Symmetries
Symmetries play a fundamental role in the study of the dynamics of physical systems, because they give rise to conserved quantities. In the Hamiltonian context, these conserved quantities are encoded in what is called a momentum mapping. This mapping gene
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Symmetries
Symmetries play a fundamental role in the study of the dynamics of physical systems, because they give rise to conserved quantities. These can be used to eliminate a number of variables. In the Hamiltonian context, this procedure is called symplectic reduction of systems with symmetries.1 It is the aim of this chapter to present this procedure in a systematic way. In the Hamiltonian context, the conserved quantities corresponding to the symmetry of the system are encoded in a mapping from the phase space of the system to the dual space of the Lie algebra of the symmetry group. It is called momentum mapping, because it generalizes well-known constants of motion, like momentum or angular momentum etc. In Sect. 10.1 we discuss this notion in detail, including a number of examples. Next, in Sect. 10.2, we present some algebraic basics needed for the symmetry reduction procedure. Then, the classical result of Marsden, Weinstein and Meyer on symmetry reduction is discussed. It states that the reduced phase space, obtained by factorizing a level set of the momentum mapping with respect to the freely acting residual symmetry group, carries a natural symplectic structure and that the dynamics of the systems reduces to this space. In particular, also the relation to orbit reduction is studied. In Sect. 10.4 we present the Symplectic Tubular Neighbourhood Theorem.2 This is an important technical tool for generalizing the above classical result to the so-called singular case, where the assumption about the free action of the residual symmetry group is removed. The theory of singular reduction is presented in detail in Sect. 10.5. In Sects. 10.6 and 10.7 the reader will find a large number of applications. First, we discuss the following examples from mechanics: the geodesic flow on the threesphere, the Kepler problem (including the Moser regularization), the Euler top and the spherical pendulum. Section 10.7 contains a model of gauge theory, which can be viewed as obtained from approximation of gauge theory on a finite lattice. Finally, we give an introduction to the study of qualitative dynamics of systems with symmetries in terms of the energy-momentum mapping. 1 In the sequel, we shall cite important contributions to the subject. However, for a quite exhaustive list of references and also for a lot of historical remarks, we refer the reader to [196] and [232]. 2 Also
called the Symplectic Slice Theorem.
G. Rudolph, M. Schmidt, Differential Geometry and Mathematical Physics, 491 Theoretical and Mathematical Physics, DOI 10.1007/978-94-007-5345-7_10, © Springer Science+Business Media Dordrecht 2013
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Symmetries
10.1 Momentum Mappings Recall that an action Ψ of a Lie group G on a symplectic manifold (M, ω) is called symplectic if Ψg∗ ω = ω
(10.1.1)
for all g ∈ G and that, in this case, the tuple (M, ω, Ψ ) is called a symplectic Gmanifold (Definition 8.6.2). Correspondingly, an action3 ψ of a Lie algebra g on (M, ω) is called symplectic if the vector field ψ(A) is symplectic for all A ∈ g. Note that t
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