8.2 Moment maps: definition and examples
3 min read•july 30, 2024
Moment maps are the secret sauce of symplectic geometry, linking symmetries to conserved quantities. They're like a GPS for symmetries, mapping a symplectic manifold to the dual of a Lie algebra and revealing hidden connections.
These maps are crucial for understanding Hamiltonian group actions and . They bridge symplectic geometry and Lie theory, encode physical conserved quantities, and unlock the structure of through their convexity properties.
Moment Maps and Symplectic Actions
Definition and Fundamental Properties
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- Moment maps encode conserved quantities associated with symmetries of a symplectic manifold
- Smooth map μ: M → g* where M represents symplectic manifold, g denotes Lie algebra of Lie group G acting on M, and g* is dual of g
- Satisfies defining equation d⟨μ,ξ⟩ = iξ#ω (ξ element of g, ξ# corresponding vector field on M, ω symplectic form on M)
- Equivariant with respect to group action on M and coadjoint action on g*
- Existence closely related to Hamiltonian group actions on symplectic manifolds
- Image provides information about orbits of group action and symplectic reduction of manifold
Mathematical Formulation and Implications
- Bridges symplectic geometry and Lie theory through symmetry encoding
- Level sets relate to orbits of group action
- Plays crucial role in symplectic reduction and quotient constructions
- Represents conserved quantities in classical mechanics ()
- Convexity properties () impact structure of symplectic manifolds
- Essential in study of integrable systems providing geometric interpretation of constants of motion
- Contributes to by constructing quantum operators for classical observables
Examples of Moment Maps
Standard Symplectic Structures
- R^2n with U(1) rotation action: μ(x,y) =
- Cotangent bundle TQ of manifold Q with natural symplectic structure and lifted action of Lie group G: momentum map J: TQ → g*
- Complex projective space CP^n with Fubini-Study symplectic form and SU(n+1) action relates to Hopf fibration
- Coadjoint orbits of Lie group G with Kirillov-Kostant-Souriau symplectic form use inclusion map into g* as
Specialized Symplectic Manifolds
- Toric moment map for toric symplectic manifold encodes combinatorial data of associated polytope
- Hamiltonian actions on symplectic vector spaces construct moment maps using quadratic functions
- Sphere S^2 with standard area form and SO(3) rotation action: angular momentum vector
- Symplectic torus T^2 with translation action: winding numbers of curves
Geometric and Physical Significance of Moment Maps
Geometric Interpretations
- Bridge between symplectic geometry and Lie theory through symmetry encoding
- Level sets relate to orbits of group action
- Play crucial role in symplectic reduction and quotient constructions
- Convexity properties (Atiyah-Guillemin-Sternberg convexity theorem) impact structure of symplectic manifolds
- Essential in study of integrable systems providing geometric interpretation of constants of motion
Physical Applications
- Represent conserved quantities in classical mechanics (Noether's theorem)
- Contribute to geometric quantization by constructing quantum operators for classical observables
- Describe collective motion in Hamiltonian systems leading to symplectic reduction concept
- Provide framework for understanding stability of relative equilibria in Hamiltonian systems with symmetry
- Support energy-momentum method for analyzing stability of Hamiltonian systems
Moment Maps vs Hamiltonian Systems
Connections and Interpretations
- Arise naturally in Hamiltonian systems representing conserved quantities associated with symmetries
- Components interpreted as Hamiltonian functions generating infinitesimal symmetries of system
- structure on symplectic manifold relates to Lie algebra structure of symmetry group through moment map
- Describe collective motion in Hamiltonian systems leading to symplectic reduction concept
- on structure of integrable systems formulated using moment maps for torus actions
Applications in System Analysis
- Provide geometric framework for understanding stability of relative equilibria in Hamiltonian systems with symmetry
- Support energy-momentum method for analyzing stability of Hamiltonian systems
- Allow formulation of Marsden-Weinstein reduction for Hamiltonian systems with symmetry
- Enable study of bifurcations in Hamiltonian systems with symmetry using equivariant singularity theory
- Facilitate analysis of Hamiltonian normal forms near equilibria or periodic orbits
Key Terms to Review (20)
Arnold-Liouville Theorem: The Arnold-Liouville Theorem states that in a Hamiltonian system with a sufficient number of independent constants of motion, the system can be transformed into action-angle coordinates, leading to integrable behavior. This theorem bridges the understanding of Hamiltonian dynamics and integrable systems, emphasizing the significance of symplectic structures and conservation laws in mechanics.
Atiyah-Guillemin-Sternberg Convexity Theorem: The Atiyah-Guillemin-Sternberg Convexity Theorem establishes a connection between symplectic geometry and convex geometry, asserting that the image of a moment map from a symplectic manifold under the action of a compact Lie group is a convex set. This theorem is crucial in understanding how symmetries in a physical system translate into geometric properties and offers deep insights into the structure of phase spaces in mechanics.
Canonical moment map: A canonical moment map is a mathematical tool used in symplectic geometry that associates a symplectic manifold with a Lie group action, capturing the essence of how the system evolves under this action. It provides a way to understand conserved quantities and symmetries in Hamiltonian systems, allowing for the identification of Lagrangian submanifolds and facilitating the study of their properties. This concept bridges the connection between geometric structures and physical systems, making it essential for understanding dynamics in a symplectic context.
