🔵Symplectic Geometry Unit 4 – Darboux's Theorem: Local Canonical Coordinates

Key Concepts and Definitions

• Symplectic manifold a smooth manifold $M$ equipped with a closed, non-degenerate 2-form $\omega$
• Symplectic form the 2-form $\omega$ on a symplectic manifold satisfying $d\omega = 0$ and $\omega^n \neq 0$
  • Closedness ensures the conservation of symplectic volume
  • Non-degeneracy implies the existence of a unique vector field for every function on the manifold
• Hamiltonian vector field the unique vector field $X_H$ associated with a function $H$ on a symplectic manifold, defined by $\omega(X_H, \cdot) = dH$
• Poisson bracket an operation $\{f, g\}$ on functions $f$ and $g$ on a symplectic manifold, measuring their failure to commute under the Hamiltonian flow
• Canonical coordinates a set of local coordinates $(q_1, \ldots, q_n, p_1, \ldots, p_n)$ on a symplectic manifold in which the symplectic form takes the standard form $\omega = \sum_{i=1}^n dq_i \wedge dp_i$
• Symplectomorphism a diffeomorphism between symplectic manifolds that preserves the symplectic form
• Lagrangian submanifold a submanifold $L$ of a symplectic manifold $(M, \omega)$ of half the dimension of $M$ on which the symplectic form vanishes ($\omega|_L = 0$)

Historical Context and Motivation

• Symplectic geometry originated from the study of classical mechanics and the phase space of a dynamical system
  • Phase space consists of positions and momenta of particles in the system
  • Evolution of the system is governed by Hamilton's equations, which have a symplectic structure
• Darboux's theorem, named after Jean Gaston Darboux, was formulated in the late 19th century
• Motivated by the desire to simplify the local description of symplectic manifolds and Hamiltonian systems
  • Canonical coordinates allow for a standard representation of the symplectic form
  • Simplifies the analysis of Hamiltonian dynamics and the study of conserved quantities
• Darboux's theorem is a powerful tool in the geometric formulation of classical mechanics
  • Enables the use of canonical transformations to simplify the equations of motion
  • Facilitates the study of integrable systems and the construction of action-angle variables
• The theorem has far-reaching consequences in modern mathematical physics, including quantum mechanics and field theory

Statement of Darboux's Theorem

• Let $(M, \omega)$ be a $2n$-dimensional symplectic manifold and $p \in M$ a point on the manifold
• Darboux's theorem states that there exists a neighborhood $U$ of $p$ and a local coordinate system $(q_1, \ldots, q_n, p_1, \ldots, p_n)$ on $U$ such that the symplectic form $\omega$ takes the standard form: $\omega = \sum_{i=1}^n dq_i \wedge dp_i$
• The coordinates $(q_1, \ldots, q_n, p_1, \ldots, p_n)$ are called canonical coordinates or Darboux coordinates
• The theorem asserts that locally, all symplectic manifolds of the same dimension are indistinguishable
  • The local structure of a symplectic manifold is completely determined by its dimension
• Darboux's theorem is an analog of the local flatness theorem in Riemannian geometry
  • Every Riemannian manifold is locally isometric to Euclidean space
  • Similarly, every symplectic manifold is locally symplectomorphic to the standard symplectic space $(\mathbb{R}^{2n}, \omega_0)$

Proof Outline and Strategy

• The proof of Darboux's theorem relies on the Moser trick, a powerful technique in symplectic geometry
• The main idea is to construct a family of symplectic forms $\omega_t$ interpolating between the given form $\omega$ and the standard form $\omega_0$
  • Define $\omega_t = (1-t)\omega_0 + t\omega$ for $t \in [0, 1]$
  • Show that $\omega_t$ is a symplectic form for all $t$
• Use the Poincaré lemma to find a family of 1-forms $\alpha_t$ such that $\frac{d}{dt}\omega_t = d\alpha_t$
• Construct a time-dependent vector field $X_t$ satisfying $\iota_{X_t}\omega_t = -\alpha_t$, where $\iota$ denotes the interior product
  • The existence of $X_t$ follows from the non-degeneracy of $\omega_t$
• Solve the flow equation $\frac{d}{dt}\varphi_t = X_t \circ \varphi_t$ with initial condition $\varphi_0 = \text{id}$ to obtain a family of diffeomorphisms $\varphi_t$
• Show that $\varphi_t^*\omega_t = \omega_0$ for all $t$, where $\varphi_t^*$ denotes the pullback by $\varphi_t$
  • This follows from the construction of $X_t$ and the properties of the Lie derivative
• The diffeomorphism $\varphi_1$ satisfies $\varphi_1^*\omega = \omega_0$, providing the desired symplectomorphism between $\omega$ and $\omega_0$

