🔵Symplectic Geometry Unit 6 – Symplectic Vector Spaces and Linear Geometry

Key Concepts and Definitions

• Symplectic vector space consists of an even-dimensional real vector space $V$ equipped with a non-degenerate, skew-symmetric bilinear form $\omega$
• Non-degenerate bilinear form $\omega$ implies that for every non-zero vector $v \in V$, there exists a vector $w \in V$ such that $\omega(v, w) \neq 0$
• Skew-symmetry of $\omega$ means that for all vectors $v, w \in V$, $\omega(v, w) = -\omega(w, v)$
  • Consequence of skew-symmetry is that $\omega(v, v) = 0$ for all $v \in V$
• Symplectic basis is a basis $\{e_1, \ldots, e_n, f_1, \ldots, f_n\}$ of $V$ satisfying $\omega(e_i, e_j) = \omega(f_i, f_j) = 0$ and $\omega(e_i, f_j) = \delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta
• Symplectic group $\operatorname{Sp}(V, \omega)$ consists of linear transformations $\phi: V \to V$ that preserve the symplectic form, i.e., $\omega(\phi(v), \phi(w)) = \omega(v, w)$ for all $v, w \in V$
• Hamiltonian vector field $X_H$ associated with a smooth function $H: V \to \mathbb{R}$ is defined by $\omega(X_H, \cdot) = dH$, where $dH$ is the differential of $H$

Foundations of Symplectic Vector Spaces

• Symplectic vector spaces arise naturally in the study of classical mechanics, where the phase space of a system is a symplectic manifold
• Canonical example of a symplectic vector space is $(\mathbb{R}^{2n}, \omega_0)$, where $\omega_0$ is the standard symplectic form given by $\omega_0(x, y) = \sum_{i=1}^n (x_i y_{n+i} - x_{n+i} y_i)$ for $x, y \in \mathbb{R}^{2n}$
• Every symplectic vector space $(V, \omega)$ is isomorphic to $(\mathbb{R}^{2n}, \omega_0)$ for some $n$, known as the Darboux theorem
• Symplectic complement of a subspace $W \subseteq V$ is defined as $W^{\omega} = \{v \in V : \omega(v, w) = 0 \text{ for all } w \in W\}$
  • Symplectic orthogonal complement satisfies $(W^{\omega})^{\omega} = W$ and $\dim W + \dim W^{\omega} = \dim V$
• Lagrangian subspace is a subspace $L \subseteq V$ satisfying $L = L^{\omega}$, i.e., it is equal to its own symplectic complement
  • Maximal dimension of a Lagrangian subspace is half the dimension of $V$

Linear Algebra Refresher

• Linear map or linear transformation $T: V \to W$ between vector spaces $V$ and $W$ preserves vector addition and scalar multiplication, i.e., $T(u + v) = T(u) + T(v)$ and $T(\alpha v) = \alpha T(v)$ for all $u, v \in V$ and $\alpha \in \mathbb{R}$
• Kernel or null space of a linear map $T: V \to W$ is defined as $\ker T = \{v \in V : T(v) = 0\}$
• Image or range of a linear map $T: V \to W$ is defined as $\operatorname{im} T = \{T(v) : v \in V\}$
• Rank-nullity theorem states that for a linear map $T: V \to W$ between finite-dimensional vector spaces, $\dim V = \dim \ker T + \dim \operatorname{im} T$
• Dual space $V^*$ of a vector space $V$ is the space of linear functionals $\varphi: V \to \mathbb{R}$
  • Dual basis $\{e_1^*, \ldots, e_n^*\}$ of $V^*$ is defined by $e_i^*(e_j) = \delta_{ij}$, where $\{e_1, \ldots, e_n\}$ is a basis of $V$
• Adjoint of a linear map $T: V \to W$ is the linear map $T^*: W^* \to V^*$ defined by $(T^*\varphi)(v) = \varphi(T(v))$ for all $\varphi \in W^*$ and $v \in V$

