Variational principles for Lie–Poisson and Hamilton–Poincaré equations (original) (raw)
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arXiv: Differential Geometry, 2020
This manuscript is essentially a collection of lecture notes which were given by the first author at the Summer School Wisla-2019, Poland and written down by the second author. As the title suggests, the material covered here includes the Poisson and symplectic structures (Poisson manifolds, Poisson bi-vectors and Poisson brackets), group actions and orbits (infinitesimal action, stabilizers and adjoint representations), moment maps, Poisson and Hamiltonian actions. Finally, the phase space reduction is also discussed. The very last section introduces the Poisson-Lie structures along with some related notions. This text represents a brief review of a well-known material citing standard references for more details. The exposition is concise but pedagogical. The Authors believe that it will be useful as an introductory exposition for students interested in this specific topic.
On Symplectic Reduction in Classical Mechanics
Philosophy of Physics, 2007
This Chapter expounds the modern theory of symplectic reduction in finitedimensional Hamiltonian mechanics. This theory generalizes the well-known connection between continuous symmetries and conserved quantities, i.e. Noether's theorem. It also illustrates one of mechanics' grand themes: exploiting a symmetry so as to reduce the number of variables needed to treat a problem. The exposition emphasises how the theory provides insights about the rotation group and the rigid body. The theory's device of quotienting a state space also casts light on philosophical issues about whether two apparently distinct but utterly indiscernible possibilities should be ruled to be one and the same. These issues are illustrated using "relationist" mechanics.
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There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three approaches coincide. The goal of this paper is to study the general case of a symplectic action that does not admit a momentum map and one needs to use its natural generalization, a cylinder valued momentum map introduced by Condevaux, Dazord, and Molino . In this case it will be shown that the three reduced spaces mentioned above do not coincide, in general. More specifically, the Marsden-Weinstein reduced spaces are not symplectic but Poisson and their symplectic leaves are given by the optimal reduced spaces. Foliation reduction produces a symplectic reduced space whose Poisson quotient by a certain Lie group associated to the group of symmetries of the problem equals the Marsden-Weinstein reduced space. We illustrate these constructions with concrete examples, special emphasis being given to the reduction of a magnetic cotangent bundle of a Lie group in the situation when the magnetic term ensures the non-existence of the momentum map for the lifted action. The precise relation of the cylinder valued momentum map with group valued momentum maps for Abelian Lie groups is also given.
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Journal of Geometry and Physics, 1985
For a symplectic manifold the Poisson bracket on the space of functions is (uniquely) extended to a graded Lie bracket on the space of differential forms modulo exact forms. A large portion of the Hamiltonian formalism is still working. Let (M, w) be a symplectic manifold. Then there is an exact sequence of Lie algebras and Lie algebra homomorphisms 0 -~H°(M)-~C~(M)~=~(M) -+ H'(M) -~0, where H°(M), H 1(M) are the de Rham cohomology spaces,~= 0(M) is the space of all vector fields X with 0 (X)w = 0 (Lie derivative), a Lie subalgebra of the space~((M)of all vector fields. C(M) is equipped with the Poisson bracket }, and H(f) is the Hamiltonian vector field for the generating function f. 'y(x)
A variational principle for actions on symmetric symplectic spaces
Journal of Geometry and Physics, 2004
We present a definition of generating functions of canonical relations, which are real functions on symmetric symplectic spaces, discussing some conditions for the presence of caustics. We show how the actions compose by a neat geometrical formula and are connected to the hamiltonians via a geometrically simple variational principle which determines the classical trajectories, discussing the temporal evolution of such "extended hamiltonians" in terms of Hamilton-Jacobi-type equations. Simplest spaces are treated explicitly.
The ubiquity of the symplectic Hamiltonian equations in mechanics
The Journal of Geometric Mechanics, 2009
In this paper, we derive a "hamiltonian formalism" for a wide class of mechanical systems, including classical hamiltonian systems, nonholonomic systems, some classes of servomechanism... This construction strongly relies in the geometry characterizing the different systems. In particular, we obtain that the class of the so-called algebroids covers a great variety of mechanical systems. Finally, as the main result, a hamiltonian symplectic realization of systems defined on algebroids is obtained. 2000 Mathematics Subject Classification. 70H05; 70G45; 53D05; 37J60. Key words and phrases. canonical symplectic formalism, algebroid, (generalized) nonholonomic mechanics, exact and closed symplectic section. This work has been partially supported by MEC (Spain) Grants MTM 2006-03322, MTM 2007-62478, project "Ingenio Mathematica" (i-MATH) No. CSD 2006-00032 (Consolider-Ingenio 2010) and S-0505/ESP/0158 of the CAM. The authors wish to thank David Iglesias for helpful comments.
The Journal of Geometric Mechanics, 2010
In this paper, we study the underlying geometry in the classical Hamilton-Jacobi equation. The proposed formalism is also valid for nonholonomic systems. We first introduce the essential geometric ingredients: a vector bundle, a linear almost Poisson structure and a Hamiltonian function, both on the dual bundle (a Hamiltonian system). From them, it is possible to formulate the Hamilton-Jacobi equation, obtaining as a particular case, the classical theory. The main application in this paper is to nonholonomic mechanical systems. For it, we first construct the linear almost Poisson structure on the dual space of the vector bundle of admissible directions, and then, apply the Hamilton-Jacobi theorem. Another important fact in our paper is the use of the orbit theorem to symplify the Hamilton-Jacobi equation, the introduction of the notion of morphisms preserving the Hamiltonian system; indeed, this concept will be very useful to treat with reduction procedures for systems with symmetries. Several detailed examples are given to illustrate the utility of these new developments.
The symplectic reduced spaces of a Poisson action
Comptes Rendus Mathematique, 2002
During the last thirty years, symplectic or Marsden-Weinstein reduction has been a major tool in the construction of new symplectic manifolds and in the study of mechanical systems with symmetry. This procedure has been traditionally associated to the canonical action of a Lie group on a symplectic manifold, in the presence of a momentum map. In this note we show that the symplectic reduction phenomenon has much deeper roots. More specifically, we will find symplectically reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a momentum map. On other words, the right category to obtain symplectically reduced spaces is that of Poisson manifolds acted canonically upon by a Lie group.