Seminar talk, 10 May 2023

Speaker: Mark Fels

Title: Variational/Symplectic and Hamiltonian Operators

Abstract:
Given a differential equation (or system) $\Delta$ = 0 the inverse problem in the calculus of variations asks if there is a multiplier function $Q$ such that

             $Q\Delta =E(L)$,

where $E(L)=0$ is the Euler-Lagrange equation for a Lagrangian $L$. A solution to this problem can be found in principle and expressed in terms of invariants of the equation $\Delta$. The variational operator problem asks the same question but $Q$ can now be a differential operator as the following simple example demonstrates for the evolution equation $u_{t}-u_{xxx}=0$,

             $D_{x}(u_{t}-u_{xxx})=u_{tx}-u_{xxxx}=E\left(-{\frac {1}{2}}(u_{t}u_{x}+u_{xx}^{2})\right)$.

Here $D_{x}$ is a variational operator for $u_{t}-u_{xxx}=0$.

This talk will discuss how the variational operator problem can be solved in the case of scalar evolution equations and how variational operators are related to symplectic and Hamiltonian operators.

References:
arXiv:1902.08178