ISSN:

1941-4889

eISSN:

1941-4897

## Journal of Geometric Mechanics

2013 , Volume 5 , Issue 4

Special issue devoted to the Proceedings of the Third Iberoamerican Meetings on Geometry, Mechanics, and Control, which was held at the University of Salamanca (Spain) from September 3 until September 7, 2012

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2013, 5(4): i-iii
doi: 10.3934/jgm.2013.5.4i

*+*[Abstract](113)*+*[PDF](110.3KB)**Abstract:**

We would like to render our tribute to Professor Pedro Luis García , who gave a great contribution to the development of Differential Geometry and Mathematical Physics in Spain.

For more information please click the “Full Text” above.

2013, 5(4): v-vi
doi: 10.3934/jgm.2013.5.4v

*+*[Abstract](56)*+*[PDF](90.6KB)**Abstract:**

This Special Issue of the Journal of Geometric Mechanics is devoted to the Proceedings of the Third Iberoamerican Meetings on Geometry, Mechanics and Control, which was held at the University of Salamanca (Spain) from September 3 until September 7, 2012.

For more information please click the “Full Text” above.

2013, 5(4): 381-397
doi: 10.3934/jgm.2013.5.381

*+*[Abstract](95)*+*[PDF](999.3KB)**Abstract:**

We briefly review the notion of second order constrained (continuous) system (SOCS) and then propose a discrete time counterpart of it, which we naturally call discrete second order constrained system (DSOCS). To illustrate and test numerically our model, we construct certain integrators that simulate the evolution of two mechanical systems: a particle moving in the plane with prescribed signed curvature, and the inertia wheel pendulum with a Lyapunov constraint. In addition, we prove a local existence and uniqueness result for trajectories of DSOCSs. As a first comparison of the underlying geometric structures, we study the symplectic behavior of both SOCSs and DSOCSs.

2013, 5(4): 399-414
doi: 10.3934/jgm.2013.5.399

*+*[Abstract](62)*+*[PDF](464.6KB)**Abstract:**

Given an $H$-principal bundle $Q\to M$ and a (left) linear action of $H$ to a real vector space $V$, let $E\to M$ be the vector bundle associated to $Q$ and to the linear action, and $Q\times_M E$ the affine principal bundle with structure group the semidirect group $G = H Ⓢ V$. If $L v$ is a Lagrangian density defined on the 1-jet bundle $J^1(Q\times_M E)$ invariant by the subgroup $H \hookrightarrow H Ⓢ V$, the variational problem induced on $(J^1(Q\times_ME)) /H = C(Q)\times_M J^1E$, where $C(Q)$ is the bundle of connections in $Q$, is considered. We show that the reduced Lagrangian density $lv$ defines a variational problem on connections $\sigma \in \Gamma (C(Q))$ and on sections $e\in \Gamma(E)$, with constraint $\textrm{Curv }\sigma =0$, and set of admissible variations those induced on $\Gamma (C(Q))$ by the infinitesimal gauge transformations of $Q$ and on $\Gamma(E)$ by arbitrary vertical variations. The Lagrange-Poincaré equations for the critical reduced sections are obtained, as well as the reconstruction process to the unreduced problem. The Poincaré equation is interpreted as the reduction of the Noether conservation law corresponding to the $H$-symmetry of the Lagrangian density $L v$. We also study the reduced system as a Lagrange problem through a suitable choice of the Lagrange multipliers. This allows us to establish a Hamilton-Cartan formalism for this class of systems. Finally, we discuss the molecular strands, a motivating example of the theory.

