6. Hamiltonian Mechanics

Riformulazione ulteriore della meccanica di Newton, a partire dalla meccanica di Lagrange. Fornisce le basi per un approccio moderno anche in altre teorie della Fisica. dots…

Starting from Lagrange equations derived in Lagrangian mechanics,

\[\dfrac{d}{dt}\Big( \frac{\partial \mathscr{L}}{\partial \dot{q}} \Big) - \frac{\partial \mathscr{L}}{\partial q} = Q_q\]

the generalized moment is defined as

\[p_k := \frac{\partial \mathscr{L}}{\partial \dot{q}^k} \ ,\]

and the Hamiltonian function as

\[\mathscr{H}(q^k(t), p_k(t), t) := p_k \dot{q}^k - \mathscr{L}(\dot{q}^l(q^k, p_k, t), q^l(t), t) \ ,\]

its differential reads

\[\begin{split}\begin{aligned} d\mathscr{H} & = dq^k \, \frac{\partial \mathscr{H}}{\partial q^k} + dp_k \, \frac{\partial \mathscr{H}}{\partial p_k} + dt \, \frac{\partial \mathscr{H}}{\partial t} = \\ & = d p_k \, \dot{q}^k + \underbrace{ p_k \, d \dot{q}^k - d \dot{q}^k \, \frac{\partial \mathscr{L}}{\partial \dot{q}^k}}_{=0} - d q^k \, \frac{\partial \mathscr{L}}{\partial q^k} - dt \, \frac{\partial \mathscr{L}}{\partial t} \end{aligned}\end{split}\]

and thus it follows

\[\begin{split}\begin{cases} \dot{q}^k & = \dfrac{\partial \mathscr{H}}{\partial p_k} \\ \dfrac{\partial \mathscr{H}}{\partial q^k} & = - \dfrac{\partial \mathscr{L}}{\partial q^k} \\ \dfrac{\partial \mathscr{H}}{\partial t} & = - \dfrac{\partial\mathscr{L}}{\partial t} \ . \end{cases}\end{split}\]

Recasting Lagrange equations as

\[\frac{\partial \mathscr{L}}{\partial q^k} = - Q_{q^k} + \dfrac{d}{dt}\Big( \frac{\partial \mathscr{L}}{\partial \dot{q}^k} \Big) = -Q_{q^k} + \dot{p}_k\]

Hamilton equations follow

\[\begin{split}\begin{cases} \dot{q}^k & = \dfrac{\partial H}{\partial p_k} \\ \dot{p}_k & =-\dfrac{\partial H}{\partial q^k} + Q_{q^k} \ . \end{cases}\end{split}\]