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Multisymplectic geometry, variational integrators, and nonlinear PDEs

Jerrold E. Marsden, George W. Patrick, Steve Shkoller

TL;DR

This work develops a cohesive geometric framework for multisymplectic–momentum integrators for variational PDEs, unifying continuous multisymplectic geometry with Veselov-style discretizations to yield covariant, structure-preserving spacetime integrators. By deriving fundamental objects like the Lagrange 1-form, Cartan form, and the discrete multisymplectic form formula directly from a variational principle, the authors obtain variationally consistent discrete dynamics that conserve discrete multisymplectic forms and momentum maps. The approach is illustrated with the nonlinear sine-Gordon/wave equation, including detailed numerical experiments comparing triangle- and rectangle-based discretizations and contrasting with energy-conserving methods. The framework extends to general field theories, jet-extensions, and elliptic problems, with implications for long-time simulations of nonlinear PDEs and potential connections to finite-element methods and reduction theory. Overall, the paper provides a principled, covariant route to structure-preserving discretizations for a broad class of variational PDEs, offering both theoretical foundations and practical numerical schemes.

Abstract

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.

Multisymplectic geometry, variational integrators, and nonlinear PDEs

TL;DR

This work develops a cohesive geometric framework for multisymplectic–momentum integrators for variational PDEs, unifying continuous multisymplectic geometry with Veselov-style discretizations to yield covariant, structure-preserving spacetime integrators. By deriving fundamental objects like the Lagrange 1-form, Cartan form, and the discrete multisymplectic form formula directly from a variational principle, the authors obtain variationally consistent discrete dynamics that conserve discrete multisymplectic forms and momentum maps. The approach is illustrated with the nonlinear sine-Gordon/wave equation, including detailed numerical experiments comparing triangle- and rectangle-based discretizations and contrasting with energy-conserving methods. The framework extends to general field theories, jet-extensions, and elliptic problems, with implications for long-time simulations of nonlinear PDEs and potential connections to finite-element methods and reduction theory. Overall, the paper provides a principled, covariant route to structure-preserving discretizations for a broad class of variational PDEs, offering both theoretical foundations and practical numerical schemes.

Abstract

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.

Paper Structure

This paper contains 59 sections, 9 theorems, 199 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Outline of paper.
  3. Lagrangian Mechanics
  4. Hamilton's Principle.
  5. The standard geometric setting.
  6. The variational approach.
  7. Lagrangian flows are symplectic.
  8. Momentum maps.
  9. Reformulation in terms of first variations.
  10. Veselov Discretizations of Mechanics
  11. The Lagrange $1$-form.
  12. Symplecticity of the flow.
  13. Noether's Theorem.
  14. Reduction.
  15. Variational Principles for Classical Field Theory
  16. ...and 44 more sections

Key Result

Theorem 2.1

Given a $C^k$ Lagrangian $L$, $k\ge2$, there exists a unique $C^{k-2}$ mapping $D_{\hbox{\scriptsize EL}}L:\ddot Q\rightarrow T^*Q$, defined on the second order submanifold of $TTQ$, and a unique $C^{k-1}$$1$-form $\theta_L$ on $TQ$, such that, for all $C^2$ variations $q_\epsilon(t)$, where The $1$-form so defined is called the Lagrange $1$-form.

Figures (11)

  • Figure 5.1: Depiction of the heuristic interpretation of an element of $J^1Y$ when $X$ is discrete.
  • Figure 5.2: The triangles which touch $(i,j)$.
  • Figure 5.3: Symplectic flow and conservation of momentum from the discrete Noether theorem when the spatial boundary is empty and the temporal boundaries agree.
  • Figure 5.4: On the left, the method based on rectangles; on the right, a possible method based on hexagons.
  • Figure 5.5: Top left: the wave forms for a two soliton kink and antikink collision using (\ref{['102']}). Top right: the energy error. Bottom left: the wave form at time $t\approx11855$. Bottom right: the portion of the bottom left graph for spatial grid points $1\ldots 16$.
  • ...and 6 more figures

Theorems & Definitions (19)

  • Theorem 2.1
  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • Definition 4.4
  • Definition 4.5
  • Definition 4.6
  • Definition 4.7
  • Definition 4.8
  • Theorem 4.1: Multisymplectic form formula
  • ...and 9 more