Multisymplectic geometry, variational integrators, and nonlinear PDEs
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
1998
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| _version_ | 1866908600126930944 |
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| author | Marsden, Jerrold E. Patrick, George W. Shkoller, Steve |
| author_facet | Marsden, Jerrold E. Patrick, George W. Shkoller, Steve |
| contents | This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle. In particular, we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function, and use this structure to prove covariant generalizations of conservation of symplecticity and Noether's theorem. Natural discretization schemes for PDEs, which have these important preservation properties, then follow by choosing a discrete action functional. In the case of mechanics, we recover the variational symplectic integrators of Veselov type, while for PDEs we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries. We show that the usual notion of symplecticity along an infinite-dimensional space of fields can be naturally obtained by making a spacetime split. All of the aspects of our method are demonstrated with a nonlinear sine-Gordon equation, including computational results and a comparison with other discretization schemes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_9807080 |
| institution | arXiv |
| publishDate | 1998 |
| record_format | arxiv |
| spellingShingle | Multisymplectic geometry, variational integrators, and nonlinear PDEs Marsden, Jerrold E. Patrick, George W. Shkoller, Steve Differential Geometry Numerical Analysis Mathematical Physics Dynamical Systems Symplectic Geometry This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle. In particular, we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function, and use this structure to prove covariant generalizations of conservation of symplecticity and Noether's theorem. Natural discretization schemes for PDEs, which have these important preservation properties, then follow by choosing a discrete action functional. In the case of mechanics, we recover the variational symplectic integrators of Veselov type, while for PDEs we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries. We show that the usual notion of symplecticity along an infinite-dimensional space of fields can be naturally obtained by making a spacetime split. All of the aspects of our method are demonstrated with a nonlinear sine-Gordon equation, including computational results and a comparison with other discretization schemes. |
| title | Multisymplectic geometry, variational integrators, and nonlinear PDEs |
| topic | Differential Geometry Numerical Analysis Mathematical Physics Dynamical Systems Symplectic Geometry |
| url | https://arxiv.org/abs/math/9807080 |