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.