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Liouville Arnol'd theorem state that given an integrable Hamiltonian system and denoting with $$M_a'=\{(q,p)\in\Gamma:f_i(q,p,t)=a_i\}$$ the connected component of the level sets of all of the first integral, then the restriction of the foundamental form on $M_a'$ $$p\cdot dq\Big|_{M_a'}$$ equal $dS(q,a)$ where $S$ is the generating functional of the ...

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i will try this one. A Hamiltonian system is (fully) integrable, which means there are $n$ ($n=$ number of dimensions) independent integrals of motion. What this states in essence (and intuitively) is that the hamiltonian system of dimension $n$ can be decomposed into a cartesian product of a set of $n$ independent sub-systems (e.g in action-angle ...

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REMARK. Perhaps I wrongly interpreted the question. I interpreted it as if were referred to the total volume of phase space. The answer is negative if the question regards general changes in time of topology of the total space of phases and if you do not impose any generic restriction on the topology of the spaces, like compactness (see the final ...

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Here's something I believe is a simple proof. Unfortunately it uses a little bit of cohomology. Consider the canonical 2-form in extended phase space $T^*M \times \mathbb{R}$ $$\omega = \sum_{i=1}^N dq_i \wedge dp_i - dH(\vec{q},\vec{p},t) \wedge dt ,$$ where $N = 2 \,dim(M)$. A function $f: M \to M$ is said to be a canonical transformation iff $f^* ... 2 By the main theorem of connectedness in general topology, continuous maps preserve connectedness. Time evolution of Hamiltonian systems preserves connectedness because it is continuous. I think it is independent of from Liouville's theorem, it just requires the proving Hamiltonian time evolution is continuous. This is just a formal way of restating ... 2 To grasp the relevant physics at a sloppy level, perhaps you simply need a few examples. You know a concept is commonly constructed by the manner you refer to it together with other concepts. Symmetry breaking usually results in ground state degeneracy and long range order. Order parameter field aids you in identifying degenerate sectors with the symmetries ... 3 The standard treatment of identical particles starts with one-particle states and then imposes symmetry conditions. This is kind of backwards. If you look at a formula like $$|x_1,x_2\rangle = \frac{1}{\sqrt 2}(|x_1\rangle | x_2\rangle \pm |x_1\rangle| x_2\rangle)$$ you are saying that the state is a linear combination of states where particle A is at$x_1\$ ...

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Incompleteness of a coordinate system is not a canonical definition as that, for instance, of geodesical (in)completeness. It simply means that the domain of the coordinate system does not cover the whole manifold (and perhaps there are several inequivalent extensions of the initial manifold represented by the given domain of the coordinate system). If a ...

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