# How to calculate the conserved energy $E$ from the Lagrangian?

I am reading a PhD thesis that considers the Lagrangian $$\mathcal{L}=\partial_\mu\phi\partial^\mu\phi^\star-U(|\phi^2|)$$ where $$\phi$$ is a complex scalar field and $$U(|\phi|^2)$$ is an arbitrary potential.

The thesis states that there is a conserved energy $$E$$ associated with this Lagrangian given by $$E=\int d^2x\;\left(\frac{1}{2}|\dot{\phi}|^2 +\frac{1}{2}|\nabla\phi|^2 +U(|\phi|^2)\right).$$

This is stated without proof/derivation (probably because it is trivial) but I can't determine the justification for this form. How would I go about deriving this myself?

• You may check it is time independent by use of the E-L equations, no? Have you learned about the energy function in transitioning to the Hamiltonian? May 31 '19 at 23:18