Geometry
Symplectic Geometry
Цели урока
- Understand the symplectic form and Darboux theorem - no local invariants
- Master Hamiltonian dynamics and Poisson brackets
- Know Gromov nonsqueezing and symplectic rigidity
- Connect to quantum mechanics through the correspondence principle
Предварительные знания
- Differential forms
- Hyperbolic Geometry
- Symplectic geometry (foundations)
Why do symplectic integrators preserve Hamiltonian dynamics accurately while standard Runge-Kutta methods accumulate drift?
- Hamiltonian Monte Carlo: symplectic leapfrog integrators used in NumPyro and Stan for Bayesian inference in neural networks
- Molecular dynamics: GROMACS and NAMD use symplectic integrators to simulate proteins - from COVID spike structure to drug design
- Astrodynamics: asteroid 2023 BU orbit computed to 1 km accuracy via symplectic Verlet integrator
- Mirror symmetry: the A-side of Kontsevich HMS is built on symplectic geometry
From Hamilton to Kontsevich
William Hamilton in 1833 reformulated Newtonian mechanics through canonically conjugate variables (q, p). Elie Cartan in the 1920s formalized the geometry of forms. Arnold in the 1960-70s built symplectic topology as a unified theory. Gromov in 1985 proved the nonsqueezing theorem via pseudoholomorphic curves - creating a discipline. Kontsevich in 1994-97 connected symplectic geometry to mirror symmetry and quantum field theory - Fields Medal 1998.
Symplectic Form and Darboux Theorem
Classical mechanics is geometry. Phase spaces of pendulums, planetary orbits, molecular dynamics - all symplectic manifolds. Wolfram Research uses symplectic integrators in Mathematica to compute the orbit of asteroid 2023 BU (flyby at 3600 km from Earth) with 1 km accuracy - thanks to preserving the symplectic structure.
The Darboux theorem says: symplectic geometry has no curvature. All symplectic manifolds are locally identical. Stark contrast with Riemannian geometry, where curvature is the main invariant.
What does Darboux's theorem state?
Darboux: near every point of a symplectic manifold there exist coordinates (q_i, p_i) in which w = sum dq_i ^ dp_i. No local symplectic invariants.
Hamiltonian Dynamics and Conservation Laws
Noether's theorem - one of the most powerful in physics: every continuous symmetry corresponds to a conserved quantity. Momentum from translational symmetry. Angular momentum from rotational. Energy from time symmetry. Symplectic geometry is the mathematical language of this theorem.
Hamiltonian Monte Carlo
Symplectic integrators in Bayesian ML
HMC (Hamiltonian Monte Carlo) - the sampling algorithm in NumPyro, Stan, PyMC - simulates Hamiltonian dynamics in parameter space. The symplectic leapfrog integrator preserves phase volume exactly (not approximately), ensuring correct MCMC samples. Without symplecticity the Markov chain drifts and samples become biased.
Liouville's theorem: the flow of a Hamiltonian system preserves the symplectic form and phase volume. This makes molecular dynamics simulations reversible and guarantees correctness of HMC sampling.
What does {F, H} = 0 mean?
dF/dt = {F, H}. When {F,H}=0, F is conserved. Example: {E, H}=0 is energy conservation.
Gromov Nonsqueezing and Quantization
1985. Mikhail Gromov proves the nonsqueezing theorem. Before it, symplectic diffeomorphisms seemed almost as flexible as volume-preserving maps. Gromov showed otherwise: there is a rigid constraint that volume does not see.
Gromov introduced pseudoholomorphic curves to prove nonsqueezing - a new tool that created the field of symplectic topology. Kontsevich in 1997 proved deformation quantization is possible for any Poisson manifold - Fields Medal 1998.
In automatic differentiation (PyTorch autograd, JAX), the forward and backward passes form a pair analogous to canonically conjugate variables (q, p). Dual-number arithmetic is formally isomorphic to Poisson brackets.
What does the Gromov nonsqueezing theorem show?
Gromov's theorem: symplectic invariants exist that cannot be reduced to volume. A ball cannot fit into a thin cylinder even at arbitrarily small volume. This is the rigidity of symplectic geometry.
Connections to Other Topics
Symplectic geometry unifies classical mechanics, quantum physics, and mirror symmetry.
- Geometry and Physics — Symplectic geometry formalises the phase space of Hamiltonian mechanics
- Vector Geometry — Symplectic form is an antisymmetric bilinear form on a vector space
- Mirror Symmetry — Mirror symmetry links symplectic and complex geometry of Calabi-Yau manifolds
Итоги
- Symplectic manifold (M, w): w is a closed nondegenerate 2-form, dim M is even
- Darboux theorem: locally w = sum dq_i ^ dp_i - no local symplectic invariants
- Hamiltonian field X_H: iota_{X_H}w = -dH; df/dt = {f, H}
- {F,H}=0 - condition for F to be conserved along trajectories
- Gromov nonsqueezing: B^{2n}(r) cannot be symplectically embedded in B^2(R) x R^{2n-2} when r > R
- Quantization: {f,g} -> [F,G]/(i*hbar), [q_i, p_j] = i*hbar*delta_{ij}
Вопросы для размышления
- Why does the Darboux theorem (no local invariants) so strikingly differentiate symplectic from Riemannian geometry?
- How does HMC use symplectic structure to ensure correct Bayesian sampling, and why is leapfrog better than Runge-Kutta?
- What is the deep meaning of the correspondence {q,p}=1 -> [q,p]=i*hbar between classical and quantum mechanics?