false_vacuum comments on Classical Configuration Spaces - Less Wrong

18 Post author: Eliezer_Yudkowsky 15 April 2008 08:40AM

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Comment author: false_vacuum 25 October 2010 12:34:24PM *  2 points [-]

This is exactly right except that the space in which Liouville's theorem holds is called phase space. Phase space is the cotangent bundle over configuration space; i.e., if the configuration space is an n-dimensional manifold M, then for every point in M there is a copy of an n-dimensional vector space. These n-vectors* represent momenta, and both a configuration and a momentum are necessary to uniquely specify a state of a classical (Hamiltonian) system.


* More precisely, they are one-forms--linear functions of n-vectors; i.e. they eat n-vectors and spit out scalars. One-forms are also called covariant vectors, whence the other kind are called contravariant. They are dual to each other (for a given n), and thus (contravariant) vectors can equivalently be considered linear functions of one-forms instead.

I think it would be nice if the post were edited to reflect this distinction. It wouldn't take much effort; just a sentence inserted at the point where it switches from talking about configuration space to talking about phase space, and appropriate tweaks to a few subsequent sentences. The Wikipedia article Configuration space links here, by the way.