In classical Lagrangian mechanics, the mass $M$ is a Riemannian metric on the configuration space $Q$. Does the "arc length" of a path $\gamma : [0, 1] \to Q$, $$ \int_0^1 {\lVert{\gamma'(t)}\rVert}_{M}\,dt $$
have a physical meaning?
When there's just one particle, this arc length is just the distance that the particle travels, multiplied by the square root of the particle's mass.
