  We introduce a new approach to studying spherical spin glass dynamics based
on differential inequalities for one-time observables. Using this approach, we
obtain an approximate phase diagram for the evolution of the energy $H$ and its
gradient under Langevin dynamics for spherical $p$-spin models. We then derive
several consequences of this phase diagram. For example, at any temperature,
uniformly over all starting points, the process must reach and remain in an
absorbing region of large negative values of $H$ and large (in norm) gradients
in order 1 time. Furthermore, if the process starts in a neighborhood of a
critical point of $H$ with negative energy, then both the gradient and energy
must increase macroscopically under this evolution, even if this critical point
is a saddle with index of order $N$. As a key technical tool, we estimate
Sobolev norms of spin glass Hamiltonians, which are of independent interest.
