Dirac Structures in Vakonomic Mechanics

TL;DR

This paper develops a geometric framework using Dirac structures to describe vakonomic mechanics, clarifying its relation to nonholonomic mechanics and formulating implicit Euler-Lagrange equations via a variational principle.

Contribution

It introduces a Dirac structure-based approach to vakonomic mechanics, extending the Hamilton-Pontryagin principle and providing new formulations and examples.

Findings

Established a Dirac structure on an extended Pontryagin bundle for vakonomic systems.

Formulated implicit vakonomic Euler-Lagrange equations using a variational principle.

Illustrated the theory with examples like the vakonomic skate and rolling coin.

Abstract

In this paper, we explore dynamics of the nonholonomic system called vakonomic mechanics in the context of Lagrange-Dirac dynamical systems using a Dirac structure and its associated Hamilton-Pontryagin variational principle. We first show the link between vakonomic mechanics and nonholonomic mechanics from the viewpoints of Dirac structures as well as Lagrangian submanifolds. Namely, we clarify that Lagrangian submanifold theory cannot represent nonholonomic mechanics properly, but vakonomic mechanics instead. Second, in order to represent vakonomic mechanics, we employ the space $TQ\times V^{\ast}$, where a vakonomic Lagrangian is defined from a given Lagrangian (possibly degenerate) subject to nonholonomic constraints. Then, we show how implicit vakonomic Euler-Lagrange equations can be formulated by the Hamilton-Pontryagin variational principle for the vakonomic Lagrangian on the extended Pontryagin bundle $(TQ\oplus T^{\ast}Q)\times V^{\ast}$. Associated with this variational principle, we establish a Dirac structure on $(TQ\oplus T^{\ast}Q)\times V^{\ast}$ to define an intrinsic vakonomic Lagrange-Dirac system. Furthermore, we establish another construction for the vakonomic Lagrange-Dirac system using a Dirac structure on $T^{\ast}Q\times V^{\ast}$, where we introduce a vakonomic Dirac differential. Lastly, we illustrate our theory of vakonomic Lagrange-Dirac systems by some examples such as the vakonomic skate and the vertical rolling coin.