[2.4] What is phase space? Phase space is the collection of possible states of a dynamical system. A ph ase space can be finite (e.g. for the ideal coin toss, we have two states he ads and tails), countably infinite (e.g. state variables are integers), or u ncountably infinite (e.g. state variables are real numbers). Implicit in the notion is that a particular state in phase space specifies the system compl etely; it is all we need to know about the system to have complete knowledge of the immediate future. Thus the phase space of the planar pendulum is two -dimensional, consisting of the position (angle) and velocity. According to Newton, specification of these two variables uniquely determines the subsequ ent motion of the pendulum. Note that if we have a non-autonomous system, where the map or vector field depends explicitly on time (e.g. a model for plant growth depending on solar flux), then according to our definition of phase space, we must include tim e as a phase space coordinate--since one must specify a specific time (e.g. 3PM on Tuesday) to know the subsequent motion. Thus dz/dt = F(z,t) is a dyna mical system on the phase space consisting of (z,t), with the addition of th e new dynamics dt/dt = 1. The path in phase space traced out by a solution of an initial value problem is called an orbit or trajectory of the dynamical system. If the state vari ables take real values in a continuum, the orbit of a continuous-time system is a curve, while the orbit of a discrete-time system is a sequence of poin ts. -- 在細雨的午後 書頁裡悉哩哩地傳來 " 週期3 = ? " 然而我知道 當我正在日耳曼深處的黑森林 繼續發掘海森堡未曾做過的夢時 康德的諾言早已遠離......... 遠來的傳教士靜靜地看著山澗不斷反覆疊代自己的 過去 現在 和 未來 於是僅以 一顆量子渾沌 一本符號動力學 祝那發生在週一下午的新生 --

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