> S, action, is the arc-length of a world-line. L = dS/dt. It is a natural law that systems move along "straight" world-lines, geodesics, in the absence of int

> S, action, is the arc-length of a world-line. L = dS/dt. It is a natural law that systems move along "straight" world-lines, geodesics, in the absence of interactions. This is a roughly correct description of GR, where the curvature is given by the Einstein field equations. It's not true in classical electromagnetism, or in QM. > The "future-looking" nature of the least action principle is unsatisfying: sure, the action is minimized over an interval in time. Since L = dS/dt, the E-L equations are that same principle expressed at a single moment in time--right? Yes, although the global principle is that the action is stationary, not minimal. > Is there some sense in which L is "orthogonal" to the space spanned by constant energy and momentum? Not quite, but you're on the right track. Conserved quantities of integrable systems are "orthogonal" to the Hamiltonian H, in that the poisson bracket `{f, H} = -df/dt` and so `{f, H} = 0` for constant `f`. The Lagrangian arises as the Legendre transform of H.