2013, 5(4): 415-432
doi: 10.3934/jgm.2013.5.415

*+*[Abstract](78)*+*[PDF](478.5KB)**Abstract:**

Given a regular optimal control problem with Lagrangian density $\mathcal{L} (t,x^\alpha,u^i)dt$ and constraints $\phi^\alpha\equiv \dot x^\alpha-f^\alpha(t,x^\beta,u^i)=0$, $1\le \alpha,\beta\le n$, $1\le i\le m$, we study the discretization defined for each pair $I_k=(k-1,k)$, $1\le k\le N$ by the functions: $$ \begin{aligned} L_{I_k}(x^\beta_{k-1},u^i_{k-1},x^\beta_k,u^i_k) = \mathcal{L} (t_{I_k},x^\alpha_{I_k},u^i_{I_k})h\\ \phi ^\alpha_{I_k}(x^\beta_{k-1},u^i_{k-1},x^\beta_k,u^i_k)=&\left(\frac{x^\alpha_{k}-x^\alpha_{k-1}}{h}-f^\alpha(t_{I_k},x^\beta_{I_k},u^i_{I_k}) \right)h \end{aligned} $$ where $t_k-t_{k-1}=h\in\mathbb{R}^+$ is fixed, and where: $$ \begin{aligned} t_{I_k}=&\epsilon t_{k-1}+(1-\epsilon) t_k=t_0+h(k-\epsilon)\\ x^\alpha_{I_k}=&\epsilon x^\alpha_{k-1}+(1-\epsilon)x^\alpha_k\\ u^i_{I_k}=&\epsilon u^i_{k-1}+(1-\epsilon)u^i_k \end{aligned}\quad 0\le \epsilon \le 1. $$ We prove that for $\epsilon\ne 0, 1$, the discrete Lagrange problems so defined are non singular in the sense of the discrete vakonomic mechanics admitting as infinitesimal symmetries the vector fields $D^i_k=\frac1{\epsilon h}\left(-\frac\epsilon{1-\epsilon}\right)^k\frac{\partial}{\partial u^i_k}$, $1\le i\le m$. The Noether invariants associated to these symmetries are used to construct the corresponding variational integrators. Finally, the theory is illustrated with two examples: the optimal regulator problem and the Heisenberg optimal control problem.

2013, 5(4): 433-444
doi: 10.3934/jgm.2013.5.433

*+*[Abstract](68)*+*[PDF](407.7KB)**Abstract:**

This work revisits, from a geometric perspective, the notion of discrete connection on a principal bundle, introduced by M. Leok, J. Marsden and A. Weinstein. It provides precise definitions of discrete connection, discrete connection form and discrete horizontal lift and studies some of their basic properties and relationships. An existence result for discrete connections on principal bundles equipped with appropriate Riemannian metrics is proved.

2013, 5(4): 445-472
doi: 10.3934/jgm.2013.5.445

*+*[Abstract](83)*+*[PDF](593.1KB)**Abstract:**

A geometric approach to dynamical equations of physics, based on the idea of the Tulczyjew triple, is presented. We show the evolution of these concepts, starting with the roots lying in the variational calculus for statics, through Lagrangian and Hamiltonian mechanics, and ending with Tulczyjew triples for classical field theories illustrated with a few important examples.

2013, 5(4): 473-491
doi: 10.3934/jgm.2013.5.473

*+*[Abstract](64)*+*[PDF](463.0KB)**Abstract:**

This article presents a Poincaré lemma for the Kostant complex, used to compute geometric quantisation, when the polarisation is given by a Lagrangian foliation defined by an integrable system with nondegenerate singularities.

2013, 5(4): 493-510
doi: 10.3934/jgm.2013.5.493

*+*[Abstract](120)*+*[PDF](521.8KB)**Abstract:**

After reviewing the Lagrangian-Hamiltonian unified formalism (i.e, the Skinner-Rusk formalism) for higher-order (non-autonomous) dynamical systems, we state a unified geometrical version of the Variational Principles which allows us to derive the Lagrangian and Hamiltonian equations for these kinds of systems. Then, the standard Lagrangian and Hamiltonian formulations of these principles and the corresponding dynamical equations are recovered from this unified framework..

2013, 5(4): 511-530
doi: 10.3934/jgm.2013.5.511

*+*[Abstract](85)*+*[PDF](592.9KB)**Abstract:**

Originally a model for wave propagation on the line, the Toda lattice is a wonderful case study in mechanics and symplectic geometry. In Flaschka's variables, it becomes an evolution given by a Lax pair on the vector space of real, symmetric, tridiagonal matrices. Its very special asymptotic behavior was studied by Moser by introducing norming constants, which play the role of discrete inverse variables in analogy to the solution by inverse scattering of KdV. It is a completely integrable system on the coadjoint orbit of the upper triangular group. Recently, bidiagonal coordinates, which parameterize also non-Jacobi tridiagonal matrices, were used to reduce asymptotic questions to local theory. Larger phase spaces for the Toda lattice lead to the study of isospectral manifolds and different coadjoint orbits. Additionally, the time one map of the associated flow is computed by a familiar algorithm in numerical linear algebra.

The text is mostly expositive and self contained, presenting alternative formulations of familiar results and applications to numerical analysis.

2016 Impact Factor: 0.857

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