## Myth of eternal return and Salamon-Zehnder theory

April 28, 2014 by zulfahmed

The actual universe, being a scaled 4-sphere, is compact and one might ask whether the mathematical results on periodic orbits of Hamiltonian systems for example of SalamonZehnderMorseTheory and others applies directly . The compactness of the configuration space in this case is nice but the classical phase space for a single particle is the cotangent bundle which is not compact. But since light speed is finite, dynamics is actually limited to a ‘c-ball bundle’ which is a compact manifold with boundary and so the mathematical theories for compact symplectic manifolds still does not apply directly. We can take the cue from the theory of convex optimizations and in particular the method it uses to produce ‘interior point methods’, which is to put a barrier to the boundary. Mathematically this means we could choose a metric that is complete on the cotangents space beyond the c-ball, say the 2c-ball. There are many other options for this problem.

Now what happens if we assume that the results for compact symplectic manifolds applies in an S4 universe and we also assume that classical physics is a correct description of nature? It says that reasonable Hamiltonian systems have plentiful periodic orbits where ‘reasonable’ can be described in a mathematical precise manner. Broadly it means that physical systems will have periodic orbit solutions at all scales — of course we have to take account of the issue of which systems can be modeled as classical systems. Thus these results can be considered quantitative claims regarding the ‘myth of eternal return’ in an infinite time universe.

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