高阶力学系统纤维导数的正则性
Regularity properties of fiber derivatives associated with higher-order mechanical systems
摘要：
本工作的目的是研究与描述高阶自治动力系统动力学的拉格朗日函数和哈密顿函数相关的纤维导数。更准确地说，给定$T^*T^{(k-1)}Q$中的函数，我们找到了这样的函数描述k阶自治动力系统的动力学的充分必要条件，从而是k阶Hamilton函数。然后，我们给出了一个合适的定义（超）正则性的高阶Hamilton函数的纤维导数。此外，我们还研究了拉格朗日子流形中的动力学的高阶Euler-Lagrange方程的解的另一种表征。
英文摘要：
The aim of this work is to study fiber derivatives associated to Lagrangian and Hamiltonian functions describing the dynamics of a higher-order autonomous dynamical system. More precisely, given a function in $T^*T^{(k-1)}Q$, we find necessary and sufficient conditions for such a function to describe the dynamics of a kth-order autonomous dynamical system, thus being a kth-order Hamiltonian function. Then, we give a suitable definition of (hyper)regularity for these higher-order Hamiltonian functions in terms of their fiber derivative. In addition, we also study an alternative characterization of the dynamics in Lagrangian submanifolds in terms of the solutions of the higher-order Euler-Lagrange equations.
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