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I will yet add another formalism: Lets start with the hamiltonian form of Hamilton's Principle. Let $c: \mathbb R \longrightarrow T^*Q; t\mapsto (q(t),p(t))$ be the trajectory of a particle in the phase space of the configuration space $Q$, we define a subset of $Q$, $C$, where no contac occur between the particles, and $\partial C$ is te set of ponts wer ...

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Building on the responses from ACuriousMind and Gennaro Tedesco, I will make an attempt to provide a satisfactory, though not mathematically rigorous, answer. Question: Does there exist a nontrival non-Legendre transformation T such that the function defined by F(q,p,t)=T[L(q,q˙,t)] contains the full dynamics of the system? Yes, any invertible ...

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Question: Does there exist a nontrival non-Legendre transformation T such that the function defined by F(q,p,t)=T[L(q,q˙,t)] contains the full dynamics of the system? Answer: any function that produces the equations of motion under some sort of rules that you state is an allowed function to describe the dynamics. In particular any function that you can ...

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In principle one can always write down the differential equations for a system of $N$ particles and attempt to solve them: as you pointed out, there is no general solution if usually $N>2$. As such, nevertheless the need to describe features of general systems remains. The key point here is understanding that, as a matter of fact, whenever dealing with ...

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