Geodesic Equations

Geodesics are the "straightest possible curves" on a manifold — the generalization of straight lines to curved spaces. A curve $\gamma(t)$ is a geodesic if its velocity vector is parallel-transported along itself:

$\nabla_{\dot\gamma} \dot\gamma = 0$

In coordinates, this becomes the geodesic equation:

$\frac{d^2 x^k}{dt^2} + \Gamma^k_{ij}\frac{dx^i}{dt}\frac{dx^j}{dt} = 0$

Connection to the Lagrangian

This is exactly the Euler-Lagrange equation for the Lagrangian $L = \frac{1}{2}g_{ij}\dot{x}^i\dot{x}^j$! The Christoffel coefficients appear in the Lagrange equations for free motion — the deep connection between geometry and dynamics.

Integration

Introduce the velocity $v^k = dx^k/dt$ as an auxiliary variable:

$\frac{dx^k}{dt} = v^k, \qquad \frac{dv^k}{dt} = -\Gamma^k_{ij}\, v^i\, v^j$

This is a first-order ODE system we can integrate with RK4.

Examples

Flat $\mathbb{R}^2$: All $\Gamma = 0$, so $\ddot{x} = 0$: straight lines.

Unit sphere: Geodesics are great circles — the equator, meridians, and any circle obtained by slicing the sphere with a plane through the center.

The equator $\gamma(t) = (\pi/2, t)$ is a geodesic: at $\theta = \pi/2$, $\Gamma^\theta_{\phi\phi} = -\sin(\pi/2)\cos(\pi/2) = 0$, so no acceleration.

Your Task

Implement integrate_geodesic(p0, v0, christoffel_fn, t_end, steps) using RK4 integration. Return the final position.