Theory

Ordinary differential equations in choreo

First order systems

First order ordinary equations have the form:

\[\frac{\mathrm{d} x(t)}{\mathrm{d}t} = f(t,x(t))\]

Hamiltonian systems

A Hamiltonian system is a specific case of first order system of ordinary differential equations. Given a sufficiently regular function \(H(p,q)\) called the Hamiltonian, a Hamiltonian system reads:

\[\begin{split}\frac{\mathrm{d} q}{\mathrm{d}t}(t) &= \frac{\partial H}{\partial p}(q(t),p(t)) \\ \frac{\mathrm{d} p}{\mathrm{d}t}(t) &= -\frac{\partial H}{\partial q}(q(t),p(t)) \\\end{split}\]

Partitioned Hamiltonian systems

If the Hamiltonian of the system has the form \(H(p,q) = T(p) + V(q)\), then the system is called a partitioned Hamiltonian system. This specific case is amenable to simulation optimizations.

Example: a system of \(n\) classical interacting point masses interacting under Newtonian gravity is a partitioned Hamiltonian system, where:

\[\begin{split}T(p) &= \sum_{i=1}^{n} \frac{p_i^2}{m_i} \\ V(q) &= \sum_{i=1}^{n-1} \sum_{j=i+1}^{n} G \frac{m_i m_j}{|q_i-q_j|} \\\end{split}\]