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One instance, where classical mechanics has to be treated with 'functional analysis' are infinite dimensional systems. The prototypical example is the Korteweg-de Vries equation $$u_t + u_{xxx} + 6 u u_x = 0$$ which a priori looks like a non-linear PDE. The key now is that it is completely integrable, which means that one can associate to an equivalent evolution for operators on Hilbert spaces. Define $$L(t) = - \frac{d^2}{dx^2} + u(x,t)$$ as an operator on $L^2(\mathbb{R})$. Then this operator obeys $$L_t = [P, L],$$ where $P$ is another operator, one can construct from $u$. (The specific form doesn't matter). The operators $P$ and $L$ are known as Lax Pair. (The $P$ stands for Peter not for Pair ☺ ). This is just the Heisenberg picture of quantum mechanics, so one can use the tools developed there, i.e. functional analysis, to investigate this equation. Of special importance is something known as scattering theory.

Just on a final point: KdV is a limit of Navier--Stokes, which is a classical system.

P.S.: In shameless self-promotion for some details on another system, the Toda Lattice, where it is easier to see that it is classical mechanics (one can write down the Hamiltonian easily), see here. I just made the post about KdV, since it is well-known.

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One instance, where classical mechanics has to be treated with 'functional analysis' are infinite dimensional systems. The prototypical example is the Korteweg-de Vries equation $$u_t + u_{xxx} + 6 u u_t u_x = 0$$ which a priori looks like a non-linear PDE. The key now is that it is completely integrable, which means that one can associate to an equivalent evolution for operators on Hilbert spaces. Define $$L(t) = - \frac{d^2}{dx^2} + u(x,t)$$ as an operator on $L^2(\mathbb{R})$. Then this operator obeys $$L_t = [P, L],$$ where $P$ is another operator, one can construct from $u$. (The specific form doesn't matter). The operators $P$ and $L$ are known as Lax Pair. (The $P$ stands for Peter not for Pair ). This is just the Heisenberg picture of quantum mechanics, so one can use the tools developed there, i.e. functional analysis, to investigate this equation. Of special importance is something known as scattering theory.

Just on a final point: KdV is a limit of Navier--Stokes, which is a classical system.

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One instance, where classical mechanics has to be treated with 'functional analysis' are infinite dimensional systems. The prototypical example is the Korteweg-de Vries equation $$u_t + u_{xxx} + 6 u u_t = 0$$ which a priori looks like a non-linear PDE. The key now is that it is completely integrable, which means that one can associate to an equivalent evolution for operators on Hilbert spaces. Define $$L(t) = - \frac{d^2}{dx^2} + u(x,t)$$ as an operator on $L^2(\mathbb{R})$. Then this operator obeys $$L_t = [P, L],$$ where $P$ is another operator, one can construct from $u$. (The specific form doesn't matter). This is just the Heisenberg picture of quantum mechanics, so one can use the tools developed there, i.e. functional analysis, to investigate this equation. Of special importance is something known as scattering theory.

Just on a final point: KdV is a limit of Navier--Stokes, which is a classical system.