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

## TL;DR

This paper unifies Hamiltonian Monte Carlo methods on Lie groups with geometric mechanics, extending algorithms to non-compact groups and formulating mechanics on homogeneous spaces as constrained systems.

## Contribution

It establishes a geometric framework linking HMC on Lie groups with classical mechanics, enabling extensions to non-compact groups and constrained systems on homogeneous spaces.

## Key findings

- Recovered HMC on compact Lie groups from geometric mechanics.
- Extended HMC algorithms to non-compact Lie groups.
- Formulated mechanics on homogeneous spaces as constrained systems.

## Abstract

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 invariances. Finally we explain how mechanics on homogeneous spaces can be formulated as a constrained system over their associated Lie groups, and how in some important cases the constraints can be naturally handled by the symmetries of the Hamiltonian.