Hamiltonian Systems
NASA sends probes to Jupiter using gravitational slingshots along constant-energy surfaces in the phase space of the three-body problem. Without understanding symplectic geometry and integrability such maneuvers would be impossible.
- **Molecular dynamics:** symplectic integrators (Verlet, Yoshida) are used for simulating proteins and materials; they conserve energy over billions of steps
- **Celestial mechanics:** the stability of the Solar System (over billions of years) is linked to the preservation of adiabatic invariants (action variables)
- **Quantum mechanics:** Bohr's correspondence principle: H(q,p) → Ĥ, {f,g} → (1/iħ)[f̂,ĝ]; quantization follows directly from symplectic structure
Предварительные знания
The Hamiltonian
**A pendulum swings. A ball traces a parabola. Planets orbit the Sun.** Behind all these motions lies a single structure: Hamiltonian mechanics. Instead of Newton's F = ma, the total energy function H(q, p) takes the leading role, from which the equations of motion follow automatically.
**Hamiltonian system**: defined by the Hamiltonian H(q, p) (total energy), where q are generalized coordinates and p are generalized momenta. Equations of motion: **dq_i/dt = ∂H/∂p_i, dp_i/dt = −∂H/∂q_i**. Key property: H is conserved along trajectories: **dH/dt = 0**.
| System | H(q, p) | Degrees of freedom |
|---|---|---|
| Harmonic oscillator | p²/2m + mω²q²/2 | 1 |
| Pendulum | p²/2ml² − mgl·cos(q) | 1 |
| Kepler problem (orbit) | p²/2m − k/|q| | 3 |
| N particles | Σpᵢ²/2mᵢ + V(q₁,...,qₙ) | N |
Hamilton and the Reform of Mechanics
William Rowan Hamilton proposed his formalism in 1833-1835, initially for optics and then extended to mechanics. The revolution was not in computation (Newton's equations give the same answers) but in structure: symmetries → conservation laws, phase space, principle of least action. This formalism became the foundation of quantum mechanics 100 years later.
For the pendulum H = p²/2 − cos(q). The equations of motion dq/dt and dp/dt are:
Symplectic Structure
**Hamiltonian flow is not merely a collection of trajectories; it is a special geometric transformation.** It preserves "symplectic area", a generalization of ordinary area to phase space. This property is what distinguishes conservative mechanics from dissipative mechanics.
**Symplectic space**-(ℝ²ⁿ, ω) where ω = Σ dqᵢ ∧ dpᵢ is the symplectic form. The Hamiltonian flow is a symplectomorphism: it preserves ω. The Jacobian matrix J satisfies: **J^T·Ω·J = Ω**, where Ω is the standard symplectic matrix. This is the symplectic counterpart of the orthogonality relation O^T·O = I.
| Property | Euclidean space | Symplectic space |
|---|---|---|
| Metric | g(u,v) = uᵀv (inner product) | ω(u,v) = uᵀΩv (antisymmetric) |
| Transformations | Orthogonal: OᵀO = I | Symplectic: JᵀΩJ = Ω |
| Invariant | Lengths and angles | Symplectic area |
| Dimension | Any | Always even: 2n |
**Symplectic integrators** are numerical methods specifically designed for Hamiltonian systems. Standard methods (Runge-Kutta) do not preserve symplecticity; after a million steps the orbit "spirals" inward or outward. The Verlet method and symplectic Runge-Kutta schemes preserve the structure and are used in molecular dynamics and celestial mechanics.
Why can Hamiltonian systems not have attractors (stable attracting sets)?
Canonical Transformations
**A striking feature of Hamiltonian systems is the freedom to choose coordinates.** It is possible to switch to any "canonical variables" and the equations of motion retain their form. This allows choosing coordinates in which the system looks as simple as possible, all the way to full integrability.
