Editing Hamiltonian mechanics (section)

=== From stationary action principle to Hamilton's equations ===
Let <math> \mathcal P(a,b,\boldsymbol x_a,\boldsymbol x_b)</math> be the set of smooth paths <math>\boldsymbol q: [a,b] \to M</math> for which <math>\boldsymbol q(a) = \boldsymbol x_a</math> and <math>\boldsymbol q(b) = \boldsymbol x_{b}. </math> The [[action (physics)|action functional]] <math> \mathcal S : \mathcal P(a,b,\boldsymbol x_a,\boldsymbol x_b) \to \Reals</math> is defined via
<math display="block"> \mathcal S[\boldsymbol q] = \int_a^b \mathcal L(t,\boldsymbol q(t),\dot{\boldsymbol q}(t))\, dt = \int_a^b \left(\sum^n_{i=1} p_i\dot q^i - \mathcal H(\boldsymbol{p},\boldsymbol{q},t) \right)\, dt,</math>
where {{tmath|1= \boldsymbol{q} = \boldsymbol{q}(t) }}, and <math>\boldsymbol{p} = \partial \mathcal L/\partial \boldsymbol{\dot q}</math> (see above). A path <math>\boldsymbol q \in \mathcal P(a,b,\boldsymbol x_a,\boldsymbol x_b)</math> is a [[Lagrangian mechanics|stationary point]] of <math> \mathcal S</math> (and hence is an equation of motion) if and only if the path <math>(\boldsymbol{p}(t),\boldsymbol{q}(t))</math> in phase space coordinates obeys the Hamilton equations.