## Potential, force, momentum, velocity Like the idea of force field, the potential term $V$ gives the force, the relation is just $-\vec{\nabla_i} V = \vec{f_i} = m_i\ddot{\vec{v_i}}=ma$ This is a very important relation, but also very intuitive. Parameters like generalized momentum can be obtained. $p_i = \frac{\partial L}{\partial \dot{q_i}}$ >[!note] >This can be easily shown as $\frac{\partial L}{\partial \dot{q_i}} = \frac{\partial (T-V)}{\partial v} = \frac{\partial T}{\partial v} = \frac{\partial (\frac{1}{2}mv^2)}{v} = mv = p_i$ and $H = T+V = 2T-(T-V) = 2T-\mathcal{L} = \sum_i p_i \dot{q_i} - \mathcal{L}(q, \dot{q})$ From this we have $\dot{p_k} = -\frac{\partial H}{\partial q_k}, \dot{q_k} = \frac{\partial H}{\partial p_k}$ >[!notice] >The above form is also shown in OCW lectures. > >If $H$ is independent of position, then $\vec{p}$ conserved. > >This also gives force as $\dot{p} = -\frac{\partial H}{\partial q_k} = \frac{\partial (mv)}{\partial t} = ma = F$ >So we have generalized velocity and force. Since H is a function of $q, p$, we'd better rewrite it in a $(q, p)$ space, that is what we typically say, the ***phase space***. This set of $(q,p)$ is called *canonical coordinates* and $H = T+V = \frac{p^2}{2m} +V(q)$. ^phase-space