Hamiltonian Function

Use Legendre transformation, the Lagrangian Function could create Hamiltonian Function

Notes

The Lagrange function derived Euler-Lagrange equation is 2 times derivative. As Euler-Lagrange equation is:

$$\begin{array}{r}\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}=\frac{\partial L}{\partial q}\\ \frac{dp}{dt}=\frac{\partial L}{\partial q}\\ p=\frac{\partial L}{\partial \dot{q}}\end{array}$$

Using Legrenger transformation

$$\begin{array}{r}L(q,\dot{q},t)\to H(q,p,t)\\ H=\sum p_i\dot{q_i}-L\end{array}$$

now H is a function of p, q and t,

$$\begin{array}{r}\frac{\partial H}{\partial q}=-\dot{p}\\ \frac{\partial H}{\partial p}=\dot{q}\\ \frac{dH}{dt}=\frac{\partial H}{\partial t}\end{array}$$