Editing Action (physics) (section)

=== Hamilton's characteristic function ===
When the total energy ''E'' is conserved, the [[Hamilton–Jacobi equation]] can be solved with the [[Hamilton–Jacobi equation#Separation of variables|additive separation of variables]]:<ref name="handfinch" />{{rp|225}}
<math display="block">S(q_1, \dots, q_N, t) = W(q_1, \dots, q_N) - E \cdot t,</math>
where the time-independent function ''W''(''q''<sub>1</sub>, ''q''<sub>2</sub>, ..., ''q<sub>N</sub>'') is called ''Hamilton's characteristic function''. The physical significance of this function is understood by taking its total time derivative

<math display="block">\frac{d W}{d t} = \frac{\partial W}{\partial q_i} \dot q_i = p_i \dot q_i.</math>

This can be integrated to give

<math display="block">W(q_1, \dots, q_N) = \int p_i\dot q_i \,dt = \int p_i \,dq_i,</math>

which is just the [[#Abbreviated action (functional)|abbreviated action]].<ref name=Goldestein3/>{{rp|434}}