7.3 Relationship between symplectic and Poisson structures
4 min read•july 30, 2024
Symplectic and Poisson structures are fundamental in geometric mechanics. They provide frameworks for describing physical systems, with symplectic structures offering a more rigid, even-dimensional approach, while Poisson structures allow for greater flexibility and broader applications.
Understanding the relationship between these structures is crucial. Every has an associated Poisson structure, but not all Poisson manifolds are symplectic. This connection reveals deep insights into the nature of physical systems and their mathematical representations.
Symplectic vs Poisson Manifolds
Fundamental Structures and Definitions
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- Symplectic manifolds consist of even-dimensional smooth manifolds equipped with closed, 2-forms (symplectic forms)
- Poisson manifolds comprise smooth manifolds with Poisson brackets (bilinear, skew-symmetric operations on smooth functions)
- Poisson brackets satisfy Leibniz rule and Jacobi identity
- Every symplectic manifold has an associated Poisson structure
- Not all Poisson manifolds are symplectic
- characterizes relationship between symplectic and Poisson structures acts as inverse of when it exists
Connection and Distinctions
- Symplectic manifolds represent special cases of Poisson manifolds
- Symplectic forms on symplectic manifolds induce Poisson brackets
- Poisson structures on symplectic manifolds derive from symplectic forms
- Symplectic manifolds maintain constant rank throughout
- Poisson manifolds allow varying rank across the manifold
- Poisson structures enable broader applications in physics and mechanics
- Symplectic structures always preserve volume ()
- Poisson structures generally lack volume preservation unless symplectic
Examples and Applications
- utilizes symplectic manifolds (phase spaces of Hamiltonian systems)
- Quantum mechanics employs Poisson manifolds (quantum phase spaces)
- Fluid dynamics incorporates both structures (vorticity in ideal fluids)
- Statistical mechanics uses Poisson structures (Vlasov equation in plasma physics)
- Integrable systems often involve both symplectic and Poisson structures (Toda lattice)
Symplectic Foliation of Poisson Manifolds
Concept and Theorem
- decomposes Poisson manifolds into symplectic submanifolds
- Symplectic foliation theorem partitions Poisson manifolds into (immersed submanifolds)
- Symplectic leaves acquire symplectic structures induced by ambient manifold's Poisson structure
- Leaf dimension at a point equals rank of Poisson bivector field at that point
- Characteristic distribution spans
- Symplectic leaves form integral submanifolds of characteristic distribution
- Understanding symplectic foliation reveals global structure and dynamics of Poisson manifolds
Process and Implementation
- Identify maximal integral submanifolds of characteristic distribution
- Determine Hamiltonian vector fields associated with smooth functions
- Compute rank of Poisson bivector field at each point
- Construct symplectic leaves by following integral curves of Hamiltonian vector fields
- Induce symplectic structure on each leaf from ambient Poisson structure
- Verify that leaves form a partition of the
- Analyze global properties of foliation (leaf dimension, topology, regularity)
Examples and Applications
- Lie-Poisson structures on dual spaces of Lie algebras (coadjoint orbits as symplectic leaves)
- Linear Poisson structures on vector spaces (symplectic leaves as affine subspaces)
- Symplectic foliation of the Euler top (concentric spheres as symplectic leaves)
- Foliation of the Toda lattice (isospectral manifolds as symplectic leaves)
- Symplectic leaves in deformation (Moyal-Weyl product on )
Symplectic Manifolds as Poisson Manifolds
Proof Outline
- Establish definitions of symplectic and Poisson manifolds emphasizing structures
- Show symplectic form on symplectic manifold induces natural on smooth functions
- Define Poisson bracket using Hamiltonian vector fields and associated with functions and
- Prove induced Poisson bracket satisfies required properties (bilinearity, skew-symmetry, Leibniz rule, Jacobi identity)
- Demonstrate Jacobi identity follows from closure of symplectic form ()
- Show non-degeneracy of symplectic form ensures non-degenerate Poisson structure
- Conclude symplectic manifolds naturally carry Poisson structures
Key Steps and Considerations
- Verify bilinearity and skew-symmetry of induced Poisson bracket
- Prove Leibniz rule using properties of Hamiltonian vector fields
- Establish Jacobi identity using Cartan's magic formula and closure of symplectic form
- Demonstrate non-degeneracy of Poisson structure using non-degeneracy of symplectic form
- Show consistency between symplectic and Poisson structures (Hamiltonian flows, conserved quantities)
- Discuss implications of proof for understanding relationship between symplectic and Poisson geometry
Examples and Applications
