Proof of Hamiltonian equations

[Daaavde]

So, I should prove that:
$- \frac{\partial H}{\partial q_i} = \dot{p_i}$
And it is shown that:
$- \frac{\partial H}{\partial q_i} = - p_j \frac{\partial \dot{q_j}}{\partial q_i} + \frac{\partial \dot{q_j}}{\partial q_i} \frac{\partial L}{\partial \dot{q_j}} + \frac{\partial L}{\partial q_i} = \dot{p_i}$
Where the first two terms delete each other $(\frac{\partial L}{\partial \dot{q_j}} = p_j)$ and the third one is equal to $\dot{p_i}$ because of the Lagrange equation.

The problem is that when I take the partial derivative of $H = \sum \dot{q_i}p_i - L$, I get:
$- \frac{\partial H}{\partial q_i} = - p_j \frac{\partial \dot{q_j}}{\partial q_i} - \dot{q_j} \frac{\partial p_j}{\partial q_i} + \frac{\partial L}{\partial q_i} = \dot{p_i}$
Because I derive a product.

Now, my second term is completely different (even the sign doesn't match). Why is that?

 

[voko]

Have a look here: https://www.physicsforums.com/threads/hamiltonian-mechanics-h-t.769278/#post-4843651

 

[Daaavde]

Sorry, I can't see how that answers the question.
There are no time derivatives here or maybe I overlooked something.

 

[voko]

Ignore the time part. Make sure you understand the technique that yields the differential of the Hamiltonian that only has the differentials of the canonical variables (and time). From that, the partial derivatives w.r.t. canonical variables follow easily.

 

[PJC01]

I would first like to clarify that the Hamiltonian equations are a set of equations that describe the evolution of a dynamical system in terms of its position and momentum variables. The proof of these equations involves the use of the Hamiltonian function, which is defined as H = \sum \dot{q_i}p_i - L, where L is the Lagrangian of the system.

In the provided content, the individual is attempting to prove the first Hamiltonian equation, which states that \frac{\partial H}{\partial q_i} = \dot{p_i}. However, there seems to be some confusion in the derivation of this equation.

The first step in the proof is to take the partial derivative of the Hamiltonian function with respect to the position variable q_i. This is correctly shown in the first part of the provided content. However, in the next step, the individual seems to have made a mistake by taking the partial derivative of the momentum variable p_j with respect to the position variable q_i. This is incorrect because the momentum variable is not a function of the position variable q_i.

The correct way to approach this is to use the chain rule to take the partial derivative of the Hamiltonian function with respect to the position variable q_i. This will give the desired result of \frac{\partial H}{\partial q_i} = - p_j \frac{\partial \dot{q_j}}{\partial q_i} + \frac{\partial L}{\partial q_i}.

The second part of the provided content also seems to have an error in the derivation. The individual has taken the partial derivative of the Hamiltonian function with respect to the momentum variable p_j, which is incorrect because the momentum variable is not a function of itself. The correct approach is to use the definition of the Hamiltonian function and the Lagrange equation, which states that \frac{\partial L}{\partial q_i} = \dot{p_i} - \frac{\partial p_i}{\partial t}. This will result in the desired result of \frac{\partial H}{\partial p_i} = \dot{q_i}.

In summary, the proof of the Hamiltonian equations involves careful application of the chain rule and the use of the Lagrange equation. It is important to be mindful of the variables that are being derived with respect to and to use the correct equations in the derivation process.