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When I was studying Classical Mechanics, particularly Lagrangian formulation and Hamiltonian formulation. I always wondering how to understand the meaning of independence of parameters used of describing motion of object? For instance, we have the generalized coordinates and generalized velocities to be independent variables of $L$ in Lagrangian formulation, on the other hand, we have generalized coordinates and canonical momenta to be independent variables of $H$ in Hamiltonian formulation.

I have read through many related explanations on stack; however, I still can't see through it with physical points of view, that is, I sometimes think that I am reading math but not physics. I need the explanation from mathmetics aspect (like manifold it belongs to or so) and the reason why we put some many emphases on canonical transformation (it seems like math to me).

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  • $\begingroup$ I like Frederic Schuller's quip about this (paraphrasing): "If the dynamics depended on our coordinate system, physics would be called telekinesis" $\endgroup$
    – Johnny
    Aug 20, 2023 at 18:40

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Its very easy. While the solution of the equation of motions yields $v=\frac{dx}{dt}$ in the experiment, preparing the starting point and the velocity to start with, are two independent sets of variables.

This fits neatly with the mathematics of Newtons equation of motion, that are second order.

As a simple fact, with the substitiuon $v=dx/dt$, they are systems of 6 equations for six functions of the time with six free constants of integration.

With the definition of $p=\partial_v L(x,v)$, the same is true for the set $p,x$,

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You are correct. Coordinates are mathematical objects: they don't exist in reality. Mathematics is a human invention: mathematical objects exist only in human minds.

This is a dirty secret of physics. It's claimed that mathematics is "unreasonably effective" as a tool for understanding the phenomena. The claim is nonsense. In Newtonian orbits, even something as simple as the three body problem is intractable. Then we get to GR, which the theorists all say is "beautiful", yet it makes even the two body problem intractable. Numerical methods are more widely applicable than symbolic methods, but they scale so badly that supercomputers are often needed for modeling of simple phenomena.

The things that mathematics does are an essential part of physics, but you are wise to seek more of a connection to the fundamental phenomena.

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