- Cotangent bundles of smooth manifolds (canonical symplectic structure induces Poisson bracket)
- Kähler manifolds (complex structure compatible with symplectic form yields Poisson structure)
- Symplectic vector spaces (linear symplectic structure induces constant Poisson structure)
- Symplectic tori (periodic symplectic structure yields Poisson structure on functions)
- Symplectic reduction (quotient of symplectic manifold by symmetry group inherits Poisson structure)
Symplectic vs Poisson Structures
Dimensional and Degeneracy Differences
- Symplectic structures exist only on even-dimensional manifolds
- Poisson structures occur on manifolds of any dimension
- Symplectic structures remain non-degenerate throughout manifold
- Poisson structures allow degeneracy varying across manifold
- Symplectic manifolds possess global symplectic forms
- Poisson manifolds may have only local symplectic structures on symplectic leaves
- Rank of symplectic structures stays constant across manifold
- Poisson structure rank can vary leading to stratification by symplectic leaves of different dimensions
Casimir Functions and Conserved Quantities
- Poisson manifolds admit (functions Poisson-commuting with all other functions)
- Symplectic manifolds lack non-constant Casimir functions
- Casimir functions relate to degeneracy of Poisson structure
- Casimir functions indicate presence of symmetries or conserved quantities in system
- Examples of Casimir functions include angular momentum magnitude in rigid body motion
- Casimir functions play crucial role in Hamiltonian reduction and stability analysis
Reduction and Symmetry
- Poisson geometry allows more general and flexible study of reduction and symmetry
- in symplectic geometry has Poisson counterpart
- Poisson reduction preserves Poisson structure on quotient space
- Symmetry groups of Poisson manifolds can have non-trivial isotropy subgroups
- Momentum maps in Poisson geometry generalize those in symplectic geometry
- Examples include reduction of angular momentum in n-body problem (Poisson) vs two-body problem (symplectic)
Key Terms to Review (21)
Canonical Transformation: A canonical transformation is a change of coordinates in phase space that preserves the symplectic structure of Hamiltonian mechanics. This means that if you transform the coordinates and momenta of a dynamical system, the new coordinates still satisfy Hamilton's equations, reflecting the underlying physics. These transformations are crucial because they allow for the simplification of problems, reveal conserved quantities, and maintain the relationships defined by symplectic geometry.
Casimir Functions: Casimir functions are specific smooth functions on a symplectic manifold that are constant along the orbits of a Poisson bracket. They play a crucial role in understanding the relationship between symplectic and Poisson structures, as they help to characterize the center of the Poisson algebra and identify the invariants of Hamiltonian systems.
Classical Mechanics: Classical mechanics is a branch of physics that deals with the motion of objects and the forces acting on them, typically described by Newton's laws. It serves as the foundation for understanding physical systems, providing insight into energy conservation, the dynamics of motion, and the relationships between different physical quantities.
Closed Form: A closed form is a differential form that satisfies the condition of having a vanishing exterior derivative, meaning it can be expressed as the differential of some other form. This concept is important in the study of symplectic and Poisson structures because closed forms can give rise to conserved quantities and play a key role in defining the geometry of phase space.
Cotangent Bundle: The cotangent bundle of a manifold is the vector bundle that consists of all the cotangent spaces at each point of the manifold, effectively capturing the linear functionals on the tangent spaces. This construction plays a crucial role in symplectic geometry as it provides a natural setting for defining symplectic structures and studying Hamiltonian dynamics.
Differential Forms: Differential forms are mathematical objects that generalize the concept of functions and can be integrated over manifolds, allowing for a powerful framework in calculus on manifolds. They are crucial for expressing concepts such as integration, orientation, and the generalization of the notion of volume in higher dimensions. In symplectic geometry, differential forms play a key role in defining symplectic structures and understanding the geometric properties of manifolds, while also connecting to Poisson structures through their relationship with Hamiltonian mechanics.
Hamiltonian Mechanics: Hamiltonian mechanics is a reformulation of classical mechanics that emphasizes the use of Hamiltonian functions, which describe the total energy of a system, to analyze the evolution of dynamical systems. This framework connects deeply with symplectic geometry and offers insights into the conservation laws and symmetries that govern physical systems.
