It is known that when the hamiltonian is time independent, it also does not vary with time. That is, $\frac{\partial \mathcal{H}}{\partial {t}}=0$ implies $\frac{\mathrm{d} \mathcal{H}}{\mathrm{d} {t}}=0$ on solutions of the hamiltonian equations. This is an easy calculation using the hamiltonian equations and the chain rule: $\frac{\mathrm{d} \mathcal{H}}{\mathrm{d} {t}}= \frac{\partial \mathcal{H}}{\partial {p}} \frac{\mathrm{d} \mathcal{q}}{\mathrm{d} {t}} + […]

This is from PDE Evans, 2nd edition: Chapter 8, Exercise 3: The elliptic regularization of the heat equation is the PDE $$ u_t – \Delta u -\epsilon u_{tt}=0 \quad \text{in }U_T, \tag{$*$}$$ where $\epsilon > 0$ and $U_T = U \times (0,t]$. Show that $(*)$ is the Euler-Lagrange equation corresponding to an energy functional $I_\epsilon[w] […]

It turns out that Yuri’s answer to my earlier question, whilst correct (and I thank him for his effort), was not quite what I desired. I had not posed the question properly, so I have chosen to re-ask as I am still struggling to implement the strict inequality constraints. Let me begin by reposing the […]

I have the following metric $$ ds^2 = dt^2-dx^2 $$ and I wanted to prove to myself that the shortest path for this metric is straight. I used the following relation $x=f(t)$ and $$ S = \int_{t_1}^{t_2} \sqrt{1-[f'(t)]^2} \ dt $$ If I minimize $S$, I obtain the condition $f”(t)=0$, which implies $x=ct+x_0$, being a […]

This question already has an answer here: A problem from Evans' PDEs book: find a Lagrangian for a given Euler-Lagrange equation 1 answer

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