前辛结构与内禀拉格朗日量
Presymplectic structures and intrinsic Lagrangians
摘要：
众所周知，拉格朗日函数在方程流形（静止表面，被理解为相应喷射空间的子流形）上推导出兼容的前辛形式。给定一个方程流形及其相容的前辛形式，我们定义了一阶拉格朗日系统，该系统是根据方程流形的内禀几何构造的。它具有前辛AKSZ西格玛模型的结构，其中方程流形，配备了前辛形式和水平微分，作为目标空间。对于广泛的一类系统（但不是所有），我们表明，如果前辛结构起源于一个给定的拉格朗日，建议的一阶拉格朗日是相当于初始的，因此拉格朗日本身可以完全编码的内在几何的固定表面。如果相容的前辛结构是通用的，建议拉格朗日量只是一个部分的意义上，它的固定表面包含初始方程流形，但不一定与它重合。
英文摘要：
It is well-known that a Lagrangian induces a compatible presymplectic form on the equation manifold (stationary surface, understood as a submanifold of the respective jet-space). Given an equation manifold and a compatible presymplectic form therein, we define the first-order Lagrangian system which is formulated in terms of the intrinsic geometry of the equation manifold. It has a structure of a presymplectic AKSZ sigma model for which the equation manifold, equipped with the presymplectic form and the horizontal differential, serves as the target space. For a wide class of systems (but not all) we show that if the presymplectic structure originates from a given Lagrangian, the proposed first-order Lagrangian is equivalent to the initial one and hence the Lagrangian per se can be entirely encoded in terms of the intrinsic geometry of its stationary surface. If the compatible presymplectic structure is generic, the proposed Lagrangian is only a partial one in the sense that its stationary surface contains the initial equation manifold but does not necessarily coincide with it.
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