Abstract
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 generalizes well-known constants of motion, like momentum or angular momentum. We start with a discussion of momentum mappings and the related Witt-Artin decomposition of the tangent space. Then, we present the classical result of Marsden, Weinstein and Meyer on symmetry reduction in the case of a free action. To generalize this result to nonfree actions, we prove the Symplectic Tubular Neighbourhood Theorem (or Symplectic Slice Theorem), derive the Marle-Guillemin-Sternberg normal form of the momentum mapping and present the theory of singular reduction in detail. Thereafter, we apply this theory to a number of examples: the geodesic flow on the three-sphere, the Kepler problem (including the Moser regularization), the Euler top, the spherical pendulum and a model of gauge theory, which can be viewed as obtained from an 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.
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- 1.
- 2.
Also called the Symplectic Slice Theorem.
- 3.
See Definition 6.2.6.
- 4.
- 5.
If a momentum mapping J is fixed, we may include it in the tuple, thus writing (M,ω,Ψ,J).
- 6.
If Ψ is a right action, J is a homomorphism.
- 7.
Since
is an initial submanifold of \(\mathfrak{g}^{\ast}\), J restricts to a smooth mapping 
, denoted by the same symbol. - 8.
And δ is called the coboundary operator.
- 9.
Note that this decomposition need not be orthogonal with respect to the G m -invariant scalar product chosen above. If one wants to have an orthogonal Witt-Artin decomposition, one has to redefine the original scalar product by choosing a scalar product on each component and taking the orthogonal direct sum.
- 10.
When working with scalar products, this would require the scalar product on \({\rm T}_{\varPsi _{g}(m_{0})} M\) to be defined by \(\langle\varPsi_{g^{-1}}' \cdot, \varPsi_{g^{-1}}' \cdot \rangle\), which makes sense, because 〈⋅,⋅〉 is \(G_{m_{0}}\)-invariant.
- 11.
Alternatively, one may observe that, on Killing vector fields, \((\iota_{\mu}^{\ast}\omega)_{m}\) coincides with the pull-back of ω to the orbit G⋅m and apply Corollary 10.1.5.
- 12.
In the sequel, for convenience, we write V≡V m .
- 13.
Induced by the injection \(\mathfrak{g}_{m} \to \mathfrak{g}\).
- 14.
As before, for simplicity we denote H≡G m .
- 15.
- 16.
Note that \(\mathfrak{h}^{0}\) coincides with the subspace of H-invariant elements of the annihilator of \(\mathfrak{h}\) in \(\mathfrak{g}\).
- 17.
Which need not be equivariant, see Remark 10.5.8/ 2.
- 18.
Here, a subtlety occurs: whereas J −1(μ) can be simply viewed as a topological subspace of M,
has to be endowed with the initial topology induced by the mapping J. If μ is regular and G acts freely on M, this is consistent with endowing 
with the initial submanifold structure provided by the Transversal Mapping Theorem 1.8.2. - 19.
A Poisson space is a topological space together with a Poisson algebra of continuous functions.
- 20.
More precisely, we have faithful representations of these Lie algebras in the Poisson algebra of smooth functions on phase space.
- 21.
We use the simplified notation
for the element of \(\mathfrak {so}(3)\) corresponding to the vector 
. - 22.
A pair (L,ω), fulfilling this differential equation, is called a Lax pair and the equation is called Lax equation, see Sect. 11.2.
- 23.
With the gravitational acceleration set equal to 1.
- 24.
A subset of ℝn defined by equations and inequalities is said to be semialgebraic.
- 25.
- 26.
See [161].
- 27.
In particular, \({\hat{P}}\) is an orbifold.
- 28.
Cf. Remark 6.6.2/5.
- 29.
Resetting \(H = \tilde{H}\) and \(J = \tilde{J}\).
is an initial submanifold of 
, denoted by the same symbol.
has to be endowed with the initial topology induced by the mapping J. If μ is regular and G acts freely on M, this is consistent with endowing 
with the initial submanifold structure provided by the Transversal Mapping Theorem 1.8.2.
for the element of 
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Rudolph, G., Schmidt, M. (2013). Symmetries. In: Differential Geometry and Mathematical Physics. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5345-7_10
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