Theory

Poincaré integral invariants are built from two ingredients: differential forms, which are the objects that can be integrated over families of trajectories, and integral invariants, which are the differential forms whose integrals are preserved by a dynamical flow. This page condenses the minimum theory required to state the first and second invariants computed by this package. A more detailed treatment of differential forms can be found in Abraham & Marsden.

Differential forms

Let the phase space be $P = \mathbb{R}^d$ with cartesian coordinates $z^i$, $i = 1, \dots, d$. The coordinate differentials $dz^i$ are treated as $d$ independent, anti-commuting quantities, so that $dz^i \wedge dz^j = - \, dz^j \wedge dz^i$, where $\wedge$ denotes the (wedge) product. A homogeneous polynomial of degree $k$ in these differentials is an algebraic $k$-form, and a differential $k$-form (or just $k$-form) is an assignment of an algebraic $k$-form to each point $\bm{z} \in P$,

\[\alpha (\bm{z}) = \sum_{i_1 < \dots < i_k} \alpha_{i_1 \dots i_k} (\bm{z}) \, dz^{i_1} \wedge \dots \wedge dz^{i_k} ,\]

with smooth real-valued coefficients $\alpha_{i_1 \dots i_k} (\bm{z})$. By anti-commutativity the only $k$-form with $k > d$ is $0$.

A $k$-form can be integrated over a parametrized $k$-dimensional submanifold $S \subset P$. Suppose $S$ is parametrized by smooth functions $\bm{s} (\bm{u}) = (s^1 (\bm{u}), \dots, s^d (\bm{u}))$ with $\bm{u} = (u^1, \dots, u^k)$ ranging over a parameter set $U \subset \mathbb{R}^k$. Applying the chain rule to $z^i = s^i (\bm{u})$ pulls the coordinate differentials back to the parameter differentials,

\[dz^i = \sum_{j=1}^{k} \frac{\partial s^i}{\partial u^j} \, du^j ,\]

and each order-$k$ monomial collapses to a Jacobian determinant,

\[dz^{i_1} \wedge \dots \wedge dz^{i_k} = \frac{\partial (s^{i_1}, \dots, s^{i_k})}{\partial (u^1, \dots, u^k)} \, du^1 \wedge \dots \wedge du^k .\]

The integral of $\alpha$ over $S$ is then the ordinary integral over $U$,

\[\int_S \alpha = \int_U \left( \sum_{i_1 < \dots < i_k} \alpha_{i_1 \dots i_k} (\bm{s} (\bm{u})) \, \frac{\partial (s^{i_1}, \dots, s^{i_k})}{\partial (u^1, \dots, u^k)} \right) du^1 \dots du^k .\]

Although this definition uses a particular parametrization, the change-of-variables rule for Jacobian determinants guarantees that the value of the integral is independent of the parametrization chosen for $S$.

Integral invariants

Consider a general non-autonomous ordinary differential equation on $P$,

\[\dot{\bm{z}} = X_t (\bm{z}) ,\]

where $X_t$ is a time-dependent vector field. An absolute integral invariant is a time-dependent $k$-form $\alpha_t$ such that the integral

\[I_t = \int_{S_t} \alpha_t\]

is independent of time whenever $S_t$ is a compact $k$-dimensional submanifold advected along solutions of the ODE. A relative integral invariant is one for which $I_t$ is constant only when $S_t$ is in addition closed (compact and without boundary).

The classic example is Kelvin's circulation theorem for the barotropic Euler equations: the $1$-form $\bm{u}_t (\bm{x}) \cdot d\bm{x}$ is a relative integral invariant, while its exterior derivative, the vorticity $2$-form built from $\bm{\omega}_t = \nabla \times \bm{u}_t$, is an absolute integral invariant. This illustrates a general fact used repeatedly below: the exterior derivative of a relative integral invariant is always an absolute integral invariant.

Canonical Hamiltonian systems

Let $P = \mathbb{R}^n \times \mathbb{R}^n \ni (\bm{q}, \bm{p})$ with coordinates $q^i, p_i$. A canonical Hamiltonian system arises as the Euler–Lagrange equations of the action

\[S[(\bm{q}, \bm{p})] = \int_{t_1}^{t_2} \left( \sum_{i=1}^{n} p_i \, \dot{q}^i - H_t (\bm{q}, \bm{p}) \right) dt ,\]

that is, Hamilton's equations

\[\dot{q}^i = \frac{\partial H_t}{\partial p_i} , \qquad \dot{p}_i = - \frac{\partial H_t}{\partial q^i} .\]

The Hamiltonian $H_t$ is a conserved scalar only when it is time-independent, but canonical Hamiltonian systems always admit integral invariants, regardless of any time-dependence of $H_t$. The fundamental one is the order-one relative invariant, the Lagrangian one-form

\[\vartheta = \sum_{i=1}^{n} p_i \, dq^i ,\]

whose exterior derivative is the order-two absolute invariant, the symplectic two-form

\[\omega = \sum_{i=1}^{n} dq^i \wedge dp_i .\]

Higher-order invariants are obtained by taking wedge powers: at each odd order a relative invariant is given by $\vartheta \wedge \omega$, $\vartheta \wedge \omega \wedge \omega$, $\dots$, and at each even order an absolute invariant by $\omega$, $\omega \wedge \omega$, $\dots$. In fact, canonical Hamiltonian systems are exactly those ODEs that admit $\vartheta$ as a relative integral invariant.

Noncanonical Hamiltonian systems

A broader class arises as the Euler–Lagrange equations of the action

\[S[\bm{z}] = \int_{t_1}^{t_2} \left( \sum_{i=1}^{d} \vartheta_{ti} (\bm{z}) \, \dot{z}^i - H_t (\bm{z}) \right) dt ,\]

for a time-dependent one-form $\vartheta_t = \sum_i \vartheta_{ti} \, dz^i$ and function $H_t$. As in the canonical case, the order-one relative integral invariant is $\vartheta_t$ and the order-two absolute integral invariant is its exterior derivative

\[\omega_t = \sum_{i < j} \left( \frac{\partial \vartheta_{tj}}{\partial z^i} - \frac{\partial \vartheta_{ti}}{\partial z^j} \right) dz^i \wedge dz^j ,\]

with higher-order invariants again constructed from wedge products of these.

The first and second Poincaré invariants $I_1$ and $I_2$ computed by this package are the integrals of $\vartheta$ and $\omega$ over an advected loop and surface, respectively; see the Home page and the guides for their numerical evaluation.