Seminar talk, 10 May 2023

Speaker: Mark Fels

Title: Variational/Symplectic and Hamiltonian Operators

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

             $Q\mathrm{\Delta }=E(L)$

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 $\mathrm{\Delta }$. 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 $u_t-u_{xxx}=0$,

             $D_x(u_t-u_{xxx})=u_{tx}-u_{xxxx}=E(-\frac{1}{2}(u_t{u}_x+u_{xx}^2))$.

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.