Normal forms of $ω$-Hamiltonian vector fields with symmetries

Periodic orbits in Hamiltonian systems with involutory symmetries

We study the existence of families of periodic solutions in a neighbourhood of a symmetric equilibrium point in two classes of Hamiltonian systems with involutory symmetries. In both classes, involutions reverse the sign of the Hamiltonian function. In the first class we study a Hamiltonian system with a reversing involution R acting symplectically. We first recover a result of Buzzi and Lamb showing that the equilibrium point is contained in a three dimensional conical subspace which consists of a two parameter family of periodic solutions with symmetry R and there may or may not exist two fa…

接触哈密顿系统对称性的特征化

Characterization of symmetries of contact Hamiltonian systems

本文研究接触哈密顿力学中Cartan对称性、动力相似性和动力对称性之间的关系，通过引入向量场的替代分解，用张量密度描述这些对称性，并在特定条件下恢复守恒量，提出新的独立性判据。

Non-Noether symmetries in Hamiltonian Dynamical Systems

We discuss geometric properties of non-Noether symmetries and their possible applications in integrable Hamiltonian systems. Correspondence between non-Noether symmetries and conservation laws is revisited. It is shown that in regular Hamiltonian systems such a symmetries canonically lead to a Lax pairs on the algebra of linear operators on cotangent bundle over the phase space. Relationship between the non-Noether symmetries and other wide spread geometric methods of generating conservation laws such as bi-Hamiltonian formalism, bidifferential calculi and Frolicher-Nijenhuis geometry is consi…

Infinitesimal symmetries in Contact Hamiltonian systems

In this paper, we extend the well-known Noether theorem for Lagrangian systems to contact Lagrangian systems. We introduce a classification of infinitesimal symmetries and obtain the corresponding dissipated quantities. We notice that in contact dynamics, the existence of infinitesimal symmetries does not produce conserved quantities, but functions that dissipate at the same rate than the energy; so, the corresponding quotients are true conserved quantities.

Kirillov结构与通过伸缩和标准对称性的Hamilton系统约化

Kirillov structures and reduction of Hamiltonian systems by scaling and standard symmetries

本文研究了通过交换的伸缩和标准对称性对辛Hamilton系统进行约化，证明该过程生成Kirillov Hamilton系统，且约化顺序不影响等价性，并探讨了Lie群位形空间下的约化结构与系统重构。

Invariance and first integrals of canonical Hamiltonian equations

In this paper we consider the relation between symmetries and first integrals of canonical Hamiltonian equations. Based on a newly established identity (which is an analog of well known Noether's identity for Lagrangian approach), this approach provides a simple and clear way to construct first integrals with the help of symmetries of a Hamiltonian. The approach is illustrated by a number of examples, including equations of the three-dimensional Kepler motion.