**Canonical transformation**: a substitution (q, p) → (Q, P) that preserves the form of Hamilton's equations. Equivalent to: the transformation is symplectic. Generated by a generating function S: **P = ∂S/∂Q, p = ∂S/∂q**. Example: action-angle variables **(J, θ)**, in which the Hamiltonian depends only on J: H = H(J).
| Transformation | Generating function | Application |
|---|---|---|
| Identity | S = Σ qᵢPᵢ | Base case |
| Action-angle | S = ∫p dq (along orbit) | Integrable systems |
| Poincaré-Birkhoff | Series in εⁿ | Perturbation theory |
| Fourier transform | exp(iqP/ħ) | Quantum mechanics |
In action-angle variables (J, θ) for an integrable system, the equations of motion take the form:
Integrability
**Why can the two-body problem (Earth-Sun) be solved exactly, while the three-body problem cannot?** Because the two-body problem is integrable: there are enough conservation laws to fully describe the motion. Adding a third body "breaks" integrability and opens the door to chaos.
**Integrable system** (in the Liouville-Arnold sense): a Hamiltonian system with n degrees of freedom having n involutive (pairwise commuting) first integrals F₁=H, F₂, ..., Fₙ: **{Fᵢ, Fⱼ} = 0**. Liouville-Arnold theorem: trajectories lie on n-dimensional tori Tⁿ, motion is quasiperiodic.
| System | Integrable? | First integrals |
|---|---|---|
| Harmonic oscillator | Yes | E = p²/2 + ω²q²/2 |
| Kepler problem (2 bodies) | Yes | E, L (angular momentum), Runge-Lenz vector |
| 3-body problem | No | Only E, P, L; not enough |
| Pendulum (small oscillations) | Yes (approximately) | E ≈ const (linearization) |
Poincaré and the Three-Body Problem
In 1887 King Oscar II of Sweden offered a prize for proving the stability of the Solar System. Poincaré submitted a paper that he believed proved it. But during the publication process he found an error. Correcting the error led to the discovery of chaos in the three-body problem, and Poincaré ended up spending more than the prize amount on reprinting the corrected version.
Hamiltonian systems are only classical mechanics
The Hamiltonian formalism is universal: quantum mechanics (Hamiltonian operator), geometric optics, field theory, molecular dynamics all use symplectic structure.
Symplectic geometry underlies geometric quantum mechanics. Quantum operators are quantizations of classical observables: {f,g} → (1/iħ)[F̂,Ĝ]. Dirac showed that the operator commutator is the quantum analogue of the Poisson bracket.
A system with n degrees of freedom is integrable (in the Liouville sense) if it has:
Key Takeaways
- **Hamiltonian H(q,p)** defines the system through total energy; equations dq/dt=∂H/∂p, dp/dt=−∂H/∂q are automatically conservative
- **Symplecticity:** flow preserves ω = Σdqᵢ∧dpᵢ; det(J) = 1; phase volume is conserved (Liouville's theorem)
- **Canonical transformations** preserve the form of equations; action-angle variables make motion explicit (dJ/dt=0, dθ/dt=ω)
- **Liouville integrability:** n involutive integrals → motion on tori → quasiperiodicity; breaking integrability → KAM theory and chaos
Related Topics
Hamiltonian mechanics is the foundation for understanding both regular and chaotic behavior:
- Ergodic Theory — Liouville's theorem is the canonical example of measure invariance in Hamiltonian systems
- KAM Theory — KAM describes what happens to tori under small perturbations of an integrable system
- Neurodynamics — Synchronization of oscillators in neural networks uses Hamiltonian ideas
Вопросы для размышления
- Liouville's theorem says phase volume is conserved. How does this relate to the second law of thermodynamics? (Hint: entropy increases.)
- Why is the three-body problem extraordinarily difficult, even though the equations have been known since the 17th century? What exactly breaks integrability?
- Can symplectic integrators be used for neural ODEs? What advantage would they give over standard Runge-Kutta methods?