Search Results

The answer is yes, if the interval you mean is $(0,\pi/2)$. That is, in any case, the natural domain on which the Euler-Lagrange equation $EL(x)$ is expected to hold. The reason is that it reflects th …

I believe that a piecewise smooth extremum would have to satisfy the Weierstrass-Erdmann corner conditions. If these conditions ensure that the extremum is in fact $C^1$, then it solves the Euler-Lagr …

A useful and fairly complete reference on this and related questions is Morrey Multiple Integrals in the Calculus of Variations. http://books.google.com/books?id=-QNKm1PBohsC

For now I'll just mention that there's a small literature on variational principles for perfect fluids in relativity, though I'm not an expert on it. Here is a reference that discusses some approaches …

Yes. The next operator in the sequence is called the Helmholtz operator, followed by higher versions thereof. The main keyword is "variational bicomplex" and a standard reference is I. M. Anderson, “ …

Let $\mathcal{E} \subset J^{2k}E$ be the submanifold (provided that this subset is a submanifold) of all $2k$-jets sitting in the zero-level set of $\rho(E(\mathscr{L}))$, the PDE submanifold. This is …

I'm not sure that there is a right answer to your question, since what is and what is not a satisfactory generalization of symplectic geometry can be quite subjective and in the end only can judge wha …

If I'm not miscalculating, the variational equation for $\varphi$ gives $\delta E_\epsilon/\delta\varphi = \partial^k (\rho^2 \partial_k \varphi) = 0$. Writing your integral identity as a flux through …

Differentiating the left-hand side of the second condition I get $b^{-1}(f(x) + \int_x^\infty f(t) dt)$, not just $b^{-1} f(x)$, but it is still $\ge 0$. So the second condition can be replaced by its …

In the form (1), if you compute the variation $\delta S / \delta x(t) = E(t)$, you find that $E(t) = E(x(t),\dot{x}(t), \ddot{x}(t) ,t)$ is a local/differential expression (the value of $E(t)$ does no …

I think you might find interesting §4.B of Anderson's The Variational Bicomplex [1] (Theorems 4.23, 4.29 and Corollary 4.30, to be more precise). However, these results are in a sense converse to your …

In a sense, all the Lagrangians giving the same Euler-Lagrange equations are exhausted by transformations of your type (b), which adds a total derivative/total divergence/boundary term/... Transforma …

The following remarks are based on having previously gone through the literature that you've mentioned also for the purposes of figuring out these differences. It has been a while since then, but the …