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Journal of Geometric Mechanics

2010 , Volume 2 , Issue 3

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Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion
Sebastián Ferrer and Martin Lara
2010, 2(3): 223-241 doi: 10.3934/jgm.2010.2.223 +[Abstract](591) +[PDF](437.7KB)
Abstract:
The Hamilton-Jacobi equation in the sense of Poincaré, i.e. formulated in the extended phase space and including regularization, is revisited building canonical transformations with the purpose of Hamiltonian reduction and perturbation theory. We illustrate our approach dealing with attitude and orbital dynamics. Based on the use of Andoyer and Whittaker symplectic charts, for which all but one coordinates are cyclic in the Hamilton-Jacobi equation of the free rigid body motion and Kepler problem, respectively, we provide whole families of canonical transformations, among which one recognizes the familiar ones used in attitude and orbital dynamics. In addition, new canonical transformations are demonstrated.
Hamiltonian mechanical systems on Lie algebroids, unimodularity and preservation of volumes
Juan Carlos Marrero
2010, 2(3): 243-263 doi: 10.3934/jgm.2010.2.243 +[Abstract](439) +[PDF](473.6KB)
Abstract:
In this paper we discuss the relation between the unimodularity of a Lie algebroid $\tau_{A}: A \to Q$ and the existence of invariant volume forms for the dynamics of hamiltonian mechanical systems on the dual bundle $A$*. The results obtained in this direction are applied to several hamiltonian systems on different examples of Lie algebroids.
When is a control system mechanical?
Sandra Ricardo and Witold Respondek
2010, 2(3): 265-302 doi: 10.3934/jgm.2010.2.265 +[Abstract](715) +[PDF](647.8KB)
Abstract:
In this work we present a geometric setting for studying mechanical control systems. We distinguish a special class: the class of geodesically accessible mechanical systems, for which the uniqueness of the mechanical structure is guaranteed (up to an extended point transformation). We characterise nonlinear control systems that are state equivalent to a system from this class and we describe the canonical mechanical structure attached to them. Several illustrative examples are given.
The non-Riemannian dislocated crystal: A tribute to Ekkehart Kröner (1919-2000)
Nicolas Van Goethem
2010, 2(3): 303-320 doi: 10.3934/jgm.2010.2.303 +[Abstract](561) +[PDF](299.3KB)
Abstract:
This expository paper is a tribute to Ekkehart Kröner's results on the intrinsic non-Riemannian geometrical nature of a single crystal filled with point and/or line defects. A new perspective on this old theory is proposed, intended to contribute to the debate around the still open Kröner's question: "what are the dynamical variables of our theory?"

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