TY - JOUR
T1 - Proof of the Quasi-Ergodic Hypothesis
JF - Proceedings of the National Academy of Sciences
JO - Proc Natl Acad Sci USA
SP - 70
LP - 82
M3 - 10.1073/pnas.18.1.70
VL - 18
IS - 1
AU - Neumann, J. v.
Y1 - 1932/01/01
UR - http://www.pnas.org/content/18/1/70.abstract
N2 - 1. The purpose of this note is to prove and to generalize the quasi-ergodic hypothesis of classical Hamiltonian dynamics1 (or “ergodic hypothesis,” as we shall say for brevity) with the aid of the reduction, recently discovered by Koopman,2 of Hamiltonian systems to Hilbert space, and with the use of certain methods of ours closely connected with recent investigations of our own of the algebra of linear transformations in this space.3 A precise statement of our results appears on page 79.We shall employ the notation of Koopman’s paper, with which we assume the reader to be familiar. The Hamiltonian system of k degrees of freedom corresponding with the Hamiltonian function H(q1, …, qk, p1, …, pk) defines a steady incompressible flow P → Pt = StP in the space Φ of the variables (q1, …, qk, p1, …, pk) or “phase-space,” and a corresponding steady conservative flow of positive density ρ in any invariant sub-space Ω ⊂ Φ (Ω being, e.g., the set of points in Φ of equal energy). The Hilbert space ℌ consists of the class of measurable functions f(P) having the finite Lebesgue integral ∫Ω|f|2ρdω, the “inner product”4 of any two of them (f, g) and “length” ‖f‖ being defined by the equations(f,g)=∫Ωfg¯ρdω; ‖f‖=(f,f).(1)The transformation Ut is defined as follows:Utf(P)=f(StP)=f(Pt);(2)obviously it has the …
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