Applications in Symplectic Geometry

• Darboux's theorem is a fundamental result in symplectic geometry with numerous applications
• Simplifies the local study of Hamiltonian systems by providing a standard coordinate system
  • Canonical coordinates allow for the straightforward expression of Hamilton's equations
  • Facilitates the analysis of conserved quantities and symmetries
• Enables the construction of action-angle variables for integrable systems
  • Action-angle variables provide a natural description of the phase space structure
  • Useful in the study of perturbation theory and the stability of periodic orbits
• Plays a crucial role in the geometric quantization of classical systems
  • Canonical coordinates are used to define the quantum Hilbert space and operators
  • Ensures the consistency of the quantization procedure with the classical symplectic structure
• Used in the study of symplectic invariants and the classification of symplectic manifolds
  • Darboux's theorem implies that local symplectic invariants are trivial
  • Global symplectic invariants, such as symplectic capacities, are essential for distinguishing symplectic manifolds
• Relevant in the formulation of the geometric version of the Atiyah-Singer index theorem for elliptic operators on symplectic manifolds

Examples and Illustrations

• The standard symplectic space $(\mathbb{R}^{2n}, \omega_0)$, where $\omega_0 = \sum_{i=1}^n dx_i \wedge dy_i$, is the prototypical example of a symplectic manifold
  • Darboux's theorem states that any symplectic manifold is locally symplectomorphic to $(\mathbb{R}^{2n}, \omega_0)$
• The phase space of a harmonic oscillator is a symplectic manifold with canonical coordinates $(q, p)$
  • The Hamiltonian function is $H(q, p) = \frac{1}{2}(p^2 + \omega^2q^2)$, where $\omega$ is the angular frequency
  • Hamilton's equations take the simple form $\dot{q} = p$ and $\dot{p} = -\omega^2q$ in canonical coordinates
• The cotangent bundle $T^*M$ of a smooth manifold $M$ is a symplectic manifold with the canonical symplectic form $\omega = \sum_{i=1}^n dq_i \wedge dp_i$
  • Canonical coordinates on $T^*M$ are given by the local coordinates $(q_1, \ldots, q_n)$ on $M$ and the corresponding momentum coordinates $(p_1, \ldots, p_n)$
• The Kähler form on a complex manifold is a symplectic form compatible with the complex structure
  • Darboux's theorem implies the local existence of complex coordinates $(z_1, \ldots, z_n)$ in which the Kähler form takes the standard form $\omega = \frac{i}{2}\sum_{j=1}^n dz_j \wedge d\bar{z}_j$

Common Misconceptions and Pitfalls

• Darboux's theorem is a local result and does not imply global symplectic equivalence
  • Two symplectic manifolds may be locally indistinguishable but have different global structures
  • Global symplectic invariants, such as symplectic capacities, are necessary to distinguish symplectic manifolds
• The existence of canonical coordinates does not mean that all coordinates on a symplectic manifold are canonical
  • Canonical coordinates are only guaranteed to exist locally around each point
  • Different choices of canonical coordinates may not be compatible globally
• Darboux's theorem does not provide an explicit construction of canonical coordinates
  • The proof is based on the Moser trick and the existence of a symplectomorphism
  • Finding explicit canonical coordinates for a given symplectic manifold can be a challenging task
• The uniqueness of canonical coordinates is not guaranteed by Darboux's theorem
  • Different choices of canonical coordinates may be related by symplectic transformations
  • The freedom in choosing canonical coordinates is analogous to the gauge freedom in physics
• Darboux's theorem does not generalize directly to infinite-dimensional symplectic manifolds
  • The proof relies on the local compactness of finite-dimensional manifolds
  • Infinite-dimensional symplectic manifolds, such as the phase space of field theories, require more careful treatment

Connections to Other Topics

• Darboux's theorem is closely related to the Poincaré lemma in differential geometry
  • The Poincaré lemma states that closed differential forms are locally exact
  • The proof of Darboux's theorem uses the Poincaré lemma to find a primitive of the symplectic form
• The Moser trick, used in the proof of Darboux's theorem, is a powerful technique in symplectic geometry
  • It can be used to prove other important results, such as the Moser stability theorem for near-identity symplectomorphisms
  • The Moser trick has applications in the study of symplectic fibrations and symplectic reduction
• Darboux's theorem has analogues in other geometric structures
  • The Newlander-Nirenberg theorem in complex geometry states that almost complex structures are locally integrable
  • The Frobenius theorem in differential topology characterizes the local structure of integrable distributions
• Symplectic geometry has deep connections to physics, particularly classical mechanics and quantum theory
  • Darboux's theorem underlies the geometric formulation of Hamiltonian mechanics
  • Canonical coordinates are essential in the geometric quantization of classical systems
  • Symplectic techniques are used in the study of gauge theories and the geometric phase in quantum mechanics
• The study of symplectic manifolds is related to algebraic geometry through the notion of symplectic varieties
  • Symplectic varieties are algebraic varieties equipped with a symplectic form
  • The resolution of singularities in algebraic geometry often involves the construction of symplectic resolutions