## What are some interesting calculus of variation problems? [closed]

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## Minimizing proper time

I’ve started studying general relativity course and now I have a question about proper time. Consider functional S[x]=−∫BAds, where A, B are fixed points of the space-time and ds2=dt2−dx2 (let our space-time be two-dimensional without loss of generality). Finding its minimum is equivalent to maximizing proper time s=∫BAds and it a well-known fact that maximizers … Read more

## Is there an accepted Lagrangian for the transport equation?

Perhaps because it is so simple, I have not seen a lagrangian form of the transport equation (∂t+a∂x)q=0. This equation is first order, which makes obtaining it from the Euler-Lagrange equation a bit tricky. It would appear that the lagrangian m2q2t+T2qxt yields qtt+Tmqxt=0, which in turn yields the transport equation with constant forcing qt+Tmqx+c=0 after … Read more

## Is Hamilton’s principle compatible with the relativity principle?

The principle of Hamilton in classical mechanics is a fundamental one. It states that the real trajectory of a particle extremize the action ∫t2t1dτL(q,˙q,τ). In this formalism, time has a different status than space. The problem is then, is it compatible with relativity? Answer Yes, the stationary action principle, δS = 0, also known as the Hamilton’s … Read more

## Hamilton’s principle and virtual work by constraint forces

I have a question about the following page 48 from the third edition of Goldstein’s “Classical Mechanics”. I do not understand how (2.34) shows that the virtual work done by forces of constraint is zero. How does the fact that “the same Hamilton’s principle holds for both holonomic and semiholonomic systems” show that the additional … Read more

## On Cohomological Gauge theory Calculation

I am in trouble with calculation details of Witten’s Two dimensional Gauge Theories Revisited. My questions is about (3.21) and (3.27). From section 3, we have δAi=iϵψiδψi=−ϵDiϕδϕ=0δλ=iϵηδη=ϵ[ϕ,λ]δχ=ϵHδH=iϵ[ϕ,χ]V=1h2∫ΣdμTr(12χ(H−2⋆F)+gijDiλψj) Then the (3.21) is given by L=−i{Q,V}=1h2∫ΣdμTr(12(H−f)2−12f2−iχ⋆Dψ+iDiηψi+DiλDiϕ+i2χ[χ,ϕ]+i[ψi,λ]ψi) My questions are followings: 1) What is the relations between δ∙ and {Q,∙}? Is there a minus sign when Q-operator crossing … Read more

## equation of motion for the scalar field via variational principle in general relativity

I would like to find the equation of motion for the scalar field ϕ by varying the following action in General Relativity. Special Relativity: S=−12∫d4ξηab∂aϕ∂bϕ General Relativity: S=−12∫d4x√ggμν∂μϕ∂νϕ I was able to get the correct equation using the covariant derivatives. Since they are constant with respect to the metric partial integration works and one obtains … Read more

## Entropy and the principle of least action

Is there any link between the law of maximum entropy and the principle of least action. Is it possible to derive one from the other ? Answer Yes, there is a link, both are examples of extremum principles. And yes it is possible to derive the principle of least action from the law of maximum … Read more

## Is boundary well defined if variation of metric don’t vanish on the boundary?

Suppose that you want to calculate the variation δS of an action induced by some arbitrary variation δgμν of the spacetime metric : S=∫ΩL√−gdDx. The domain of integration Ω is any finite part of spacetime, and ∂Ω is its boundary. Usually, Ω and ∂Ω are fixed during the variation and we’re asking that δgμν=0 on … Read more

## Least-action classical electrodynamics without potentials

Is it possible to formulate classical electrodynamics (in the sense of deriving Maxwell’s equations) from a least-action principle, without the use of potentials? That is, is there a lagrangian which depends only on the electric and magnetic fields and which will have Maxwell’s equations as its Euler-Lagrange equations? Answer 1) Well, at the classical level, … Read more