Tensor products of Dirac structures and interconnection in Lagrangian mechanics

Henry O. Jacobs; Hiroaki Yoshimura

doi:10.3934/jgm.2014.6.67

Article Contents

2014, Volume 6, Issue 1: 67-98. Doi: 10.3934/jgm.2014.6.67

This issue Previous Article Fluid-structure interaction in the Lagrange-Poincaré formalism: The Navier-Stokes and inviscid regimes Next Article Bundle-theoretic methods for higher-order variational calculus

Tensor products of Dirac structures and interconnection in Lagrangian mechanics

Henry O. Jacobs^1, and
Hiroaki Yoshimura^2,

1.
Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ
2.
Applied Mechanics and Aerospace Engineering, Waseda University, Okubo, Shinjuku, Tokyo 169-8555

Received: September 30, 2013

Revised: January 31, 2014

Published: February 2014

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Abstract

Many mechanical systems are large and complex, despite being composed of simple subsystems. In order to understand such large systems it is natural to tear the system into these subsystems. Conversely we must understand how to invert this tearing procedure. In other words, we must understand interconnection of subsystems. Such an understanding has been already shown in the context of Hamiltonian systems on vector spaces via the port-Hamiltonian systems program, in which an interconnection may be achieved through the identification of shared variables, whereupon the notion of composition of Dirac structures allows one to interconnect two systems. In this paper, we seek to extend the program of the port-Hamiltonian systems on vector spaces to the case of Lagrangian systems on manifolds and also extend the notion of composition of Dirac structures appropriately. In particular, we will interconnect Lagrange-Dirac systems by modifying the respective Dirac structures of the involved subsystems. We define the interconnection of Dirac structures via an interaction Dirac structure and a tensor product of Dirac structures. We will show how the dynamics of the interconnected system is formulated as a function of the subsystems, and we will elucidate the associated variational principles. We will then illustrate how this theory extends the theory of port-Hamiltonian systems and the notion of composition of Dirac structures to manifolds with couplings which do not require the identification of shared variables. Lastly, we will show some examples: a mass-spring mechanical systems, an electric circuit, and a nonholonomic mechanical system.

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Mathematics Subject Classification: Primary: 70H03, 70H45; Secondary: 70F25.

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