🔵Symplectic Geometry Unit 1 – Introduction to Symplectic Geometry

Key Concepts and Definitions

• Symplectic geometry studies symplectic manifolds, which are smooth manifolds equipped with a closed, non-degenerate 2-form called the symplectic form
• The symplectic form $\omega$ is a skew-symmetric bilinear form that satisfies the condition $d\omega = 0$ (closedness) and is non-degenerate at every point
• Symplectic vector spaces are vector spaces equipped with a symplectic form, serving as the local model for symplectic manifolds
• Hamiltonian systems are dynamical systems described by a Hamiltonian function $H$ and evolve according to Hamilton's equations
  • The Hamiltonian function $H$ represents the total energy of the system (kinetic + potential)
  • Hamilton's equations: $\frac{dq_i}{dt} = \frac{\partial H}{\partial p_i}$ and $\frac{dp_i}{dt} = -\frac{\partial H}{\partial q_i}$
• Symplectomorphisms are diffeomorphisms between symplectic manifolds that preserve the symplectic form
• Poisson brackets $\{f, g\}$ are a binary operation on functions that encodes the geometry and dynamics of a Hamiltonian system

Historical Context and Motivation

• Symplectic geometry has its roots in the study of classical mechanics and the work of mathematicians like Lagrange, Poisson, and Hamilton in the 18th and 19th centuries
• The formulation of Hamiltonian mechanics using symplectic structures provided a geometric framework for understanding the dynamics of physical systems
• The concept of phase space, which combines the position and momentum variables of a system, naturally carries a symplectic structure
• Symplectic geometry gained prominence in the 20th century with the development of modern differential geometry and its applications in physics
  • The geometric formulation of mechanics using symplectic manifolds provided a unifying framework for various physical theories
• The study of symplectic geometry has led to important developments in areas such as geometric quantization, mirror symmetry, and the topology of symplectic manifolds
• Symplectic techniques have found applications in fields beyond classical mechanics, including quantum mechanics, field theory, and string theory

Symplectic Manifolds

• A symplectic manifold $(M, \omega)$ is a smooth manifold $M$ equipped with a symplectic form $\omega$
• The symplectic form $\omega$ is a closed, non-degenerate 2-form that assigns a skew-symmetric bilinear form to each tangent space
  • Closedness: $d\omega = 0$, which implies that the symplectic form is locally exact (locally, $\omega = d\theta$ for some 1-form $\theta$)
  • Non-degeneracy: For every non-zero tangent vector $v$, there exists a tangent vector $w$ such that $\omega(v, w) \neq 0$
• Examples of symplectic manifolds include:
  • Cotangent bundles $(T^*M, \omega_{\text{can}})$ with the canonical symplectic form $\omega_{\text{can}} = \sum_i dq_i \wedge dp_i$
  • Kähler manifolds with the Kähler form (a symplectic form compatible with the complex structure)
• Symplectic manifolds are always even-dimensional (the dimension is $2n$ for some integer $n$)
• Symplectic manifolds have no local invariants other than the dimension (Darboux's theorem)
• The standard example of a symplectic manifold is $(\mathbb{R}^{2n}, \omega_0)$ with the standard symplectic form $\omega_0 = \sum_i dx_i \wedge dy_i$

Symplectic Vector Spaces

• A symplectic vector space $(V, \omega)$ is a finite-dimensional real vector space $V$ equipped with a symplectic form $\omega$
• The symplectic form $\omega$ is a skew-symmetric, non-degenerate bilinear form on $V$
  • Skew-symmetry: $\omega(v, w) = -\omega(w, v)$ for all $v, w \in V$
  • Non-degeneracy: If $\omega(v, w) = 0$ for all $w \in V$, then $v = 0$
• Symplectic vector spaces are the local model for symplectic manifolds (every tangent space of a symplectic manifold is a symplectic vector space)
• The standard example of a symplectic vector space is $(\mathbb{R}^{2n}, \omega_0)$ with the standard symplectic form $\omega_0(v, w) = \sum_i (v_i w_{n+i} - v_{n+i} w_i)$
• Symplectic vector spaces have a compatible complex structure (multiplication by $i$) and a compatible inner product (the real part of the Hermitian inner product)
• Linear transformations that preserve the symplectic form are called symplectic linear maps or symplectic matrices
  • The group of symplectic matrices is denoted by $\text{Sp}(2n, \mathbb{R})$ and is a subgroup of $\text{GL}(2n, \mathbb{R})$

Darboux's Theorem and Local Structure

• Darboux's theorem states that every symplectic manifold is locally symplectomorphic to the standard symplectic vector space $(\mathbb{R}^{2n}, \omega_0)$
• In other words, around every point of a symplectic manifold, there exists a local coordinate system (Darboux coordinates) in which the symplectic form takes the standard form $\omega_0 = \sum_i dx_i \wedge dy_i$
• Darboux coordinates are also known as canonical coordinates or symplectic coordinates
• The existence of Darboux coordinates implies that symplectic manifolds have no local invariants other than the dimension
  • This is in contrast to Riemannian manifolds, which have local invariants such as curvature
• Darboux's theorem is the symplectic analog of the local flatness theorem for Riemannian manifolds
• The proof of Darboux's theorem relies on the Moser trick, which uses the flow of a time-dependent vector field to construct the desired coordinate transformation
• The local structure of symplectic manifolds is completely determined by the dimension, making symplectic geometry a global theory

