- #1

skate_nerd

- 176

- 0

The Hamiltonian is defined as

\begin{align*}

\mathcal{H} = \dot{q}_ip_i - \mathcal{L}

\end{align*}

(where the summation convention is implied), and solving for $\mathcal{L}$, we have

\begin{align*}

\mathcal{L} = \dot{q}_ip_i - \mathcal{H}

\end{align*}

Taking the partial derivative with respect to $\dot{q}_i$ on both sides of the above equation, we have

\begin{align*}

\frac{\partial\mathcal{L}}{\partial\dot{q}_i} &= \frac{\partial}{\partial\dot{q}_i}\left[\dot{q}_ip_i - \mathcal{H}\right] \\

&= p_i - \frac{\partial\mathcal{H}}{\partial\dot{q}_i}

\end{align*}

We are given that

\begin{align*}

\frac{\partial\mathcal{L}}{\partial\dot{q}_i} = p_i

\end{align*}

so going back to our definition for the Hamiltonian, we have

\begin{align*}

p_i = p_i - \frac{\partial\mathcal{H}}{\partial\dot{q}_i}

\end{align*}

So we find that

\begin{align*}

\frac{\partial\mathcal{H}}{\partial\dot{q}_i} = 0

\end{align*}

Clearly what I have tried is going nowhere, but the professor gave a hint where he says to start with the definition of the Hamiltonian and invert it to solve for the Lagrangian, which is exactly what I did. I feel like I'm at a bit of a roadblock, so any hints would be appreciated. Thanks everybody