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TOC
1 Introduction
1.1 Why classical mechanics?
1.2 Prospectus
2 Symplectic reduction: an overview
2.1 Hamiltonian mechanics and Noether’s theorem: a review
2.1.1 Symplectic manifolds; the cotangent bundle as a symplectic manifold
2.1.2 Geometric formulations of Hamilton’s equations
2.1.3 Noether’s theorem
2.2 The road ahead
2.3 Appetizer: Belot on relationist mechanics
2.3.1 Comparing two quotienting procedures
2.3.2 The spaces and group actions introduced
2.3.3 The Relationist procedure
2.3.4 The Reductionist procedure
2.3.5 Comparing the Relationist and Reductionist procedures
3 Some geometric tools
3.1 Vector fields on manifolds
3.1.1 Manifolds, vectors, curves and derivatives
3.1.2 Vector fields, integral curves and flows
3.1.3 The Lie derivative
3.2 Lie algebras and brackets
3.2.1 Lie algebras
3.2.2 The Lie bracket of two vector fields
3.3 Submanifolds and Frobenius’ theorem
3.3.1 Submanifolds
3.3.2 The theorem
3.4 Lie groups, and their Lie algebras
3.4.1 Lie groups andmatrix Lie groups
3.4.2 The Lie algebra of a Lie group
3.4.3 Examples, subgroups and subalgebras
3.4.4 The Lie algebra of the rotation group
4 Actions of Lie groups
4.1 Basic definitions and examples
4.2 Quotient structures fromgroup actions
4.3 Proper actions
4.4 Infinitesimal generators of actions
4.5 The adjoint and co-adjoint representations
4.5.1 The adjoint representation
4.5.2 The co-adjoint representation
4.6 Kinematics on Lie groups
4.6.1 Space and body coordinates generalized to G
4.6.2 Passage to the quotients
5 Poisson manifolds
5.1 Preamble: three reasons for Poisson manifolds
5.2 Basics
5.2.1 Poisson brackets
5.2.2 Hamiltonian vector fields
5.2.3 Structure functions
5.2.4 The Poisson structure on g
5.3 The symplectic foliation of Poisson manifolds
5.3.1 The Poisson structure and its rank
5.3.2 Poisson maps
5.3.3 Poisson submanifolds: the foliation theorem
5.3.4 Darboux’s theorem
5.4 The symplectic structure of the co-adjoint representation
5.5 Quotients of Poisson manifolds
6 Symmetry and conservation revisited: momentum maps
6.1 Canonical actions and momentum maps
6.1.1 Canonical actions and infinitesimal generators
6.1.2 Momentum maps introduced
6.2 Conservation of momentum maps: Noether’s theorem
6.3 Examples
6.4 Equivariance of momentum maps
6.4.1 Equivariance and infinitesimal equivariance
6.4.2 Equivariant momentum maps are Poisson
6.5 Momentum maps on cotangent bundles
6.5.1 Momentum functions
6.5.2 Momentum maps for cotangent lifted actions
6.5.3 Examples
7 Reduction
7.1 Preamble
7.2 The Lie-Poisson Reduction Theorem
7.3 Meshing with the symplectic structure on TG: invariant functions
7.3.1 Left-invariant and right-invariant functions on TG
7.3.2 Recovering the Lie-Poisson bracket
7.3.3 Deriving the Lie-Poisson bracket
7.4 Reduction of dynamics
7.5 Envoi: theMarsden-Weinstein-Meyer theorem
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