Euler-Lagrange differential equations

Question

This is used to find the shortest path between two points, however I'm a little confused, wouldn't the answer always be a straight line?

Answer 1

To explain in a little more detail:

Suppose we use the Euler-Lagrange equations to find "shortest paths" on the flat Euclidean plane. Writing the path as $x(t)$ and $y(t)$, the length of the path is $$ L[x(t), y(t)] = \int_{t_0}^{t_1} \sqrt{ \dot x^2 + \dot y^2 } \ dt.$$ If you calculate and solve the Euler-Lagrange equations for this, you'll find that all the solutions are straight lines. Very boring.

But what if we are on the surface of a sphere? In this case, we should describe points using spherical polars angles $\theta(t)$ and $\phi(t)$. The expression for the length of a path is then given by $$ L[\theta(t), \phi(t)] = \int_{t_0}^{t_1} \sqrt{\dot \theta^2 + \sin^2 \theta \ \dot \phi^2} \ dt$$ I encourage you to derive the Euler-Lagrange equations for this, and at least verify that constant-speed trajectories along great circles (for instance, along the equator, or along lines of longitude) are valid solutions.

For the most general case, we have a surface parametrised by coordinates $(u,v)$. Due to the fact that the surface is curved, the length function is given by $$ L[u(t), v(t)] = \int_{t_0}^{t_1} \sqrt{g_{uu}(u,v) \dot u^2 + 2g_{uv}(u,v) \dot u \dot v + g_{vv} (u,v)\dot v^2} \ dt,$$ where the functions $g_{uu}(u,v)$, $g_{uv}(u,v)$ and $g_{vv}(u,v)$ are properties of the surface, capturing information about distances between neighbouring points. The paths of shortest length are called the "geodesics", and the Euler-Lagrange equation that you must solve to find these shortest paths is called the "geodesic equation".

This idea is used in Einstein's general relativity, where objects freely-falling under gravity travel along geodesics in spacetime. Gravity is then interpreted as the curvature of spacetime, which causes the geodesics to deviate from straight lines.