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Journal of Geometric Mechanics

2014 , Volume 6 , Issue 1

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Aspects of reduction and transformation of Lagrangian systems with symmetry
E. García-Toraño Andrés, Bavo Langerock and Frans Cantrijn
2014, 6(1): 1-23 doi: 10.3934/jgm.2014.6.1 +[Abstract](716) +[PDF](579.8KB)
Abstract:
This paper contains results on geometric Routh reduction and it is a continuation of a previous paper [7] where a new class of transformations is introduced between Lagrangian systems obtained after Routh reduction. In general, these reduced Lagrangian systems have magnetic force terms and are singular in the sense that the Lagrangian does not depend on some velocity components. The main purpose of this paper is to show that the Routh reduction process itself is entirely captured by the application of such a new transformation on the initial Lagrangian system with symmetry.
Andoyer's variables and phases in the free rigid body
Sebastián Ferrer and Francisco J. Molero
2014, 6(1): 25-37 doi: 10.3934/jgm.2014.6.25 +[Abstract](652) +[PDF](981.0KB)
Abstract:
Using Andoyer's variables we present a new proof of Montgomery's formula by measuring $\Delta\mu$ when $\nu$ has made a rotation. Our treatment is built on the equations of the differential system of the free rigid solid, together with the explicit expression of the spherical area defined by the intersection of the surfaces given by the energy and momentum integrals. We also consider the phase $\Delta\nu$ of the moving frame when $\mu$ has made a rotation around the angular momentum vector, and we give the formula for its computation.
Fluid-structure interaction in the Lagrange-Poincaré formalism: The Navier-Stokes and inviscid regimes
Henry Jacobs and Joris Vankerschaver
2014, 6(1): 39-66 doi: 10.3934/jgm.2014.6.39 +[Abstract](811) +[PDF](571.1KB)
Abstract:
In this paper, we derive the equations of motion for an elastic body interacting with a perfect fluid via the framework of Lagrange-Poincaré reduction. We model the combined fluid-structure system as a geodesic curve on the total space of a principal bundle on which a diffeomorphism group acts. After reduction by the diffeomorphism group we obtain the fluid-structure interactions where the fluid evolves by the inviscid fluid equations. Along the way, we describe various geometric structures appearing in fluid-structure interactions: principal connections, Lie groupoids, Lie algebroids, etc. We finish by introducing viscosity in our framework as an external force and adding the no-slip boundary condition. The result is a description of an elastic body immersed in a Navier-Stokes fluid as an externally forced Lagrange-Poincaré equation. Expressing fluid-structure interactions with Lagrange-Poincaré theory provides an alternative to the traditional description of the Navier-Stokes equations on an evolving domain.
Tensor products of Dirac structures and interconnection in Lagrangian mechanics
Henry O. Jacobs and Hiroaki Yoshimura
2014, 6(1): 67-98 doi: 10.3934/jgm.2014.6.67 +[Abstract](667) +[PDF](1756.9KB)
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.
Bundle-theoretic methods for higher-order variational calculus
Michał Jóźwikowski and Mikołaj Rotkiewicz
2014, 6(1): 99-120 doi: 10.3934/jgm.2014.6.99 +[Abstract](670) +[PDF](566.0KB)
Abstract:
We present a geometric interpretation of the integration-by-parts formula on an arbitrary vector bundle. As an application we give a new geometric formulation of higher-order variational calculus.
A Hamilton-Jacobi theory on Poisson manifolds
Manuel de León, David Martín de Diego and Miguel Vaquero
2014, 6(1): 121-140 doi: 10.3934/jgm.2014.6.121 +[Abstract](823) +[PDF](443.8KB)
Abstract:
In this paper we develop a Hamilton-Jacobi theory in the setting of almost Poisson manifolds. The theory extends the classical Hamilton-Jacobi theory and can be also applied to very general situations including nonholonomic mechanical systems and time dependent systems with external forces.

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