We present a complete theory of higher-order autonomous contact mechanics, which allows us to describe higher-order dynamical systems with dissipation. The essential tools for the theory are the extended higher-order tangent bundles, ${\mathrm{T}}^{k}Q\times \mathbb{R}$, whose geometric structures are previously introduced in order to state the Lagrangian and Hamiltonian formalisms for these kinds of systems, including their variational formulation. The variational principle, the contact forms, and the geometric dynamical equations are obtained by using those structures and generalizing the standard formulation of contact Lagrangian and Hamiltonian systems. As an alternative approach, we develop a unified description that encompasses the Lagrangian and Hamiltonian equations as well as their relationship through the Legendre map; all of them are obtained from the contact dynamical equations and the constraint algorithm that is implemented because, in this formalism, the dynamical systems are always singular. Some interesting examples are finally analyzed using these geometric formulations.

我们提出了一个完整的高阶自主接触力学理论，这使我们能够描述具有耗散的高阶动力学系统。该理论的基本工具是扩展的高阶切线束，${\mathrm{Ť}}^{ķ}问\times \mathbb{[R}$，之前介绍了它们的几何结构是为了说明这些系统的拉格朗日和汉密尔顿形式主义，包括其变式形式。通过使用这些结构并推广接触拉格朗日系统和哈密顿系统的标准公式，可以获得变分原理，接触形式和几何动力学方程。作为一种替代方法，我们开发了一个统一的描述，其中涵盖了拉格朗日方程和哈密顿方程，以及它们通过勒让德图的关系。它们都是从接触动力学方程式和约束算法中获得的，因为在这种形式主义中，动力学系统总是奇异的。最后，使用这些几何公式分析了一些有趣的示例。