Coadjoint Orbit: A coadjoint orbit is the orbit of a coadjoint representation of a Lie group acting on its dual space, which encapsulates important geometric and physical information about the system. This concept connects symplectic geometry with representation theory, as coadjoint orbits arise naturally in the study of moment maps and in the reduction processes that help simplify complex systems by analyzing their behavior under symmetries.
Conservation Laws: Conservation laws are principles in physics that state certain quantities remain constant within a closed system, regardless of the processes happening inside it. These laws are fundamental in understanding symmetries in physical systems, as they connect to how systems evolve over time, influencing their dynamics and stability.
Contact Structures: Contact structures are a type of geometric structure on odd-dimensional manifolds that arise from the study of differential geometry and dynamical systems. They provide a way to define a hyperplane distribution that satisfies certain non-integrability conditions, which allows for the study of curves and flows on these manifolds, connecting them to symplectic geometry and moment maps.
Geometric Quantization: Geometric quantization is a mathematical framework that aims to derive quantum mechanical systems from classical phase spaces using symplectic geometry. This process connects classical mechanics to quantum mechanics through the use of geometric structures, incorporating concepts such as symplectomorphisms and moment maps, which are crucial for understanding the relationships between these two domains.
Hamiltonian action: Hamiltonian action refers to a smooth action of a Lie group on a symplectic manifold that preserves the symplectic structure, allowing for the formulation of classical mechanics in a geometric framework. This concept connects deeply with the behavior of physical systems under transformations and leads to the definition of moment maps, which encapsulate important information about the dynamics and symmetry of the system.
Kähler Manifolds: A Kähler manifold is a special type of complex manifold that has a Riemannian metric compatible with the complex structure, meaning it allows for the integration of symplectic geometry and complex analysis. These manifolds exhibit rich geometric properties due to the presence of a Kähler metric, which can be derived from a scalar function known as the Kähler potential. This connection makes Kähler manifolds particularly relevant when discussing moment maps, as they provide a framework for understanding symplectic actions in a complex setting.
Lie group actions: Lie group actions refer to the way a Lie group, which is a group that is also a differentiable manifold, can act smoothly on another manifold. This action can be thought of as a way to apply the symmetries of the Lie group to the geometric structure of the other manifold, helping to uncover important properties such as invariance and equivariance. Understanding these actions is key to exploring concepts like moment maps, which link the geometry of the manifold with the algebraic structure of the Lie group.
Marsden-Weinstein Theorem: The Marsden-Weinstein Theorem provides a way to construct symplectic manifolds by reducing the symplectic structure of a Hamiltonian system with a symmetry, utilizing moment maps. This theorem connects the concepts of symplectic reduction and the geometry of orbits in the presence of group actions, facilitating the study of reduced spaces in symplectic geometry.
Moment Map: A moment map is a mathematical tool used in classical mechanics to describe the symplectic structure of a Hamiltonian system, connecting the symmetries of the system to conserved quantities. It provides a way to express the action of a group of symmetries on the phase space, allowing for a systematic way to understand how physical systems behave under transformations. By mapping points in phase space to a dual space associated with the symmetries, moment maps reveal deep insights into the dynamics of the system.
Moment map for a torus action: A moment map for a torus action is a mathematical tool that associates each point in a symplectic manifold to a point in the dual of the Lie algebra of the torus, effectively encoding the geometric and dynamical properties of the action. This concept is crucial in understanding how the symplectic structure interacts with the symmetry provided by the torus, often revealing insights into conserved quantities in Hamiltonian systems.
Noether's Theorem: Noether's Theorem states that every differentiable symmetry of the action of a physical system corresponds to a conserved quantity. This fundamental principle links symmetries in physics to conservation laws, revealing deep connections between various physical phenomena and mathematical structures.
Poisson bracket: The Poisson bracket is a binary operation defined on the algebra of smooth functions over a symplectic manifold, capturing the structure of Hamiltonian mechanics. It quantifies the rate of change of one observable with respect to another, linking dynamics with the underlying symplectic geometry and establishing essential relationships among various physical quantities.
S. Sternberg: S. Sternberg is a prominent mathematician known for his contributions to symplectic geometry, particularly in relation to moment maps and their applications in physics and mathematics. His work has significantly advanced the understanding of moment maps, providing foundational results that link symplectic geometry to Hamiltonian mechanics, which is crucial for understanding collective motion and equivariant systems.
Symplectic manifolds: Symplectic manifolds are smooth, even-dimensional manifolds equipped with a closed, non-degenerate 2-form known as the symplectic form. This mathematical structure allows for the formulation of geometric concepts essential to classical mechanics and plays a crucial role in understanding dynamics through its relations to Hamiltonian systems and conservation laws.
Symplectic Moment Map: A symplectic moment map is a mathematical construct that associates a symplectic manifold with a Lie group action, capturing the way in which the group acts on the manifold while preserving its symplectic structure. This map provides a powerful tool for studying the geometry and dynamics of Hamiltonian systems, linking symplectic geometry with mechanics through the principles of conservation and invariance.
Symplectic Reduction: Symplectic reduction is a process in symplectic geometry that simplifies a symplectic manifold by factoring out symmetries, typically associated with a group action, leading to a new manifold that retains essential features of the original. This process is crucial for understanding the structure of phase spaces in mechanics and connects to various mathematical concepts and applications.
V. guillemin: In symplectic geometry, v. guillemin refers to the foundational work of Victor Guillemin, who introduced the concept of moment maps as a tool for studying symplectic manifolds and their symmetries. This concept plays a critical role in understanding how symplectic structures interact with group actions, particularly in areas such as mechanics and algebraic geometry.