Combining Lagrangian and Hamiltonian mechanics
\[S =\int L \text{d}t\] \[L = p\dot{q}-H\] \[S = \int p\dot{q}-H \text{d}t\] \[S = \int p \text{d}q - \int H \text{d}t\]$S$ thought as a function of $q$ and $t$:
\[\text{d}S = p \text{d}q - H \text{d}t\] \[\frac{\partial S}{\partial q}= p\] \[\frac{\partial S}{\partial t}= H\]Hamilton-Jacobi equation:
\[\frac{\partial S}{\partial t} - H(q,t,\nabla S) = 0\]