Variational integrators have traditionally been constructed from the perspective of Lagrangian mechanics, but there have been recent efforts to adopt discrete variational approaches to the symplectic discretization of Hamiltonian mechanics using Hamiltonian variational integrators. In this paper, we will extend these results to the setting of Hamiltonian multisymplectic field theories. We demonstrate that one can use the notion of Type II generating functionals for Hamiltonian partial differential equations as the basis for systematically constructing Galerkin Hamiltonian variational integrators that automatically satisfy a discrete multisymplectic conservation law, and establish a discrete Noether's theorem for discretizations that are invariant under a Lie group action on the discrete dual jet bundle. In addition, we demonstrate that for spacetime tensor product discretizations, one can recover the multisymplectic integrators of Bridges and Reich, and show that a variational multisymplectic discretization of a Hamiltonian multisymplectic field theory using spacetime tensor product Runge--Kutta discretizations is well-defined if and only if the partitioned Runge--Kutta methods are symplectic in space and time.
翻译:在传统上,从拉格朗杰机械学的角度来构建变异融合器,但最近曾努力采用离散的不同方法,利用汉密尔顿变异融合器,对汉密尔顿机械学的互换分解法采取分立的不同方法。在本文件中,我们将这些结果推广到汉密尔顿多分流场理论的设置。我们证明,可以使用为汉密尔顿部分分化方程式生成功能的第二类概念,作为系统构建加勒金·汉密尔顿分解器的基础,这种系统化自动满足离散多相位保护法,并在离散双机群行动下为离散的汉密尔顿机械学分解器建立一个离解器。此外,我们证明,对于空间时间的多分解器产品分解器,人们可以恢复大桥和帝国的多相位分解器的多相位分离器,并且表明,使用空间时高压分流-Kut分解器的多位化多位分解法,只有确定和空间分流式分解法,才能运行和空间分解法。