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The Liouville equation for the $N$ particle system, describes the time evolution of the phase space N-particle probability density, which you can also neatly rewrite with the Liouville operator: $f^{N}(t)= e^{-iLt}f^{N}(0).$ Now almost always we're interested in a smaller subset of only $n$ particles, for which then we have to define a reduced phase space ...

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The key point here is that, any dynamical system that is not completely integrable will exhibit chaotic regimes1. In other words not all orbits will lie on an invariant torus (Liouville's torus is the topological structure of a fully integrable system), in principle a chaotic system can even have closed stable periodic orbits (typical for regular/integrable ...

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First some terminology: A non-degenerate 2-form $\omega$ is called an almost symplectic structure. A closed 2-form $\omega$ is often called a pre-symplectic structure. If the 2-form $\omega$ is both non-degenerate and closed, it becomes a symplectic structure. In the non-degenerate case, the closedness condition $$\mathrm{d}\omega~=~0\tag{C}$$ is ...

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Comments to the question (v2): It seems that the question does not explain how a 'Hamiltonian' $H$ differs from a self-adjoint operator $A$ (presumably bounded from below). This would make OP's question a duplicate of the linked Phys.SE post. Perhaps a 'Hamiltonian' $H$ is also supposed to generate 'time'-evolution for some distinguished parameter $t$, ...

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If you know the propagator, ie. $\langle x'|e^{itH}|x\rangle\,,$ then you could differentiate with respect to time at $t=0$ to get $\langle x'|H|x\rangle\,.$ From this we have, using the resolution of the identity, $H|x\rangle=\int_{-\infty}^\infty dx'\, |x'\rangle\langle x'|H|x\rangle\,,$ from which we have $V(x)|x\rangle=\int_{-\infty}^\infty \, ... 2 The propagators themselves are not indicative for the form of the Lagrangian. They only provide information regarding the nature of the field - e.g. scalar / fermion / vector boson, etc (gravity metric?). Things that allude what the Lagrangian looks like are vertices / interactions. As a simple example: if you have a theory of field$\phi$with a 4-prong ... 0 I have a hunch that it might not be possible in the general case. Since you integrated over the fields already, it would be similar to trying to find the original integral from a real number. Also the basic path integral$Z[0] =1$no matter the field, for instance. 1 The Hamilton-Jacobi equation is a partial non-linear differential equation. A complete integral depends on$2n+1\$ arbitrary integration constants. The complete integral defines an integral surface on which there are characteristics that are solutions to a set of first order coupled ODEs. In this way we have related a 1st order, non-linear PDE to a set of ...

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The Hamilton-Jacobi equation is particularly good for describing a family of solutions with different unknown initial conditions with the unknowns being parameterized in a particularly nice way (relating to conserved quantities in a nice way) This is exactly why it is so much easier to relate to quantum mechanics, but it can be useful in other situations as ...

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People were excited about the ADM equations for two reasons. One was quantum gravity -- the equations give you a canonical coordinate and its momentum, so you might hope to promote these to quantum operators and be done with it. Decades later, little has come of this. The other reason is numerical relativity -- the study of the evolution of spacetime given ...

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You are confusing two definitions - closed system and conservation of energy. I'll clear them up for you. In classical dynamics a closed system is one where no force external to the system acts. In a closed system, the total energy, total momentum and total angular momentum must be conserved. This follows from Noether's theorem. If a has no interaction with ...

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Comments to the question (v2): First of all, be aware that there exist various different definitions of canonical transformations (CT) in the literature, cf. e.g. this Phys.SE post. What OP (v2) above refers to as a CT, we will in this answer call a symplectomorphism for clarity. What we in this answer will refer to as a CT, will just be a CT of type 2. It ...

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