Derivation of Euler-Lagrange equations w/ Hamilton's equations

[skate_nerd]

I've got a problem that asks us to derive the Euler-Lagrange equations by only using Hamilton's equations and the definition of the Hamiltonian in terms of the Lagrangian. Here's what I tried:

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

 

[GJA]

Hi skatenerd,

This is a nice question.

I've got a problem that asks us to derive the Euler-Lagrange equations by only using Hamilton's equations and the definition of the Hamiltonian in terms of the Lagrangian. Here's what I tried:

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

Your attempt was good and was in the right direction. Given your calculation, I imagine you've overlooked the same thing I did when I first encountered the relationship between Hamiltonian and Lagrangian mechanics. To simplify things, I will present things in one generalized coordinate dimension and let you work out how to extend things to the case of several variables (i.e. I won't have any $i$ indices anywhere)

Note that via the Legendre transformation, $p=p(q,\dot{q})$ and so its partial derivative with respect to $\dot{q}$ isn't zero in general. Furthermore, you can use this fact to further expand the partial derivative of the Hamiltonian in your calculation. From here you should be able to apply Hamilton's equations to derive the Euler-Lagrange equations.

Hopefully this can help you make some sense out of things this time around.

 

[skate_nerd]

Hi skatenerd,

This is a nice question.
Your attempt was good and was in the right direction. Given your calculation, I imagine you've overlooked the same thing I did when I first encountered the relationship between Hamiltonian and Lagrangian mechanics. To simplify things, I will present things in one generalized coordinate dimension and let you work out how to extend things to the case of several variables (i.e. I won't have any $i$ indices anywhere)

Note that via the Legendre transformation, $p=p(q,\dot{q})$ and so its partial derivative with respect to $\dot{q}$ isn't zero in general. Furthermore, you can use this fact to further expand the partial derivative of the Hamiltonian in your calculation. From here you should be able to apply Hamilton's equations to derive the Euler-Lagrange equations.

Hopefully this can help you make some sense out of things this time around.

Thanks for the response! I think I see what you mean. When I took the partial derivative with respect to $\dot{q}_i$ I neglected the fact that $p_i$ is a function of $\dot{q}_i$. I'll rework this keeping that in mind.