Symplectic Structures and Forms

• Symplectic form $\omega$ on a vector space $V$ induces a natural isomorphism between $V$ and its dual space $V^*$ given by $v \mapsto \omega(v, \cdot)$
• Symplectic form $\omega$ can be represented by a skew-symmetric matrix $\Omega$ with respect to a basis $\{e_1, \ldots, e_n, f_1, \ldots, f_n\}$, where $\Omega_{ij} = \omega(e_i, e_j)$ and $\Omega_{i,n+j} = \omega(e_i, f_j)$
  • Standard symplectic matrix $J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}$ represents the standard symplectic form $\omega_0$ on $\mathbb{R}^{2n}$
• Poisson bracket of two smooth functions $F, G: V \to \mathbb{R}$ is defined as $\{F, G\} = \omega(X_F, X_G)$, where $X_F$ and $X_G$ are the Hamiltonian vector fields associated with $F$ and $G$
  • Poisson bracket satisfies skew-symmetry $\{F, G\} = -\{G, F\}$, Leibniz rule $\{F, GH\} = \{F, G\}H + G\{F, H\}$, and Jacobi identity $\{\{F, G\}, H\} + \{\{G, H\}, F\} + \{\{H, F\}, G\} = 0$
• Symplectic manifold is a smooth manifold $M$ equipped with a closed, non-degenerate 2-form $\omega$, called the symplectic form
  • Closedness of $\omega$ means that its exterior derivative vanishes, i.e., $d\omega = 0$
  • Non-degeneracy of $\omega$ means that for every non-zero tangent vector $v \in T_pM$, there exists a tangent vector $w \in T_pM$ such that $\omega_p(v, w) \neq 0$

Darboux's Theorem and Local Structure

• Darboux's theorem states that every symplectic manifold $(M, \omega)$ is locally isomorphic to the standard symplectic space $(\mathbb{R}^{2n}, \omega_0)$
  • Local isomorphism means that for every point $p \in M$, there exists a neighborhood $U$ of $p$ and a diffeomorphism $\varphi: U \to \varphi(U) \subseteq \mathbb{R}^{2n}$ such that $\varphi^*\omega_0 = \omega|_U$
• Consequence of Darboux's theorem is that symplectic manifolds have no local invariants other than the dimension
• Symplectic coordinates or Darboux coordinates on a symplectic manifold $(M, \omega)$ are local coordinates $(q_1, \ldots, q_n, p_1, \ldots, p_n)$ such that $\omega = \sum_{i=1}^n dq_i \wedge dp_i$
  • Existence of symplectic coordinates is guaranteed by Darboux's theorem
• Lagrangian submanifold of a symplectic manifold $(M, \omega)$ is a submanifold $L \subseteq M$ of dimension $\frac{1}{2}\dim M$ such that $\omega|_L = 0$
  • Locally, every Lagrangian submanifold can be described as the graph of a generating function in symplectic coordinates

Subspaces and Quotients in Symplectic Geometry

• Symplectic subspace of a symplectic vector space $(V, \omega)$ is a subspace $W \subseteq V$ such that the restriction $\omega|_W$ is a symplectic form on $W$
  • Equivalent characterization is that $W \cap W^{\omega} = \{0\}$
• Symplectic reduction or Marsden-Weinstein reduction is a method for constructing lower-dimensional symplectic manifolds from a given symplectic manifold with symmetry
  • Let $(M, \omega)$ be a symplectic manifold and $G$ a Lie group acting on $M$ by symplectomorphisms, i.e., $g^*\omega = \omega$ for all $g \in G$
  • Moment map $\mu: M \to \mathfrak{g}^*$ is a $G$-equivariant map satisfying $d\mu_X = \iota_{X_M}\omega$ for all $X \in \mathfrak{g}$, where $X_M$ is the vector field on $M$ generated by $X$
  • Marsden-Weinstein quotient is the space $M // G = \mu^{-1}(0) / G$, which inherits a symplectic structure from $M$
• Coisotropic reduction is a generalization of symplectic reduction that allows for the construction of symplectic quotients from coisotropic submanifolds
  • Coisotropic submanifold $C$ of a symplectic manifold $(M, \omega)$ is a submanifold satisfying $(TC)^{\omega} \subseteq TC$
  • Coisotropic reduction theorem states that if $C$ is a coisotropic submanifold of $(M, \omega)$, then the quotient $C / (TC)^{\omega}$ inherits a symplectic structure

