Lagrangian vs Newtonian formalism

Main Question or Discussion Point

I have seen the lagrange equations derived from newtons laws in the special case, where forces were derivable from a potential.
Now with the introduction of hamiltons principle, I think my book wants to say this: We can always find a lagrangian such that the principle of least action holds. This lagrangian describes the entire system and the terms T and U, the potential and kinetic energy, need not be what we in Newtonian dynamics like to think of them of as - at least that is how I understood and I think it's correct since the lagrangian for an electrodynamic system for instance contains the term A*v (*=dot), which I don't see how you would interpret in terms of the classical definition of potential energy.
Now that means that the principle of action is regarded as a deep principle which is always true. Though we know, that the newtonian formalism is also always true for any coordinate frame and arbitrarily complicated systems. So how do you show that the two formalisms are equivalent? And how do you know that the lagrangian for a classically dynamic system always just contain the ordinary potential and kinetic energy of the system - unlike the electromagnetic case, where a new term has to be introduced.
Hope this made at least somewhat sense

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You need to understand the concept of generalised coordinates to fully appreciate Lagrange methods.

You should also be wary of the idea that potential refers standard potential energy. This is a special case of a more general statement.

The work function = The sum of the products of the generalised forces and the generalised displacements

A potential is only possible (is a necessary and suffiicient condition for) with a conservative system.

I should think I understand them. I just don't understand how the term A*v is to be interpreted in the electromagnetic lagrangian. Surely it has nothing to do with potential energy?
You can show that principle of least action -> euler lagrange equation -> independent for each generalized coordinate. But how do we know that the Lagrangian is always L = T - V? Well okay for a mechanical system I have seen the derivation that d'Alemberts principle -> L = T - V. But that only holds when the constraint forces do no work and when the potential is derivable from a scalar which only depends on (q1...qn). What if the potential is velocity dependent, will it then still be what we identify with the classical potential energy?

Have you looked at this thread?

Generalised forces and generalised displacements include electromagnetism and other agents.

The generalised displacement is of suitable dimensions to make the product equal to work in the work equation.

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