高阶导数场论中的Tulczyjew三元组
Tulczyjew Triples in Higher Derivative Field Theory
摘要：
被称为Tulczyjew三元组的几何结构已成功地用于分析力学和一阶场论中，以描述广泛的物理系统，包括具有约束和/或源的拉格朗日/哈密顿系统，或具有奇异拉格朗日的系统。从变分的第一原理出发，我们推导出任意高阶的经典场论的Tulczyjew三元组，即依赖于场的任意高阶导数。第一个三元组出现的结果，考虑高阶理论作为一阶的配置被约束为完整的射流。第二个三元组是在一个旨在摆脱非物理自由度的简化过程后得到的。我们提出的这个图像是完全协变的和完整的：它包含拉格朗日和哈密顿形式主义，特别是欧拉-拉格朗日方程。请注意，所需的射流束几何是仿射的（与切线束的线性几何相反）。因此，仿射对偶和仿射相空间的概念在我们的图片中发挥了突出的作用。特别是Tulczyjew三元组在本文中包括态射的双重仿射向量丛，而且，保持适当的前辛结构。
英文摘要：
The geometrical structure known as Tulczyjew triple has been used with success in analytical mechanics and first order field theory to describe a wide range of physical systems including Lagrangian/Hamiltonian systems with constraints and/or sources, or with singular Lagrangian. Starting from the first principles of the variational calculus we derive Tulczyjew triples for classical field theories of arbitrary high order, i.e. depending on arbitrary high derivatives of the fields. A first triple appears as the result of considering higher order theories as first order ones with configurations being constrained to be holonomic jets. A second triple is obtained after a reduction procedure aimed at getting rid of nonphysical degrees of freedom. This picture we present is fully covariant and complete: it contains both Lagrangian and Hamiltonian formalisms, in particular the Euler-Lagrange equations. Notice that, the required Geometry of jet bundles is affine (as opposed to the linear Geometry of the tangent bundle). Accordingly, the notions of affine duality and affine phase space play a distinguished role in our picture. In particular the Tulczyjew triples in this paper consist of morphisms of double affine-vector bundles which, moreover, preserve suitable presymplectic structures.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
