Seminar talk, 10 May 2023: Difference between revisions
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| abstract = Given a differential equation (or system) <math>\Delta</math> = 0 the inverse problem in the calculus of variations asks if there is a multiplier function <math>Q</math> such that | | abstract = Given a differential equation (or system) <math>\Delta</math> = 0 the inverse problem in the calculus of variations asks if there is a multiplier function <math>Q</math> such that | ||
<math>Q\Delta=E(L)</math> | <math>Q\Delta=E(L)</math> | ||
where <math>E(L)=0</math> is the Euler Lagrange equation for a Lagrangian <math>L</math>. A solution to this problem can be found in principle and expressed in terms of invariants of the equation <math>\Delta</math>. The variational operator problem asks the same question but <math>Q</math> can now be a differential operator as the following simple example demonstrates for the evolution equation <math>u_t - u_{xxx} = 0</math>, | where <math>E(L)=0</math> is the Euler Lagrange equation for a Lagrangian <math>L</math>. A solution to this problem can be found in principle and expressed in terms of invariants of the equation <math>\Delta</math>. The variational operator problem asks the same question but <math>Q</math> can now be a differential operator as the following simple example demonstrates for the evolution equation <math>u_t - u_{xxx} = 0</math>, | ||
<math>D_x(u_t - u_{xxx}) = u_{tx}-u_{xxxx}=E(-\frac12(u_tu_x+u_{xx}^2))</math>. | <math>D_x(u_t - u_{xxx}) = u_{tx}-u_{xxxx}=E\left(-\frac12(u_tu_x+u_{xx}^2)\right)</math>. | ||
Here <math>D_x</math> is a variational operator for <math>u_t - u_{xxx} = 0</math>. | Here <math>D_x</math> is a variational operator for <math>u_t - u_{xxx} = 0</math>. | ||
Revision as of 20:50, 25 April 2023
Speaker: Mark Fels
Title: Variational/Symplectic and Hamiltonian Operators
Abstract:
Given a differential equation (or system) = 0 the inverse problem in the calculus of variations asks if there is a multiplier function such that
where is the Euler Lagrange equation for a Lagrangian . A solution to this problem can be found in principle and expressed in terms of invariants of the equation . The variational operator problem asks the same question but can now be a differential operator as the following simple example demonstrates for the evolution equation ,
.
Here is a variational operator for .
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.