}{\delta a}$, by $E_a$....
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