Contact Reduction

辛、余辛、切触与余切触哈密顿系统中的余迷向约化综述

A review on coisotropic reduction in Symplectic, Cosymplectic, Contact and Co-contact Hamiltonian systems

该研究针对辛、余辛等四类哈密顿系统，基于相空间几何探讨余迷向约化，阐明其因动力学可对应拉格朗日/勒让德子流形且能被余迷向子流形约化而具有研究意义。

约化：预接触与预辛结构

Reductions: precontact versus presymplectic

本文将接触约化纳入 Marsden-Weinstein-Meyer 及常秩辛约化框架，通过辛覆盖把接触同胚群作用提升为哈密顿作用，并推广到预接触与预辛情形，揭示预接触流形与齐次预辛流形的一一对应。

Momentum mapping and reduction in contact Hamiltonian systems

Due to the emergence of symplectic geometry, the geometric treatment of mechanics underwent a great development during the last century. In this scenario the pressence of symmetries in Hamiltonian systems leads naturally to the existence of conserved quantities. This integrals of motions are described by the well-known momentum mapping. Furthermore, the equations of motion can be simplified by a process known as reduction, if the system is invariant under the action of a certain group. This process can also be considered in the framework of contact geometry, a much more recent field of study. …

Scaling Symmetries, Contact Reduction and Poincaré's dream

A symplectic Hamiltonian system admitting a scaling symmetry can be reduced to an equivalent contact Hamiltonian system in which some physically-irrelevant degree of freedom has been removed. As a consequence, one obtains an equivalent description for the same physical phenomenon, but with fewer inputs needed, thus realizing "Poincaré's dream" of a scale-invariant description of the universe. This work is devoted to a thorough analysis of the mathematical framework behind such reductions. We show that generically such reduction is possible and the reduced (fundamental) system is a contact Hami…

接触几何与Lagrange-Poincaré-Herglotz方程中的对称性约化与重构

Symmetry reduction and reconstruction in contact geometry and Lagrange-Poincaré-Herglotz equations

该研究针对对称群下不变的接触拉格朗日系统，基于Herglotz向量场、其投影场及准速度推导了Lagrange-Poincaré-Herglotz约化方程的显式坐标形式，并给出了相关示例，丰富了接触几何下的对称约化理论。

接触李系统在黎曼和洛伦兹空间上的应用：从缩放对称性到曲率依赖的降维

Contact Lie systems on Riemannian and Lorentzian spaces: from scaling symmetries to curvature-dependent reductions

本文提出了一种将缩放对称性应用于李-哈密顿系统的方法，用于构造接触李系统，并展示了其在不同空间中的应用及降维效果。

奇异余切球丛约化

Singular cosphere bundle reduction

本文研究动量映射零值下余切球丛的奇异切触约化，提出一种更适配丛结构的更精细奇异商分层，其层由余切球丛及对应切触分量的余迷向、勒让德子流形组成。

Contact Hamiltonian Systems

In this paper we study Hamiltonian systems on contact manifolds, which is an appropriate scenario to discuss dissipative systems. We prove a coisotropic reduction theorem similar to the one in symplectic mechanics.

A dual pair for the contact group

Generalizing the canonical symplectization of contact manifolds, we construct an infinite dimensional non-linear Stiefel manifold of weighted embeddings into a contact manifold. This space carries a symplectic structure such that the contact group and the group of reparametrizations act in a Hamiltonian fashion with equivariant moment maps, respectively, giving rise to a dual pair, called the EPContact dual pair. Via symplectic reduction, this dual pair provides a conceptual identification of non-linear Grassmannians of weighted submanifolds with certain coadjoint orbits of the contact group. …

Quasi-morphisms on contactomorphism groups and contact rigidity

We build homogeneous quasi-morphisms on the universal cover of the contactomorphism group for certain prequantizations of monotone symplectic toric manifolds. This is done using Givental's nonlinear Maslov index and a contact reduction technique for quasi-morphisms. We show how these quasi-morphisms lead to a hierarchy of rigid subsets of contact manifolds. We also show that the nonlinear Maslov index has a vanishing property, which plays a key role in our proofs. Finally we present applications to orderability of contact manifolds and Sandon-type metrics on contactomorphism groups.

作为辛-接触约化与相对信息熵的非平衡热力学

Nonequilibrium thermodynamics as a symplecto-contact reduction and relative information entropy

本文从统计相空间和动力学理论相空间的辛结构出发，通过 Marsden-Weinstein 约化与相对信息熵生成函数，将非平衡热力学相空间构造为接触流形，并用辛-接触几何解释 Maxwell 等面积法则与相变。