Variational calculus with constraints on general algebroids

TL;DR

This paper develops a variational calculus framework on general algebroids, deriving Euler-Lagrange equations for constrained systems, unifying many first-order Lagrangian models including classical mechanics.

Contribution

It introduces a generalized variational calculus on algebroids with constraints, extending classical mechanics to a broader geometric setting.

Findings

Derived constrained Euler-Lagrange equations for holonomic, vakonomic, and nonholonomic constraints.

Unified framework covering most first-order Lagrangian systems.

Reduces to classical variational calculus on tangent bundles.

Abstract

Variational calculus on a vector bundle E equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler-Lagrange equations. Constrained systems are introduced in the variational and in the geometrical setting. The constrained Euler-Lagrange equations are derived for analogs of holonomic, vakonomic and nonholonomic constraints. This general model covers majority of first-order Lagrangian systems which are present in the literature and reduces to the standard variational calculus and the Euler-Lagrange equations in Classical Mechanics for E=TM.