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Journal of Geometric Mechanics

2011 , Volume 3 , Issue 1

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Superposition rules and second-order Riccati equations
José F. Cariñena and Javier de Lucas Araujo
2011, 3(1): 1-22 doi: 10.3934/jgm.2011.3.1 +[Abstract](783) +[PDF](494.3KB)
Abstract:
A superposition rule is a particular type of map that enables one to express the general solution of certain systems of first-order ordinary differential equations, the so-called Lie systems, out of generic families of particular solutions and a set of constants. The first aim of this work is to propose various generalisations of this notion to second-order differential equations. Next, several results on the existence of such generalisations are given and relations with the theories of Lie systems and quasi-Lie schemes are found. Finally, our methods are used to study second-order Riccati equations and other second-order differential equations of mathematical and physical interest.
Reduction of invariant constrained systems using anholonomic frames
Mike Crampin and Tom Mestdag
2011, 3(1): 23-40 doi: 10.3934/jgm.2011.3.23 +[Abstract](557) +[PDF](382.8KB)
Abstract:
We analyze two reduction methods for nonholonomic systems that are invariant under the action of a Lie group on the configuration space. Our approach for obtaining the reduced equations is entirely based on the observation that the dynamics can be represented by a second-order differential equations vector field and that in both cases the reduced dynamics can be described by expressing that vector field in terms of an appropriately chosen anholonomic frame.
Clebsch optimal control formulation in mechanics
François Gay-Balmaz and Tudor S. Ratiu
2011, 3(1): 41-79 doi: 10.3934/jgm.2011.3.41 +[Abstract](827) +[PDF](597.2KB)
Abstract:
This paper introduces and studies a class of optimal control problems based on the Clebsch approach to Euler-Poincaré dynamics. This approach unifies and generalizes a wide range of examples appearing in the literature: the symmetric formulation of $N$-dimensional rigid body and its generalization to other matrix groups; optimal control for ideal flow using the back-to-labels map; the double bracket equations associated to symmetric spaces. New examples are provided such as the optimal control formulation for the $N$-Camassa-Holm equation and a new geodesic interpretation of its singular solutions.
A theoretical framework for backward error analysis on manifolds
Anders C. Hansen
2011, 3(1): 81-111 doi: 10.3934/jgm.2011.3.81 +[Abstract](451) +[PDF](540.8KB)
Abstract:
Backward Error Analysis (BEA) has been a crucial tool when analyzing long-time behavior of numerical integrators, in particular, one is interested in the geometric properties of the perturbed vector field that a numerical integrator generates. In this article we present a new framework for BEA on manifolds. We extend the previously known "exponentially close" estimates from $\mathbb{R}^n$ to smooth manifolds and also provide an abstract theory for classifications of numerical integrators in terms of their geometric properties. Classification theorems of type "symplectic integrators generate symplectic perturbed vector fields" are known to be true in $\mathbb{R}^n.$ We present a general theory for proving such theorems on manifolds by looking at the preservation of smooth $k$-forms on manifolds by the pull-back of a numerical integrator. This theory is related to classification theory of subgroups of diffeomorphisms. We also look at other subsets of diffeomorphisms that occur in the classification theory of numerical integrators. Typically these subsets are anti-fixed points of group homomorphisms.
On the $k$-symplectic, $k$-cosymplectic and multisymplectic formalisms of classical field theories
Narciso Román-Roy, Ángel M. Rey, Modesto Salgado and Silvia Vilariño
2011, 3(1): 113-137 doi: 10.3934/jgm.2011.3.113 +[Abstract](729) +[PDF](593.6KB)
Abstract:
The objective of this work is twofold: First, we analyze the relation between the $k$-cosymplectic and the $k$-symplectic Hamiltonian and Lagrangian formalisms in classical field theories. In particular, we prove the equivalence between $k$-symplectic field theories and the so-called autonomous $k$-cosymplectic field theories, extending in this way the description of the symplectic formalism of autonomous systems as a particular case of the cosymplectic formalism in non-autonomous mechanics. Furthermore, we clarify some aspects of the geometric character of the solutions to the Hamilton-de Donder-Weyl and the Euler-Lagrange equations in these formalisms. Second, we study the equivalence between $k$-cosymplectic and a particular kind of multisymplectic Hamiltonian and Lagrangian field theories (those where the configuration bundle of the theory is trivial).
Book review: Marcelo Epstein, The Geometrical Language of Continuum Mechanics
Reuven Segev
2011, 3(1): 139-143 doi: 10.3934/jgm.2011.3.139 +[Abstract](794) +[PDF](210.1KB)
Abstract:
Intended mainly for continuum mechanicists, Epstein's book introduces modern geometry and some of its applications to theoretical continuum mechanics. Thus, examples for the mathematical objects introduced are chosen from the realm of mechanics. In particular, differentiable manifolds, tangent and cotangent bundles, Riemannian manifolds, Lie derivatives, Lie groups, Lie algebras, differential forms and integration theory are presented in the main part of the book. Once the reader's familiarity with continuum mechanics is used for the introduction of basic geometry, geometry is used in order to generalize notions of continuum mechanics. Integration of differential forms is used to formulate flux theory on manifolds devoid of a Riemannian structure. More specialized topics, namely, Whitney's geometric integration theory and Sikorski's differential spaces are used to relax smoothness assumptions for bodies and fields defined on them. Finally, an overview is given of the work that Epstein and co-workers carried out in recent years where the theory of inhomogeneity of constitutive relations is developed using the geometry of principal fiber bundles, G-structures and connections.

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