# Variational reduction of Hamiltonian systems with general constraints

## TL;DR

This paper extends variational reduction techniques to higher order constrained Hamiltonian systems, including nonholonomic systems, providing a unified framework that generalizes existing methods and recovers known equations in special cases.

## Contribution

It introduces a variational reduction procedure for all higher order constrained Hamiltonian systems, encompassing nonholonomic and generalized cases, and recovers Hamilton-Poincaré equations in the unconstrained limit.

## Key findings

- Developed a reduction procedure for higher order constrained Hamiltonian systems.
- Unified treatment of nonholonomic and generalized Hamiltonian systems.
- Illustrated the method with a case involving trivial principal bundles.

## Abstract

In the Hamiltonian formalism, and in the presence of a symmetry Lie group, a variational reduction procedure has already been developed for Hamiltonian systems without constraints. In this paper we present a procedure of the same kind, but for the entire class of the higher order constrained systems (HOCS), described in the Hamiltonian formalism. Last systems include the standard and generalized nonholonomic Hamiltonian systems as particular cases. When restricted to Hamiltonian systems without constraints, our procedure gives rise exactly to the so-called Hamilton-Poincar\'e equations, as expected. In order to illustrate the procedure, we study in detail the case in which both the configuration space of the system and the involved symmetry define a trivial principal bundle.