Based on developments in the theory of variational and Hamiltonian control systems by Crouch and van der Schaft (1987), this paper answers two questions: given an input-output differential equation description of a nonlinear system, what is the adjoint variational system in input-output differential form and what are the conditions for the system to be Hamiltonian, i.e., such that the variational and the adjoint variational systems coincide? This resulting set of conditions is then used to generalize classical conditions such as the well-known Helmholtz conditions for the inverse problem in classical mechanics.
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机译：基于Crouch和van der Schaft（1987）对变分和哈密顿控制系统理论的发展，本文回答了两个问题：给定非线性系统的输入输出微分方程描述，输入-输入中的伴随变分系统是什么？输出微分形式以及该系统成为哈密顿量的条件是什么，即使得变分系统和伴随变分系统重合？然后，将所得的一组条件用于概括经典条件，例如针对经典力学中反问题的众所周知的亥姆霍兹条件。
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