Boolean Algebras on a Classical Phase Space

Abstract

We begin with a brief account of the usual description of a classical mechanical system with a finite number of degrees of freedom. Associated with such a system there is an integer n, and an open set M of the n-dimensional space R ⁿ of n-tuples (x ₁, x ₂ , ... , x _n) of real numbers. n is called the number of degrees of freedom of the system. The points of M represent the possible configurations of the system. A state of the system at any instant of time is specified completely by giving a 2n-tuple (x ₁, x ₂, ... , x _n , p ₁, ... , p _n ) such that (x ₁, ... , x _n ) represents the configuration and (p ₁, ... , p _n ) the momentum vector, of the system at that instant of time. The possible states of the system are thus represented by the points of the open set M × R ⁿ of R ²ⁿ. The law of evolution of the system is specified by a smooth function H on M × R ⁿ, called the Hamiltonian of the system. If (x ₁ (t), ... , x _n (t), p ₁ (t), ... , p _n (t)) represents the state of the system at time t, then the functions x _i (·), p _i (·), i =1, 2, ... , n, satisfy the well known differential equations:

EquationSource% MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % WGKbGaamiEamaaBaaaleaacaaIXaaabeaaaOqaaiaadsgacaWG0baa % aiabg2da9maalaaabaGaeyOaIyRaamisaaqaaiabgkGi2kaadchada % WgaaWcbaGaaGymaaqabaaaaOGaaiilaiaaywW7caWGPbGaeyypa0Ja % aGymaiaacYcacaaIYaGaaiilaiaac6cacaGGUaGaaiOlaiaacYcaca % WGUbGaaiilaaaa!4CC2! \[\frac{{d{x_1}}} {{dt}} = \frac{{\partial H}} {{\partial {p_1}}},\quad i = 1,2,...,n,\] $$

((1))

EquationSource% MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % WGKbGaamiCamaaBaaaleaacaaIXaaabeaaaOqaaiaadsgacaWG0baa % aiabg2da9iabgkHiTmaalaaabaGaeyOaIyRaamisaaqaaiabgkGi2k % aadIhadaWgaaWcbaGaaGymaaqabaaaaOGaaiilaiaaywW7caWGPbGa % eyypa0JaaGymaiaacYcacaaIYaGaaiilaiaac6cacaGGUaGaaiOlai % aacYcacaWGUbaaaa!4CFF! \[\frac{{d{p_1}}} {{dt}} = - \frac{{\partial H}} {{\partial {x_1}}},\quad i = 1,2,...,n\] $$

((1))