Hamiltonian Mechanics

For chaotic systems or long-duration simulations, the Hamiltonian formulation offers superior energy conservation properties compared to Lagrangian mechanics.

Legendre Transform

The compiler performs a symbolic Legendre transform to convert user-defined Lagrangians into Hamiltonians:

\[ \begin{align}\begin{aligned}\mathcal{H}(q, p, t) = \sum_i \dot{q}_i p_i - \mathcal{L}\\p_i = \frac{\partial \mathcal{L}}{\partial \dot{q}_i}\end{aligned}\end{align} \]

Symplectic Integration

The generated C++ code for Hamiltonian systems uses symplectic integrators which preserve the phase-space volume, ensuring that energy drift is bounded even over millions of time steps. This is critical for orbital mechanics (e.g., the Figure-8 solution).