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Journal of Geometric Mechanics

2017 , Volume 9 , Issue 1

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Second-order constrained variational problems on Lie algebroids: Applications to optimal control
Leonardo Colombo
2017, 9(1): 1-45 doi: 10.3934/jgm.2017001 +[Abstract](40) +[HTML](0) +[PDF](680.5KB)
The aim of this work is to study, from an intrinsic and geometric point of view, second-order constrained variational problems on Lie algebroids, that is, optimization problems defined by a cost function which depends on higher-order derivatives of admissible curves on a Lie algebroid. Extending the classical Skinner and Rusk formalism for the mechanics in the context of Lie algebroids, for second-order constrained mechanical systems, we derive the corresponding dynamical equations. We find a symplectic Lie subalgebroid where, under some mild regularity conditions, the second-order constrained variational problem, seen as a presymplectic Hamiltonian system, has a unique solution. We study the relationship of this formalism with the second-order constrained Euler-Poincaré and Lagrange-Poincaré equations, among others. Our study is applied to the optimal control of mechanical systems.
Covariant Hamiltonian field theories on manifolds with boundary: Yang-Mills theories
Alberto Ibort and  Amelia Spivak
2017, 9(1): 47-82 doi: 10.3934/jgm.2017002 +[Abstract](44) +[HTML](0) +[PDF](785.0KB)
The multisymplectic formalism of field theories developed over the last fifty years is extended to deal with manifolds that have boundaries. In particular, a multisymplectic framework for first-order covariant Hamiltonian field theories on manifolds with boundaries is developed. This work is a geometric fulfillment of Fock's formulation of field theories as it appears in recent work by Cattaneo, Mnev and Reshetikhin [11]. This framework leads to a geometric understanding of conventional choices for boundary conditions and relates them to the moment map of the gauge group of the theory.It is also shown that the natural interpretation of the Euler-Lagrange equations as an evolution system near the boundary leads to a presymplectic Hamiltonian system in an extended phase space containing the natural configuration and momenta fields at the boundary together with extra degrees of freedom corresponding to the transversal components at the boundary of the momenta fields of the theory. The consistency conditions for evolution at the boundary are analyzed and the reduced phase space of the system is shown to be a symplectic manifold with a distinguished isotropic submanifold corresponding to the boundary data of the solutions of Euler-Lagrange equations. This setting makes it possible to define well-posed boundary conditions, and provides the adequate setting for the canonical quantization of the system.The notions of the theory are tested against three significant examples: scalar fields, Poisson σ-model and Yang-Mills theories.
The $2$-plectic structures induced by the Lie bialgebras
Mohammad Shafiee
2017, 9(1): 83-90 doi: 10.3934/jgm.2017003 +[Abstract](21) +[HTML](0) +[PDF](332.0KB)
In this paper we show that if the Lie algebra $\mathfrak{g}$ admits a Lie bialgebra structure and $\mathcal{D}$ is a Lie group with Lie algebra $\mathfrak{d}$, the double of $\mathfrak{g}$, then $\mathcal{D}$ or its quotient by a suitable Lie subgroup admits a $2$-plectic structure. In particular it is shown that the imaginary part of the Killing form on $\mathfrak{sl}(n, \mathbb{C})$ (as a real Lie algebra) induces a $2$-plectic structure on $SL(n, \mathbb{C})$.
Uniform motions in central fields
Martin Swaczyna and  Petr Volný
2017, 9(1): 91-130 doi: 10.3934/jgm.2017004 +[Abstract](51) +[HTML](0) +[PDF](879.7KB)
We present a theoretical problem of uniform motions, i.e. motions with constant magnitude of the velocity in central fields as a nonholonomic system of one particle with a nonlinear constraint. The concept of the article is in analogy with the recent paper [21]. The problem is analysed from the kinematic and dynamic point of view. The corresponding reduced equation of motion in the Newtonian central gravitational field is solved numerically. Appropriate trajectories for suitable initial conditions are presented. Symmetries and conservation laws are investigated using the concept of constrained Noetherian symmetry [9] and the corresponding constrained Noetherian conservation law. Isotachytonic version of the conservation law of mechanical energy is found as one of the corresponding constraint Noetherian conservation law of this nonholonomic system.

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