9.1 Marsden-Weinstein reduction theorem

The Marsden-Weinstein reduction theorem is a powerful tool in symplectic geometry. It allows us to simplify complex systems with symmetry by reducing their dimension while preserving key geometric properties. This process is crucial for understanding the dynamics of many physical systems.

The theorem shows how to construct a reduced phase space that captures the essential features of a system with symmetry. By quotienting out the symmetry group action, we obtain a lower-dimensional space that still retains the symplectic structure. This simplified space often reveals deeper insights into the system's behavior.

Marsden-Weinstein Reduction Theorem

Theorem Statement and Key Assumptions

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• Marsden-Weinstein reduction theorem provides a method for reducing the dimension of a symplectic manifold with symmetry
• Assumes existence of a Lie group G acting on a symplectic manifold M in a Hamiltonian fashion
• Requires an equivariant momentum map $J: M \rightarrow g^*$, where $g^*$ denotes the dual of the Lie algebra of G
• Necessitates $\mu \in g^*$ as a regular value of the momentum map J
• Demands free and proper action of G on $J^{-1}(\mu)$
• Defines reduced phase space as quotient space $J^{-1}(\mu)/G_\mu$, with $G_\mu$ representing the isotropy subgroup of $\mu$
• States reduced phase space inherits unique symplectic structure from original manifold M
  • Inherited structure preserves key geometric properties (Poisson brackets, Hamiltonian vector fields)
• Assumes symplectic manifold M equipped with symplectic form $\omega$
  • $\omega$ closed and non-degenerate 2-form on M
• Requires G-action to preserve symplectic structure
  • For all $g \in G$ and $x \in M$, $g^*\omega = \omega$
• Momentum map J satisfies specific properties
  • $\langle J(x), \xi \rangle = H_\xi(x)$ for all $x \in M$ and $\xi \in g$
  • $H_\xi$ denotes Hamiltonian function corresponding to infinitesimal generator $\xi_M$

Theorem Application Process

• Identify symplectic manifold M and its symplectic form $\omega$ for theorem application
• Determine Lie group G acting on M in Hamiltonian fashion, preserving symplectic structure
  • Consider rotational symmetries (SO(3)) or translational symmetries (R^n)
• Construct equivariant momentum map $J: M \rightarrow g^*$ associated with group action
  • For angular momentum, J maps phase space points to angular momentum vectors
• Choose regular value $\mu$ of momentum map J and identify level set $J^{-1}(\mu)$
  • Regular value ensures $J^{-1}(\mu)$ submanifold of M
• Verify free and proper G-action on $J^{-1}(\mu)$ for well-defined quotient space
  • Free action no fixed points, proper action closed orbits
• Compute isotropy subgroup $G_\mu$ and form quotient space $J^{-1}(\mu)/G_\mu$
  • $G_\mu = \{g \in G : Ad^*_g \mu = \mu\}$, where $Ad^*$ coadjoint action
• Demonstrate reduced space inherits symplectic structure from M
  • Construct reduced symplectic form $\omega_{red}$ satisfying $\pi^*\omega_{red} = i^*\omega$
  • $\pi: J^{-1}(\mu) \rightarrow J^{-1}(\mu)/G_\mu$ projection, $i: J^{-1}(\mu) \rightarrow M$ inclusion

Applying the Reduction Theorem

Specific Examples of Reduction

• Apply theorem to rigid body rotation
  • M phase space T*SO(3), G rotation group SO(3)
  • J angular momentum map, $\mu$ fixed angular momentum value
  • Reduced space diffeomorphic to S^2 (angular momentum sphere)
• Utilize theorem for n-particle system with translational symmetry
  • M phase space T*(R^3n), G translation group R^3
  • J linear momentum map, $\mu$ fixed total linear momentum
  • Reduced space represents relative motions of particles
• Implement reduction for axisymmetric systems
  • M phase space of axisymmetric body, G rotation group SO(2)
  • J angular momentum about symmetry axis, $\mu$ fixed value
  • Reduced space describes dynamics in rotating frame

