PX285 - D1 - the hamiltonian

consider a particle of mass, $m$ , moving on a line, $x$ , with a velocity, $\dot{x}$
$T = \frac{1}{2} m {\dot{x}}^{2}$
$V = V (x)$
$⟹ L = \frac{1}{2} m {\dot{x}}^{2} - V (x) = \frac{1}{2} m v^{2} - V (x)$
the hamiltonian is defined as:

\begin{aligned} H & = v \frac{\partial L}{\partial v} - L \\ = m v^{2} - (\frac{1}{2} m v^{2} + V (x)) \\ = \frac{1}{2} m v^{2} + V (x) \\ ∴ H & = T + V \end{aligned}