The <Euler-Lagrange equation> is
$$
\boxed{\phi_{tt}-\phi_{xx}+U'(\phi)=0}.
$$
Time-translation invariance gives the conserved energy
$$
\boxed{E=\int_{\mathbb R}\left[\frac12\phi_t^2+\frac12\phi_x^2+U(\phi)\right]dx}.
$$
Indeed, after integrating the spatial term by parts and imposing finite-energy boundary conditions,
$$
\frac{dE}{dt}
=\int_{\mathbb R}\phi_t(\phi_{tt}-\phi_{xx}+U'(\phi))\,dx=0.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 313 / 1 / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-313/1/solution