多辛几何、协变哈密顿算子和水波
Multisymplectic geometry, covariant Hamiltonians, and water waves
摘要：
本文讨论非线性偏微分方程的多辛拉格朗日和哈密顿形式及其应用。在这个理论中，偏微分方程的解是在维度为$n$ $+$ 1的基流形$X$上的纤维丛$Y$的截面，通常被视为时空。给定$Y$上的连接，协变汉密尔顿密度${\mathcal H}$然后在主约束流形$P_{\mathcal L}$（Legendre变换的多辛版本的图像）上本质地定义。人们把$P_{\mathcal L}$看作是$J^1(Y)^\star$的一个子丛，它是$J^1(Y)$的仿射对偶，是$Y$的第一个喷丛。然后定义了一个规范的多辛（$n$ $+$ 2）形式$Ω_{\mathcal H}$，由此得到了一个多辛Hamilton微分方程组，它等价于原偏微分方程和相应的拉格朗日量的Euler-Lagrange方程.我们证明了Bridges [1997]定义的$n$ $+$ 1 2-形式$ω^{(μ)}$是单个多辛（$n$ $+$ 2）-形式的特殊坐标表示，并且在对称性存在的情况下，可以组装成$Ω_{\mathcal H}$ .一个广义的Hamilton Noether理论，然后构建恢复的$n$ $+$ 1不同的动量映射中定义的桥梁[1997]和矢量的发散消失，当应用到水波，恢复惠特姆的波作用守恒。我们还表明，该理论的实用程序在研究周期性图案的形成和波的不稳定性。
英文摘要：
This paper concerns the development and application of the multisymplectic Lagrangian and Hamiltonian formalism for nonlinear partial differential equations. In this theory, solutions of a PDE are sections of a fiber bundle $Y$ over a base manifold $X$ of dimension $n$$+$1, typically taken to be spacetime. Given a connection on $Y$, a covariant Hamiltonian density ${\mathcal H}$ is then intrinsically defined on the primary constraint manifold $P_{\mathcal L}$, the image of the multisymplectic version of the Legendre transformation. One views $P_{\mathcal L}$ as a subbundle of $J^1(Y)^\star$, the affine dual of $J^1(Y)$, the first jet bundle of $Y$. A canonical multisymplectic ($n$$+$2)-form $Ω_{\mathcal H}$ is then defined, from which we obtain a multisymplectic Hamiltonian system of differential equations that is equivalent to both the original PDE as well as the Euler-Lagrange equations of the corresponding Lagrangian. We show that the $n$$+$1 2-forms $ω^{(μ)}$ defined by Bridges [1997] are a particular coordinate representation for a single multisymplectic ($n$$+$2)-form, and in the presence of symmetries, can be assembled into $Ω_{\mathcal H}$. A generalized Hamiltonian Noether theory is then constructed which recovers the vanishing of the divergence of the vector of $n$$+$1 distinct momentum mappings defined in Bridges [1997] and, when applied to water waves, recovers Whitham's conservation of wave action. We also show the utility of this theory in the study of periodic pattern formation and wave instability.
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