Hamiltonian reduction and unfolding of dynamical systems with gauge symmetries
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amiltonian Reduction and Unfolding of Dynamical Systems with Gauge Symmetries1 Mihai Visinescu Department of Theoretical Physics, National Institute for Physics and Nuclear Engineering, P.O.Box M.G.6, Magurele, Bucharest, Romania email: [email protected] Abstract—We investigate the reduction and unfolding of dynamical systems with gauge symmetries. An application is provided by a non relativistic point charge in the field of a Dirac monopole. The corresponding dynamical system possessing a Kepler type symmetry is associated with the TaubNUT metric using a reduc tion procedure of symplectic manifolds with symmetries. The reverse of the reduction procedure is done by stages performing the unfolding of the gauge transformation followed by the Eisenhart lift in connection with scalar potentials. DOI: 10.1134/S1063779612050383 1
INTRODUCTION In the case of a symplectic manifold on which a group of symmetries acts symplectically, it is possible to reduce the original phase space to another symplec tic manifold in which the symmetries are divided out. Such a situation arises when one has a particle moving in an electromagnetic field [1]. On the other hand the reverse of the reduction pro cedure can be used to investigate complicated systems. It is possible to use a sort of unfolding of the initial dynamics by imbedding it in a larger one which is eas ier to integrate [2]. Sometimes the equations of motion in a higher dimensional space are quite trans parent, e.g. geodesic motions, but the equations of motion of the reduced system appear more compli cated [3]. As an illustration of the reduction of a symplectic manifold with symmetries and the opposite procedure of oxidation of a dynamical system we shall consider the principal bundle π: 4 – {0} → 3 – {0} with structure group U(1). The Hamiltonian function on the cotangent bundle T*(4 – {0}) is invariant under the U(1) action and the reduced Hamiltonian system proves to describe the threedimensional Kepler prob lem in the presence of a centrifugal potential and Dirac’s monopole field. Concerning the unfolding of the reduced Hamilto nian system we shall perform it by stages. In a first stage of unfolding we use an opposite procedure to the reduction by an U(1) ⯝ S1 action to a fourdimen sional generalized Kepler problem. Finally we resort to the method introduced by Eisenhart [4] who added one or two extra dimensions to configuration space to represent trajectories by geodesics. 1 The article is published in the original.
1. HAMILTONIAN REDUCTION Let us start to consider the principal fiber bundle π : 4 – {0} → 3 – {0} with structure group U(1). The U(1) action is lifted to a symplectic action on T*(4 – {0}) equipped with the standard symplectic form dΘ. Let Ψ : T*(4 – {0}) → be the moment map associated with the U(1) action 1 Ψ ( x, y ) = ( – x 2 y 1 + x 1 y 2 – x 4 y 3 + x 3 y 4 ), 2
(1)
where (x, y) ∈ (4 – {0}) × 4. The reduced phasespace Pμ is defined through –1
–1
π μ : Ψ ( μ ) → P μ := Ψ ( μ )/U ( 1 ),
(2)
which is diffeomorphic
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