Verifying Assumptions and Constructing Reduced Space

• Confirm symplectic structure of manifold M
  • Check $\omega$ closed (d$\omega$ = 0) and non-degenerate
• Validate Hamiltonian G-action on M
  • Ensure $g^*\omega = \omega$ for all $g \in G$
• Construct momentum map J explicitly
  • Use formula $\langle J(x), \xi \rangle = H_\xi(x)$ for $\xi \in g$
• Prove chosen $\mu$ regular value of J
  • Check rank(dJ) maximal at points in $J^{-1}(\mu)$
• Demonstrate free and proper G-action on $J^{-1}(\mu)$
  • Verify no fixed points and closed orbits
• Compute isotropy subgroup $G_\mu$
  • Find elements $g \in G$ satisfying $Ad^*_g \mu = \mu$
• Construct quotient space $J^{-1}(\mu)/G_\mu$ explicitly
  • Identify equivalence classes under $G_\mu$-action

Reduced Phase Space Construction

Level Set and Quotient Space Formation

• Begin with level set $J^{-1}(\mu)$ of momentum map, submanifold of M
  • Characterize $J^{-1}(\mu)$ geometry and topology
• Analyze action of isotropy subgroup $G_\mu$ on $J^{-1}(\mu)$ to understand orbit structure
  • Determine orbits of $G_\mu$-action on $J^{-1}(\mu)$
• Identify equivalence classes of points in $J^{-1}(\mu)$ under $G_\mu$-action
  • Describe representatives of each equivalence class
• Construct quotient space $J^{-1}(\mu)/G_\mu$, forming reduced phase space
  • Characterize topology and geometry of quotient space
• Verify reduced phase space smooth manifold
  • Check quotient map submersion (surjective differential)
• Define projection map $\pi: J^{-1}(\mu) \rightarrow J^{-1}(\mu)/G_\mu$ and study properties
  • Analyze $\pi$ continuity, differentiability, and fiber structure
• Construct reduced symplectic form $\omega_{red}$ on quotient space
  • Ensure $\pi^*\omega_{red} = i^*\omega$, with $i: J^{-1}(\mu) \rightarrow M$ inclusion map

Symplectic Structure Inheritance

• Prove existence of unique reduced symplectic form $\omega_{red}$
  • Use Marsden-Weinstein theorem guarantee
• Construct $\omega_{red}$ explicitly using symplectic reduction procedure
  • Start with $i^*\omega$ on $J^{-1}(\mu)$, project to quotient space
• Verify $\omega_{red}$ well-defined on quotient space
  • Show independence of choice of representatives
• Demonstrate $\omega_{red}$ closed and non-degenerate
  • Inherit properties from original symplectic form $\omega$
• Analyze relationship between $\omega_{red}$ and original $\omega$
  • Understand how symplectic reduction preserves key geometric features
• Compute dimension of reduced phase space
  • Use formula dim($J^{-1}(\mu)/G_\mu$) = dim(M) - 2dim(G) + dim($G_\mu$)
• Investigate Poisson structure on reduced phase space
  • Derive reduced Poisson brackets from $\omega_{red}$

Physical Significance of Reduced Phase Space

Symmetry and Conservation

• Understand reduced phase space represents physically distinct states after accounting for symmetries
  • Eliminates redundant configurations related by symmetry transformations
• Recognize dimension of reduced phase space reflects removal of redundant degrees of freedom
  • Dimension formula dim($J^{-1}(\mu)/G_\mu$) = dim(M) - 2dim(G) + dim($G_\mu$)
• Interpret points in reduced phase space as equivalence classes of dynamical states related by symmetry group action
  • Each point represents family of states indistinguishable under symmetry
• Analyze how conserved quantities associated with symmetry group incorporated into reduced phase space structure
  • Momentum map values become parameters defining reduced space
• Explain how reduced dynamics on quotient space relates to original dynamics on M
  • Preservation of conservation laws and symmetry-related invariants
• Discuss how reduced phase space simplifies system analysis by eliminating symmetry-associated variables
  • Reduced equations of motion often more tractable than original system

Applications and Physical Interpretations

• Relate reduced phase space to physical concepts
  • Relative equilibria (fixed points in reduced space)
  • Stability analysis in systems with symmetry
• Apply reduction to specific physical systems
  • Rigid body dynamics reduced to motion on angular momentum sphere
  • N-body problem reduced to study of shape dynamics
• Interpret reduced phase space in context of gauge theories
  • Reduction analogous to gauge fixing in field theories
• Analyze bifurcations in reduced phase space
  • Correspond to symmetry-breaking phenomena in original system
• Discuss role of reduction in perturbation theory
  • Simplifies analysis of nearly symmetric systems
• Explore connections between reduction and adiabatic invariants
  • Reduced phase space often related to slow variables in system
• Investigate quantum analogues of symplectic reduction
  • Quantum reduction and its relation to classical limit