> I'm guessing it's the one that says kinetic minus potential energy, but that seems arbitrary
It is arbitrary, in some sense: Lagrangians are specifications of a physical system in the same way that laws of motion are. Instead of directly describing a trajectory as the solution to some set of differential equations, the Lagrangian approach describes it as a stationary point (an extremum or a saddle point) of the action, which is the integral of the Lagrangian over time. The resulting equations of motion are then given by the Euler-Lagrange equation. L = K - V is (one formulation of) the Lagrangian for a system that obeys Newton's laws. Other systems have other Lagrangians, from which we get other laws of motion.
The other important formulation of classical mechanics is the Hamiltonian one, which very roughly speaking reparameterizes the Euler-Lagrange equation in terms of (generalized) positions and momenta, instead of just position. This turns your n-dimensional second order differential equation into a 2n-dimensional first order one, which is often more convenient to work with.
For simple "ideal billiard balls in a vacuum" situations it makes very little difference what approach you use, but Newtonian mechanics generalizes poorly. All of modern physics is based on the Lagrangian and Hamiltonian approaches.
It is arbitrary, in some sense: Lagrangians are specifications of a physical system in the same way that laws of motion are. Instead of directly describing a trajectory as the solution to some set of differential equations, the Lagrangian approach describes it as a stationary point (an extremum or a saddle point) of the action, which is the integral of the Lagrangian over time. The resulting equations of motion are then given by the Euler-Lagrange equation. L = K - V is (one formulation of) the Lagrangian for a system that obeys Newton's laws. Other systems have other Lagrangians, from which we get other laws of motion.
The other important formulation of classical mechanics is the Hamiltonian one, which very roughly speaking reparameterizes the Euler-Lagrange equation in terms of (generalized) positions and momenta, instead of just position. This turns your n-dimensional second order differential equation into a 2n-dimensional first order one, which is often more convenient to work with.
For simple "ideal billiard balls in a vacuum" situations it makes very little difference what approach you use, but Newtonian mechanics generalizes poorly. All of modern physics is based on the Lagrangian and Hamiltonian approaches.