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Journal of Geometric Mechanics

2010 , Volume 2 , Issue 1

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Dirac constraints in field theory and exterior differential systems
Santiago Capriotti
2010, 2(1): 1-50 doi: 10.3934/jgm.2010.2.1 +[Abstract](25) +[PDF](477.9KB)
The usual treatment of a (first order) classical field theory such as electromagnetism has a little drawback: It has a primary constraint submanifold that arise from the fact that the dynamics is governed by the antisymmetric part of the jet variables. So it is natural to ask if there exists a formulation of this kind of field theories which avoids this problem, retaining the versatility of the known approach. The following paper deals with a family of variational problems, namely, the so called non standard variational problems, which intends to capture the data necessary to set up such a formulation for field theories. A multisymplectic structure for the family of non standard variational problems will be found, and it will be related with the (pre)symplectic structure arising on the space of sections of the bundle of fields. In this setting the Dirac theory of constraints will be studied, obtaining among other things a novel characterization of the constraint manifold which arises in this theory, as generators of an exterior differential system associated to the equations of motion and the chosen slicing. Several examples of application of this formalism will be discussed.
Geodesic boundary value problems with symmetry
Colin J. Cotter and  Darryl D. Holm
2010, 2(1): 51-68 doi: 10.3934/jgm.2010.2.51 +[Abstract](50) +[PDF](572.3KB)
This paper shows how commuting left and right actions of Lie groups on a manifold may be used to complement one another in a variational reformulation of optimal control problems as geodesic boundary value problems with symmetry. In such problems, the endpoint boundary condition is only specified up to the right action of a symmetry group. In this paper we show how to reformulate the problem by introducing extra degrees of freedom so that the endpoint condition specifies a single point on the manifold. We prove an equivalence theorem to this effect and illustrate it with several examples. In finite-dimensions, we discuss geodesic flows on the Lie groups $SO(3)$ and $SE(3)$ under the left and right actions of their respective Lie algebras. In an infinite-dimensional example, we discuss optimal large-deformation matching of one closed oriented curve to another embedded in the same plane. In the curve-matching example, the manifold Emb$(S^1, \mathbb{R}^2)$ comprises the space $S^1$ embedded in the plane $\mathbb{R}^2$. The diffeomorphic left action Diff$(\mathbb{R}^2)$ deforms the curve by a smooth invertible time-dependent transformation of the coordinate system in which it is embedded, while leaving the parameterisation of the curve invariant. The diffeomorphic right action Diff$(S^1)$ corresponds to a smooth invertible reparameterisation of the $S^1$ domain coordinates of the curve. As we show, this right action unlocks an important degree of freedom for geodesically matching the curve shapes using an equivalent fixed boundary value problem, without being constrained to match corresponding points along the template and target curves at the endpoint in time.
Lagrangian reduction of nonholonomic discrete mechanical systems
Javier Fernández , Cora Tori and  Marcela Zuccalli
2010, 2(1): 69-111 doi: 10.3934/jgm.2010.2.69 +[Abstract](59) +[PDF](524.8KB)
In this paper we propose a process of lagrangian reduction and reconstruction for nonholonomic discrete mechanical systems where the action of a continuous symmetry group makes the configuration space a principal bundle. The result of the reduction process is a discrete dynamical system that we call the discrete reduced system. We illustrate the techniques by analyzing two types of discrete symmetric systems where it is possible to go further and obtain (forced) discrete mechanical systems that determine the dynamics of the discrete reduced system.
An elementary derivation of the Montgomery phase formula for the Euler top
José Natário
2010, 2(1): 113-118 doi: 10.3934/jgm.2010.2.113 +[Abstract](43) +[PDF](148.1KB)
We give an elementary derivation of the Montgomery phase formula for the motion of an Euler top, using only basic facts about the Euler equation and parallel transport on the $2$-sphere (whose holonomy is seen to be responsible for the geometric phase). We also give an approximate geometric interpretation of the geometric phase for motions starting close to an unstable equilibrium point.

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