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>. | ||
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. | 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. | ||
| video = | | video = https://video.gdeq.org/GDEq-zoom-seminar-20230510-Mark_Fels.mp4 | ||
| slides = | | slides = [[Media:SEMINAR-2023-05-10.pdf|SEMINAR-2023-05-10.pdf]] | ||
| references = | | references = {{arXiv|1902.08178}} | ||
| 79YY-MM-DD = 7976-94-89 | | 79YY-MM-DD = 7976-94-89 | ||
}} | }} | ||
Latest revision as of 08:40, 4 January 2025
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.
Video
Slides: SEMINAR-2023-05-10.pdf
References:
arXiv:1902.08178