Vector Hamiltonians in Nambu mechanics

Hamiltonian Mechanics on Duals of Generalized Lie Algebroids

A new description, different by the classical theory of Hamiltonian Mechanics, in the general framework of generalized Lie algebroids is presented. In the particular case of Lie algebroids, new and important results are obtained. We present the \emph{dual mechanical systems} called by use, \emph{dual mechanical}$(ρ,η) $\emph{-systems, Hamilton mechanical}$(ρ,η) $\emph{-systems} or \emph{% Cartan mechanical}$(ρ,η) $\emph{-systems.} and we develop their geometries. We obtain the canonical $(ρ,η) $\emph{-}semi(spray) associated to a dual mechanical $(ρ,η) $-system. The Hamilton mechanical $(ρ,η)$…

连续介质力学的端口-哈密顿结构

The port-Hamiltonian structure of continuum mechanics

本文提出在端口-哈密顿框架下，利用Dirac结构和哈密顿约化理论，系统推导固体与流体力学在物质、空间及对流表示中的几何模型，并涵盖超弹性和粘性流动的本构关系。

Lagrangian and Hamiltonian Geometries. Applications to Analytical Mechanics

The aim of the present text is twofold: to provide a compendium of Lagrangian and Hamiltonian geometries and to introduce and investigate new analytical Mechanics: Finslerian, Lagrangian and Hamiltonian. The fundamental equations (or evolution equations) of these Mechanics are derived from the variational calculus applied to the integral of action and these can be studied by using the methods of Lagrangian or Hamiltonian geometries. More general, the notions of higher order Lagrange or Hamilton spaces have been introduced and developed by the present author. The applications led to the notions…

Measurand Spaces and Dimensioned Hamiltonian Mechanics

In this paper we introduce a generalization of Hamiltonian mechanics that replaces configuration spaces, conventionally regarded simply as smooth manifolds, with line bundles over smooth manifolds. Classical observables are then identified with the sections of these (generically non-trivial) line bundles. This generalization, mathematically articulated with theory of Jacobi manifolds, is motivated by a conceptual revision of the mathematical foundations of the notion of measurand and unit of measurement in practical science. We prove several technical results for the contact structures present…

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.

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.

用闭微分形式推广哈密顿力学

Generalizing Hamiltonian Mechanics with Closed Differential Forms

本研究针对经典哈密顿力学隐藏其他不变量、Nambu力学缺乏合适几何框架的问题，通过2次以上微分形式的可逆性建立广义哈密顿力学与多辛几何的新对应，重新解释核心定理并提出两个关键定理，推进力学几何框架的推广。

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…

Phase flows and vectorial lagrangians in $J^3(π)$

On the basis of Liouville theorem the generalization of the Nambu mechanics is considered. For three-dimensional phase space the concept of vector hamiltonian and vector lagrangian is entered.