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

• MHB
• skate_nerd
In summary, I tried to derive the Euler-Lagrange equations by only using Hamilton's equations and the definition of the Hamiltonian in terms of the Lagrangian, but I'm stuck. Any help would be much appreciated!
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

Hi skatenerd,

This is a nice question.

skatenerd said:
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.

GJA said:
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.

## 1. What is the significance of the Euler-Lagrange equations and Hamilton's equations in physics?

The Euler-Lagrange equations and Hamilton's equations are fundamental principles in classical mechanics. They provide a mathematical framework for understanding the motion of particles and systems, and they are used to derive the equations of motion for a wide range of physical phenomena, such as planetary orbits, pendulum motion, and fluid dynamics.

## 2. How are the Euler-Lagrange equations and Hamilton's equations related?

The Euler-Lagrange equations and Hamilton's equations are closely related and are often used together to describe the dynamics of a system. The Euler-Lagrange equations are used to derive the equations of motion from a given Lagrangian function, while Hamilton's equations describe the evolution of the system in terms of Hamiltonian equations of motion.

## 3. What is the role of the Lagrangian function in the derivation of Euler-Lagrange equations?

The Lagrangian function is a mathematical function that represents the energy of a system in terms of its position and velocity. It plays a crucial role in the derivation of the Euler-Lagrange equations, as it provides a mathematical description of the system's behavior and allows for the determination of its equations of motion.

## 4. Can the Euler-Lagrange equations and Hamilton's equations be applied to all systems?

Yes, the Euler-Lagrange equations and Hamilton's equations can be applied to any system, as long as the system can be described by a Lagrangian function. This includes both classical and quantum systems, making these equations incredibly versatile and widely applicable in the field of physics.

## 5. How are the Euler-Lagrange equations and Hamilton's equations used in practical applications?

The Euler-Lagrange equations and Hamilton's equations have numerous practical applications in physics, engineering, and other fields. They are used to model and analyze the behavior of physical systems, design control systems for autonomous vehicles, and develop numerical methods for solving complex problems in mechanics and dynamics.

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