Hamiltonian theory of constrained impulsive motion

Momentum section on Courant algebroid and constrained Hamiltonian mechanics

We propose a generalization of the momentum map on a symplectic manifold with a Lie algebra action to a Courant algebroid structure. The theory of a momentum section on a Lie algebroid is generalized to the theory compatible with a Courant algebroid. As an example, we identify the momentum section in a constrained Hamiltonian mechanics with Courant algebroid symmetry. Moreover, we construct cohomological formulations by considering the BFV and BV formalism of this Hamiltonian system. The Weil algebra for this structure is constructed.

Hamiltonian Monte Carlo On Lie Groups and Constrained Mechanics on Homogeneous Manifolds

In this paper we show that the Hamiltonian Monte Carlo method for compact Lie groups constructed in \cite{kennedy88b} using a symplectic structure can be recovered from canonical geometric mechanics with a bi-invariant metric. Hence we obtain the correspondence between the various formulations of Hamiltonian mechanics on Lie groups, and their induced HMC algorithms. Working on $\G\times \g$ we recover the Euler-Arnold formulation of geodesic motion, and construct explicit HMC schemes that extend \cite{kennedy88b,Kennedy:2012} to non-compact Lie groups by choosing metrics with appropriate invar…

哈密顿动力学与约束变分法：连续与离散情形

Hamiltonian dynamics and constrained variational calculus: continuous and discrete settings

本文利用方便辛流形的拉格朗日子流形和Tulczyjew三元组，研究哈密顿动力学与约束变分法之间的关系，并推广到离散动力学和非完整力学，应用于哈密顿系统的几何积分。

Lagrangian and Hamiltonian Formalism for Constrained Variational Problems

We consider solutions of Lagrangian variational problems with linear constraints on the derivative. These solutions are given by curves $γ$ in a differentiable manifold $M$ that are everywhere tangent to a smooth distribution $\mathcal D$ on $M$; such curves are called horizontal. We study the manifold structure of the set $Ω_{P,Q}(M,\mathcal D)$ of horizontal curves that join two submanifolds $P$ and $Q$ of $M$. We consider an action functional $\mathcal L$ defined on $Ω_{P,Q}(M,\mathcal D)$ associated to a time-dependent Lagrangian defined on $\mathcal D$. If the Lagrangian satisfies a suita…

Momentum sections in Hamiltonian mechanics and sigma models

We show a constrained Hamiltonian system and a gauged sigma model have a structure of a momentum section and a Hamiltonian Lie algebroid theory recently introduced by Blohmann and Weinstein. We propose a generalization of a momentum section on a pre-multisymplectic manifold by considering gauged sigma models on a higher dimensional manifold.

On the relation of Lie algebroids to constrained systems and their BV/BFV formulation

We observe that a system of irreducible, fiber-linear, first class constraints on T*M is equivalent to the definition of a foliation Lie algebroid over M. The BFV formulation of the constrained system is given by the Hamiltonian lift of the Vaintrob description (E[1],Q) of the Lie algebroid to its cotangent bundle T*E[1]. Affine deformations of the constraints are parametrized by the first Lie algebroid cohomology H^1_Q and lead to irreducible constraints also for much more general Lie algebroids such as Dirac structures; the modified BFV function follows by the addition of a representative of…