Hamiltonian Vector Fields: Hamiltonian vector fields are special vector fields associated with Hamiltonian functions in symplectic geometry, representing the flow of a dynamical system. These vector fields arise from the Hamiltonian formulation of mechanics, where they describe how a system evolves over time in phase space, connecting symplectic structures and Poisson structures through their properties.
Liouville's Theorem: Liouville's Theorem states that in Hamiltonian mechanics, the volume of phase space occupied by a set of initial conditions remains constant over time as the system evolves. This theorem provides a fundamental insight into the conservation properties of Hamiltonian systems, connecting symplectic geometry with classical mechanics.
Marsden-Weinstein Reduction: Marsden-Weinstein reduction is a process in symplectic geometry that allows for the simplification of symplectic manifolds by reducing the system with respect to group actions and associated moment maps. This technique connects geometric structures with physical systems, making it possible to analyze and understand complex dynamical systems by focusing on the behavior of trajectories and invariant properties under symmetry transformations.
Moser's Theorem: Moser's Theorem states that if a symplectic manifold possesses a symplectic structure that is compatible with a Hamiltonian function, then any two Hamiltonian systems that are sufficiently close can be smoothly connected through symplectomorphisms. This concept has deep implications in the study of dynamics and stability in symplectic geometry.
Non-degenerate: In symplectic geometry, a non-degenerate structure refers to a bilinear form that does not have any non-zero vectors that are annihilated by it. This concept is crucial because it ensures the existence of a unique symplectic orthogonal complement for every subspace and allows for the establishment of a well-defined symplectic manifold. A non-degenerate symplectic form guarantees that the dynamics of a system can be properly described and facilitates the transition from geometric to analytical perspectives in various mathematical and physical contexts.
Poisson Bivector Field: A Poisson bivector field is a geometric structure that provides a way to define a Poisson bracket on a manifold, thereby allowing for the study of Hamiltonian dynamics. It consists of a skew-symmetric bilinear map that satisfies the Jacobi identity and acts as a generalization of symplectic structures, enabling one to describe both integrable systems and more complex dynamical behaviors in mathematical physics.
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.
Poisson manifold: A Poisson manifold is a smooth manifold equipped with a Poisson bracket, which is a bilinear operation that satisfies certain properties, turning the manifold into a geometric structure that encapsulates both symplectic and algebraic properties. It allows for a general framework to study Hamiltonian dynamics in a broader context than just symplectic geometry. Understanding Poisson manifolds enables connections between classical mechanics and modern geometry.
Quantization: Quantization refers to the process of transitioning from classical physics to quantum physics by discretizing physical quantities. This concept is crucial as it bridges the gap between classical mechanics, often described using symplectic geometry, and quantum mechanics, where observables are represented as operators on a Hilbert space. The implications of quantization extend to understanding how symplectic structures relate to Poisson structures, emphasizing the fundamental connections between classical and quantum theories.
Symplectic Foliation: Symplectic foliation refers to a decomposition of a symplectic manifold into a collection of disjoint symplectic submanifolds called leaves. Each leaf in the foliation preserves the symplectic structure, meaning that the symplectic form remains closed and non-degenerate when restricted to these submanifolds. This concept ties into how symplectic geometry relates to Poisson structures, showcasing the interplay between different geometric frameworks.
Symplectic Form: A symplectic form is a closed, non-degenerate 2-form defined on a differentiable manifold, which provides a geometric framework for the study of Hamiltonian mechanics and symplectic geometry. It plays a crucial role in defining the structure of symplectic manifolds, facilitating the formulation of Hamiltonian dynamics, and providing insights into the conservation laws in integrable systems.
Symplectic leaves: Symplectic leaves are the connected components of the symplectic foliation in a Poisson manifold. They can be thought of as the 'slices' or 'layers' of the manifold where the symplectic structure is well-defined and behaves nicely. Understanding symplectic leaves is crucial for exploring the relationship between symplectic geometry and Poisson structures, as they reveal how these structures can vary across the manifold.
Symplectic Manifold: A symplectic manifold is a smooth, even-dimensional differentiable manifold equipped with a closed, non-degenerate differential 2-form called the symplectic form. This structure allows for a rich interplay between geometry and physics, especially in the formulation of Hamiltonian mechanics and the study of dynamical systems.
Symplectomorphism: A symplectomorphism is a smooth, invertible mapping between two symplectic manifolds that preserves their symplectic structure. This means that if you have a symplectic form on one manifold, the image of that form under the mapping will still be a symplectic form on the other manifold, ensuring the preservation of geometric and physical properties between these spaces.