Dynamical systems with dissipative behaviour can be described in terms of contact manifolds and a modified version of Hamilton's equations. Dissipation terms can also be added to field equations, as showed in a recent paper where we introduced the notion of k-contact structure, and obtained a modified version of the De Donder–Weyl equations of covariant Hamiltonian field theory. In this paper we continue this study by presenting a k-contact Lagrangian formulation for nonconservative field theories. The Lagrangian density is defined on the product of the space of k-velocities times a k-dimensional Euclidean space with coordinates s^α, which are responsible for the dissipation. We analyze the regularity of such Lagrangians; only in the regular case we obtain a k-contact Hamiltonian system. We study several types of symmetries for k-contact Lagrangian systems, and relate them with dissipation laws, which are analogous to conservation laws of conservative systems. Several examples are discussed: we find contact Lagrangians for some kinds of second-order linear partial differential equations, with the damped membrane as a particular example, and we also study a vibrating string with a magnetic-like term.

具有耗散行为的动力系统可以用接触流形和哈密顿方程的修改版本来描述。耗散项也可以添加到场方程中，如最近的一篇论文所示，我们引入了k接触结构的概念，并获得了协变哈密顿场论的 De Donder-Weyl 方程的修改版本。在本文中，我们通过提出非保守场论的k接触拉格朗日公式来继续这项研究。拉格朗日密度定义为k速度空间乘以k维欧几里得空间与坐标s ^α的乘积，它们负责耗散。我们分析了这些拉格朗日函数的规律性；只有在常规情况下，我们才能获得k接触哈密顿系统。我们研究了k接触拉格朗日系统的几种类型的对称性，并将它们与耗散定律联系起来，耗散定律类似于保守系统的守恒定律。讨论了几个例子：我们找到了一些二阶线性偏微分方程的接触拉格朗日，以阻尼膜为例，我们还研究了具有类磁项的振动弦。