Hamiltonian Method: Getting Final Equation of Motion

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The discussion centers on clarifying the process of deriving the final equation of motion using Lagrangian and Hamiltonian mechanics. The user understands the transition from Lagrangian to Hamiltonian but is confused about the necessity of taking time derivatives to arrive at the final equations of motion. They seek an explanation for this step, noting that some examples incorporate time derivatives at the end. The user references a specific resource for further clarification. Understanding the role of time derivatives is crucial for completing the derivation of equations of motion in this context.
oldspice1212
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Hey, I was hoping someone could clear this up for me. When using this method, how do you get the final equation of motion, that's where I am confused.

So I know I start off using Lagrangian (T - U) -> momentum (partial L/ partial q dot) -> Hamiltonian T+U, and then using the hamiltonian equation's of motion, we find the equations ( I can do this part). This is just a quick summary of the procedure. But, for some examples, they are taking time derivatives at the end for some equations, and finding the equation of motion. I don't understand why and where they come from?
Could someone please explain, thanks.
 
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