Hamiltonian Systems and Dynamics

• A Hamiltonian system is a triple $(M, \omega, H)$, where $(M, \omega)$ is a symplectic manifold and $H: M \to \mathbb{R}$ is a smooth function called the Hamiltonian
• The Hamiltonian function $H$ represents the total energy of the system (kinetic + potential) and generates the dynamics of the system via Hamilton's equations
  • Hamilton's equations: $\frac{dq_i}{dt} = \frac{\partial H}{\partial p_i}$ and $\frac{dp_i}{dt} = -\frac{\partial H}{\partial q_i}$, where $(q_i, p_i)$ are canonical coordinates on the phase space $M$
• The flow of a Hamiltonian system is a one-parameter family of symplectomorphisms $\phi_t: M \to M$ that describes the evolution of the system over time
  • The flow satisfies the equation $\frac{d}{dt} \phi_t = X_H \circ \phi_t$, where $X_H$ is the Hamiltonian vector field associated with $H$
• Hamiltonian systems have several important properties:
  • Conservation of energy: The Hamiltonian function $H$ is constant along the flow of the system
  • Symplectic invariance: The flow of a Hamiltonian system preserves the symplectic form $\omega$
  • Liouville's theorem: The flow of a Hamiltonian system preserves the phase space volume (Liouville measure)
• Poisson brackets $\{f, g\}$ are a binary operation on functions that encode the geometry and dynamics of a Hamiltonian system
  • The Poisson bracket satisfies properties such as skew-symmetry, bilinearity, and the Jacobi identity
  • The time evolution of an observable $f$ is given by $\frac{df}{dt} = \{f, H\}$

Applications in Physics and Mechanics

• Symplectic geometry provides a natural framework for formulating and studying classical mechanics
  • The phase space of a mechanical system (consisting of position and momentum variables) is a symplectic manifold
  • The dynamics of the system are governed by Hamilton's equations, which are derived from a Hamiltonian function
• Symplectic techniques have been successfully applied to various areas of physics, including:
  • Celestial mechanics: Studying the motion of celestial bodies, such as planets and satellites, using symplectic integrators
  • Optics and wave optics: Describing the propagation of light using the symplectic structure of the phase space (ray optics) or the wave front (wave optics)
  • Quantum mechanics: Formulating quantum mechanics using the symplectic structure of the phase space (Weyl quantization) or the projective Hilbert space (geometric quantization)
• Symplectic geometry has also found applications in the study of integrable systems and the KAM (Kolmogorov-Arnold-Moser) theory
  • Integrable systems are Hamiltonian systems with a sufficient number of conserved quantities (first integrals) that allow for explicit solutions
  • The KAM theory describes the stability of quasi-periodic motions in nearly integrable Hamiltonian systems under small perturbations
• Other areas where symplectic techniques have been fruitful include:
  • Fluid dynamics and geophysical fluid dynamics
  • Plasma physics and magnetohydrodynamics
  • Field theory and gauge theory
  • String theory and M-theory

Advanced Topics and Current Research

• Symplectic topology is the study of global properties of symplectic manifolds using techniques from differential topology and algebraic topology
  • Important invariants in symplectic topology include symplectic capacities, displacement energy, and Gromov-Witten invariants
  • The study of symplectic embeddings and symplectic packing problems has led to deep connections with algebraic geometry and combinatorics
• Moment maps and Hamiltonian group actions play a central role in the study of symmetries in symplectic geometry
  • A moment map is a equivariant map from a symplectic manifold with a group action to the dual of the Lie algebra of the group, generalizing conserved quantities
  • The Marsden-Weinstein reduction (symplectic reduction) allows for the construction of new symplectic manifolds by "quotienting out" symmetries
• Floer homology is a powerful tool in symplectic geometry that combines ideas from Morse theory and pseudoholomorphic curve theory
  • Floer homology groups are invariants of symplectic manifolds that capture information about the dynamics of Hamiltonian systems and the intersection properties of Lagrangian submanifolds
  • Variants of Floer homology include Hamiltonian Floer homology, Lagrangian Floer homology, and symplectic field theory
• Mirror symmetry is a conjectural duality between symplectic geometry and complex geometry that originated in string theory
  • Mirror symmetry relates the symplectic invariants of a Calabi-Yau manifold (A-model) to the complex invariants of its mirror manifold (B-model)
  • The homological mirror symmetry conjecture states that the derived Fukaya category of a symplectic manifold is equivalent to the derived category of coherent sheaves on its mirror
• Current research in symplectic geometry includes topics such as:
  • Symplectic flexibility and rigidity phenomena
  • The topology and geometry of Lagrangian submanifolds
  • Fukaya categories and their applications in mirror symmetry
  • Symplectic field theory and its relations to contact geometry and low-dimensional topology
  • The interplay between symplectic geometry and gauge theory, such as in the study of moduli spaces of flat connections and the Atiyah-Floer conjecture