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I am trying to self-learn the Hamiltonian and Lagrangian mechanics and I came across thoughts to which I could not find an answer therefore I would like to try and ask them here.

My questions are as follows:

First of all, consider the Lagrangian ($1$ degree of freedom) $L \left( q, \frac{d q}{d t}, t \right)$. It's given that $q \text{ and } \frac{d q}{d t}$ are independent variables in a specific moment $t$. The explanation I found is that knowing the velocity at a specific moment can't determine the position and vice versa, which in my opinion is an intuitive explanation but I tend to like the "mathematical” ones more. Thus, what I am searching for is an explanation using a mathematical approach (if possible).

Secondly, about the same question but this time it is about the Hamiltonian $H \left( q, p, t \right)$ and about $q$ and $p$.

Best regards.

banercat
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Nitay
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    I can't post a complete answer now, but look up the following terms: tangent bundle, cotangent bundle, and maybe jet bundle. Depending on the formulation chosen, these are the mathematical objects where these variables live as variables independent from one another. – Bence Racskó Sep 19 '23 at 20:16
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    possible duplicate: https://physics.stackexchange.com/q/307794/50583 and its linked questions – ACuriousMind Sep 19 '23 at 20:52
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    I posted an answer in the spirit of your question in https://physics.stackexchange.com/q/759523/. Here the mathematical approach to Lagrangian and Hamiltonian mechanics is explained, although I assumed Lagrangian and Hamiltonian does not depend on time explicitly. – ProphetX Sep 19 '23 at 21:39

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