Canonical Hamiltonian density of a real scalar field ($\mathcal{H}=\frac{1}{2}{\pi }^2+\frac{1}{2}|\nabla \varphi {|}^2+V(\varphi )$)

For a real scalar field with Lagrangian density $\mathcal{L}={\dot{\varphi }}^2/2-|\nabla \varphi {|}^2/2-V(\varphi )$, the canonical momentum is $\pi =\dot{\varphi }$. Its Hamiltonian density is the sum of momentum energy, gradient energy and potential energy. Hamilton's equations recover the Euler-Lagrange equation.