Subdifferential condition for minimizing trajectories

Assume the hypotheses of the paper's Euler--Poisson theorem. Let $\sigma$ be a minimizing trajectory and let $v(t)$ denote its minimal velocity. Let $\nabla^-{\cal U}$ be the subdifferential specified in the paper's definition of $\nabla^-{\cal U}$.

Subdifferential condition. For almost every $t\in(0,\infty)$,

$$-|v(t)|^{p-2}v(t)\in \nabla^-{\cal U}(\sigma(t)).$$

This condition is presented as a stronger statement about minimizing trajectories. The paper proves it in a special case for simple potentials, while its validity under the full assumptions of the Euler--Poisson theorem is not established in the supplied text.