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Journal of Geometric Mechanics

2012 , Volume 4 , Issue 2

Special issue dedicated to Tudor S. Ratiu on the occasion of his 60th birthday
Guest editor: Juan-Pablo Ortega

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Foreword
Juan-Pablo Ortega
2012, 4(2): i-i doi: 10.3934/jgm.2012.4.2i +[Abstract](270) +[PDF](89.0KB)
Abstract:
This number is the second one that the Journal of Geometric Mechanics dedicates to the works presented on the occasion of the 60th birthday celebration of Tudor S. Ratiu, held at the Centre International de Rencontres Math\'ematiques. As it was the case for the first number (volume 3(4)), we hope that the articles contained in this special number will give the reader an appreciation for the importance of the contributions of Tudor S. Ratiu to the fields that are at the core of this journal.
Symmetries and reduction of multiplicative 2-forms
Henrique Bursztyn and Alejandro Cabrera
2012, 4(2): 111-127 doi: 10.3934/jgm.2012.4.111 +[Abstract](453) +[PDF](468.4KB)
Abstract:
This paper is concerned with symmetries of closed multiplicative 2-forms on Lie groupoids and their infinitesimal counterparts. We use them to study Lie group actions on Dirac manifolds by Dirac diffeomorphisms and their lifts to presymplectic groupoids, building on recent work of Fernandes-Ortega-Ratiu [11] on Poisson actions.
Linear weakly Noetherian constants of motion are horizontal gauge momenta
Francesco Fassò, Andrea Giacobbe and Nicola Sansonetto
2012, 4(2): 129-136 doi: 10.3934/jgm.2012.4.129 +[Abstract](417) +[PDF](322.6KB)
Abstract:
noindent The notion of gauge momenta is a generalization of the momentum map which is relevant for nonholonomic systems with symmetry. Weakly Noetherian functions are functions which are constants of motion of all 'natural' nonholonomic systems with a given kinetic energy and any $G$-invariant potential energy. We show that, when the action of the symmetry group on the configuration manifold is free and proper, a function which is linear in the velocities is weakly-Noetherian if anf only if it is a gauge momenta which has a horizontal generator.
Variational Integrators for Hamiltonizable Nonholonomic Systems
Oscar E. Fernandez, Anthony M. Bloch and P. J. Olver
2012, 4(2): 137-163 doi: 10.3934/jgm.2012.4.137 +[Abstract](545) +[PDF](1066.9KB)
Abstract:
We report on new applications of the Poincaré and Sundman time-transformations to the simulation of nonholonomic systems. These transformations are here applied to nonholonomic mechanical systems known to be Hamiltonizable (briefly, nonholonomic systems whose constrained mechanics are Hamiltonian after a suitable time reparameterization). We show how such an application permits the usage of variational integrators for these non-variational mechanical systems. Examples are given and numerical results are compared to the standard nonholonomic integrator results.
Dirac pairs
Yvette Kosmann-Schwarzbach
2012, 4(2): 165-180 doi: 10.3934/jgm.2012.4.165 +[Abstract](422) +[PDF](378.7KB)
Abstract:
We extend the definition of the Nijenhuis torsion of an endomorphism of a Lie algebroid to that of a relation, and we prove that the torsion of the relation defined by a bi-Hamiltonian structure vanishes. Following Gelfand and Dorfman, we then define Dirac pairs, and we analyze the relationship of this general notion with the various kinds of compatible structures on manifolds, more generally, on Lie algebroids.
A property of conformally Hamiltonian vector fields; Application to the Kepler problem
Charles-Michel Marle
2012, 4(2): 181-206 doi: 10.3934/jgm.2012.4.181 +[Abstract](425) +[PDF](524.4KB)
Abstract:
Let $X$ be a Hamiltonian vector field defined on a symplectic manifold $(M,\omega)$, $g$ a nowhere vanishing smooth function defined on an open dense subset $M^0$ of $M$. We will say that the vector field $Y=gX$ is \emph{conformally Hamiltonian}. We prove that when $X$ is complete, when $Y$ is Hamiltonian with respect to another symplectic form $\omega_2$ defined on $M^0$, and when another technical condition is satisfied, then there is a symplectic diffeomorphism from $(M^0,\omega_2)$ onto an open subset of $(M,\omega)$, which maps each orbit to itself and is equivariant with respect to the flows of the vector fields $Y$ on $M^0$ and $X$ on $M$. This result explains why the diffeomorphism of the phase space of the Kepler problem restricted to the negative (resp. positive) values of the energy function, onto an open subset of the cotangent bundle to a three-dimensional sphere (resp. two-sheeted hyperboloid), discovered by Györgyi (1968) [10], re-discovered by Ligon and Schaaf (1976) [16], is a symplectic diffeomorphism. Cushman and Duistermaat (1997) [5] have shown that the Györgyi-Ligon-Schaaf diffeomorphism is characterized by three very natural properties; here that diffeomorphism is obtained by composition of the diffeomorphism given by our result about conformally Hamiltonian vector fields with a (non-symplectic) diffeomorphism built by a variant of Moser's method [20]. Infinitesimal symmetries of the Kepler problem are discussed, and it is shown that their space is a Lie algebroid with zero anchor map rather than a Lie algebra.

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