[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. -- 在细雨的午後 书页里悉哩哩地传来 " 周期3 = ? " 然而我知道 当我正在日耳曼深处的黑森林 继续发掘海森堡未曾做过的梦时 康德的诺言早已远离......... 远来的传教士静静地看着山涧不断反覆叠代自己的 过去 现在 和 未来 於是仅以 一颗量子浑沌 一本符号动力学 祝那发生在周一下午的新生 --

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