Applications in Physics and Mechanics

• Hamiltonian mechanics formulates classical mechanics using symplectic geometry
  • Phase space of a mechanical system is a symplectic manifold $(M, \omega)$, typically the cotangent bundle $T^*Q$ of the configuration space $Q$
  • Hamiltonian function $H: M \to \mathbb{R}$ represents the total energy of the system
  • Dynamics of the system are governed by Hamilton's equations $\dot{q}_i = \frac{\partial H}{\partial p_i}$ and $\dot{p}_i = -\frac{\partial H}{\partial q_i}$, where $(q_i, p_i)$ are canonical coordinates on $M$
• Symplectic integrators are numerical methods for solving Hamilton's equations that preserve the symplectic structure of the phase space
  • Examples of symplectic integrators include the Störmer-Verlet method and the Gauss-Legendre Runge-Kutta methods
• Optics and wave optics can be formulated using symplectic geometry, with the phase space being the space of rays or wavefronts
  • Fermat's principle of least time can be interpreted as a variational principle in a symplectic framework
  • Symplectic methods have been applied to the design of optical systems and the analysis of aberrations
• Quantum mechanics can be formulated using symplectic geometry, with the phase space being the projective Hilbert space of quantum states
  • Schrödinger equation can be written as a Hamiltonian system with respect to the Fubini-Study symplectic form on the projective Hilbert space
  • Geometric quantization is a method for constructing quantum mechanical systems from classical systems using symplectic geometry

Problem-Solving Techniques and Examples

• Identifying symplectic vector spaces and their properties
  • Example: Show that the space of complex numbers $\mathbb{C}$ with the symplectic form $\omega(z_1, z_2) = \operatorname{Im}(z_1 \overline{z_2})$ is a symplectic vector space
  • Example: Prove that the direct sum of symplectic vector spaces is a symplectic vector space with the natural symplectic form
• Computing symplectic complements and Lagrangian subspaces
  • Example: Find the symplectic complement of the subspace $W = \operatorname{span}\{(1, 0, 0, 0), (0, 1, 0, 0)\}$ in $(\mathbb{R}^4, \omega_0)$
  • Example: Show that the subspace $L = \{(x, y, x, y) : x, y \in \mathbb{R}\}$ is a Lagrangian subspace of $(\mathbb{R}^4, \omega_0)$
• Constructing symplectic bases and symplectic coordinates
  • Example: Find a symplectic basis for the symplectic vector space $(\mathbb{R}^4, \omega)$, where $\omega((x_1, y_1, x_2, y_2), (x_1', y_1', x_2', y_2')) = x_1 y_1' - y_1 x_1' + 2(x_2 y_2' - y_2 x_2')$
  • Example: Given the symplectic form $\omega = dx \wedge dy + 2du \wedge dv$ on $\mathbb{R}^4$, find symplectic coordinates $(q_1, q_2, p_1, p_2)$ such that $\omega = dq_1 \wedge dp_1 + dq_2 \wedge dp_2$
• Applying symplectic reduction and coisotropic reduction
  • Example: Consider the symplectic vector space $(\mathbb{R}^4, \omega_0)$ with the action of $S^1$ given by $\theta \cdot (x_1, y_1, x_2, y_2) = (x_1 \cos \theta - y_1 \sin \theta, x_1 \sin \theta + y_1 \cos \theta, x_2, y_2)$. Find the moment map and the Marsden-Weinstein quotient.
  • Example: Let $C = \{(x, y, z, w) \in \mathbb{R}^4 : x^2 + y^2 = 1\}$ be a submanifold of $(\mathbb{R}^4, \omega_0)$. Show that $C$ is coisotropic and find the reduced symplectic manifold.
• Solving problems in Hamiltonian mechanics and other applications
  • Example: Given a Hamiltonian function $H(q, p) = \frac{1}{2}(p^2 + q