Bifurcation and Forced Symmetry Breaking In Hamiltonian Systems (original) (raw)
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African Diaspora Journal of Mathematics , 2017
This paper continues to carry out a foundational study of Banyaga's topologies of a closed symplectic manifold (M,ω) [4]. Our intention in writing this paper is to work out several “symplectic analogues” of some results found in the study of Hamiltonian dynamics. By symplectic analogue, we mean if the first de Rham's group (with real coefficients) of the manifold is trivial, then the results of this paper reduce to some results found in the study of Hamiltonian dynamics. Especially, without appealing to the positivity of the symplectic displacement energy, we point out an impact of the L∞−version of Hofer-like length in the investigation of the symplectic nature of the C0−limit of a sequence of symplectic maps. This yields a symplectic analogue of a result that was proved by Hofer-Zehnder [10] (for compactly supported Hamiltonian diffeomorphisms on R2n); then reformulated by Oh-Müller [14] for Hamiltonian diffeomorphisms in general. Furthermore, we show that Polterovich's regularization process for Hamiltonian paths extends over the whole group of symplectic isotopies, and then use it to prove the equality between the two versions of Hofer-like norms. This yields the symplectic analogue of the uniqueness result of Hofer's geometry proved by Polterovich [13]. Our results also include the symplectic analogues of some approximation lemmas found by Oh-Müller [14]. As a consequence of a result of this paper, we prove by other method a result found by McDuff-Salamon.
Symplectic bifurcation theory for integrable systems
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This paper develops a symplectic bifurcation theory for integrable systems in dimension four. We prove that if an integrable system has no hyperbolic singularities and its bifurcation diagram has no vertical tangencies, then the fibers of the induced singular Lagrangian fibration are connected. The image of this singular Lagrangian fibration is, up to smooth deformations, a planar region bounded by the graphs of two continuous functions. The bifurcation diagram consists of the boundary points in this image plus a countable collection of rank zero singularities, which are contained in the interior of the image. Because it recently has become clear to the mathematics and mathematical physics communities that the bifurcation diagram of an integrable system provides the best framework to study symplectic invariants, this paper provides a setting for studying quantization questions, and spectral theory of quantum integrable systems.
Bifurcations in symplectic space
Geometry and topology of caustics, 2008
In this paper we take new steps in the theory of symplectic and isotropic bifurcations, by solving the classification problem under a natural equivalence in several typical cases. Moreover we define the notion of coisotropic varieties and formulate also the coisotropic bifurcation problem. We consider several symplectic invariants of isotropic and coisotropic varieties, providing illustrative examples in the simplest non-trivial cases.
Infinitesimal symplectic relations and generalized hamiltonian dynamics
1978
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On symplectomorphisms and Hamiltonian flows
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We propose the construction of a sequence of time one flows of autonomous Hamiltonian vector fields, converging to a fixed near the identity C 1 symplectic diffeomorphism ψ. This convergence is proved to be uniformly exponentially fast, in a non analytic symplectic topology framework.
Relative equilibria and conserved quantities in symmetric Hamiltonian systems
2000
In this introduction, we first recall the basic phase space structures involved in Hamiltonian systems, the symplectic form, the Poisson brackets and the Hamiltonian function and vector fields, and the relationship between them. Afterwards we describe a few examples of Hamiltonian systems, both of the classical'kinetic+ potential'type as well as others using the symplectic/Poisson structure more explicitly.
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.
Hamiltonian systems with symmetry, coadjoint orbits and plasma physics
1982
The symplectic and Poisson structures on reduced phase spaces are reviewed, including the symplectic structure on coadjoint orbits of a Lie group and the Lie-Poisson structure on the dual of a Lie algebra. These results are applied to plasma physics. We show in three steps how the Maxwell-Vlasov equations for a collisionless plasma can be written in Hamiltonian form relative to a certain Poisson bracket. First, the Poisson-Vlasov equations are shown to be in Hamiltonian form relative to the Lie-Poisson bracket on the dual of the (finite dimensional) Lie algebra of infinitesimal canonical transformations. Then we write Maxwell's equations in Hamiltonian form using the canonical symplectic structure on the phase space of the electromagnetic fields, regarded as a gauge theory. In the last step we couple these two systems via the reduction procedure for interacting systems. We also show that two other standard models in plasma physics, ideal MHD and two-fluid electrodynamics, can be written in Hamiltonian form using similar group theoretic techniques. 4 and we write ξ P = X b J(ξ) . We thus obtain a mapping J : g → C ∞ (P ), such that
ON THE SYMMETRIES OF HAMILTONIAN SYSTEMS
International Journal of Modern Physics A, 1995
In this paper we show how the well-known local symmetries of Lagrangean systems, and in particular the diffeomorphism invariance, emerge in the Hamiltonian formulation. We show that only the constraints which are linear in the momenta generate transformations which correspond to symmetries of the corresponding Lagrangean system. The nonlinear constraints (which we have, for instance, in gravity, supergravity and string theory) rather generate the dynamics of the corresponding Lagrangean system. Only in a very special combination with "trivial" transformations proportional to the equations of motion do they lead to symmetry transformations. We reveal the importance of these special "trivial" transformations for the interconnection theorems which relate the symmetries of a system with its dynamics. We prove these theorems for general Hamiltonian systems. We apply the developed formalism to concrete physically relevant systems and in particular those which are diffeomorphism invariant. The connection between the parameters of the symmetry transformations in the Hamiltonian-and Lagrangean formalisms is found. The possible applications of our results are discussed.