Hamiltonian Systems

This page fixes the notation and states the two facts the rest of the package rests on: what a Hamiltonian vector field is, and what "preserving the symplectic structure" buys you. Readers who know Hamiltonian mechanics can skip to Sparse Identification.

From Lagrange to Hamilton

A mechanical system with configuration coordinates $q \in \mathbb{R}^d$ and Lagrangian $L(q, \dot q, t)$ follows the stationary points of the action, giving the Euler–Lagrange equations

\[\frac{d}{dt}\frac{\partial L}{\partial \dot q} - \frac{\partial L}{\partial q} = 0 .\]

Defining the conjugate momentum $p = \partial L / \partial \dot q$ and passing to the Legendre transform $H(q,p,t) = p \cdot \dot q - L$ turns this second-order system in $q$ into a first-order system in the $2d$ variables $z = (q, p)$:

\[\dot q = \frac{\partial H}{\partial p}, \qquad \dot p = -\frac{\partial H}{\partial q} .\]

These are Hamilton's equations. The pair $(q, p)$ are canonical conjugate coordinates, and $H$ is the Hamiltonian — for most mechanical systems, the total energy.

The canonical symplectic form

Hamilton's equations have a compact form that is the one this package uses throughout. Collect the state into $z = (q_1, \dots, q_d, p_1, \dots, p_d) \in \mathbb{R}^{2d}$ and define the canonical symplectic matrix

\[J = \begin{pmatrix} 0 & I_d \\ -I_d & 0 \end{pmatrix} .\]

$J$ is skew-symmetric ($J^\top = -J$) and orthogonal ($J^{-1} = J^\top = -J$). Hamilton's equations become

\[\dot z = J \, \nabla H(z) .\]

A vector field of this form is called a Hamiltonian vector field. Everything the package does on the Hamiltonian side is organised around this single equation.

Why the structure matters

Three consequences follow, and they are the reason to identify $H$ rather than $\dot z$ directly.

Energy is conserved. Along any solution,

\[\frac{dH}{dt} = \nabla H \cdot \dot z = \nabla H \cdot J \nabla H = 0 ,\]

because $J$ is skew-symmetric and $x^\top J x = 0$ for every $x$. Note this holds for any $H$ — it is a property of the form of the equation, not of a particular system.

The flow preserves phase-space volume (Liouville's theorem), and more strongly preserves the symplectic two-form. Trajectories cannot spiral into an attractor; a damped oscillator is not Hamiltonian, which is exactly why the damped linear oscillator appears in this documentation as a non-Hamiltonian example.

The Jacobian is constrained. Differentiating $f(z) = J\nabla H(z)$ gives $\partial f = J \, \nabla^2 H$, so

\[J^{-1} \, \partial f = \nabla^2 H \quad\text{is symmetric.}\]

This is the sharp characterisation: a vector field is (locally) Hamiltonian if and only if $J^{-1}\partial f$ is symmetric. It is also a property you can test, which is what scripts/verify_thesis_examples.jl does — see Structure preservation is exact.

What this means for identification

Suppose you fit a general vector field $\dot z = f(z)$ from data, as classical SINDy does. Nothing in that fit knows about $J$. The identified $f$ will be approximately Hamiltonian if the data is, but only approximately: the symmetry condition above will hold to within the fitting error, and integrating the identified system will drift in energy.

If instead you parametrise $H$ and derive $f = J\nabla H$, then the symmetry condition holds to machine precision for every candidate the fit ever considers, including wrong ones. The constraint is structural, not a penalty term. That distinction is the subject of Hamiltonian SINDy.

Non-canonical and Poisson systems

The package currently assumes canonical coordinates, i.e. the constant $J$ above. Many systems of interest — rigid-body rotation, point vortices, guiding-centre dynamics — are Poisson rather than canonically Hamiltonian, with a state-dependent structure matrix $J(z)$ satisfying the Jacobi identity. Identifying those requires learning $J(z)$ as well as $H$, which this package does not do; see When It Fails for what goes wrong if you assume canonical structure where there is none.