[2.5] What is a degree of freedom?
The notion of "degrees of freedom" as it is used for Hamiltonian systems mea
ns one canonical conjugate pair, a configuration, q, and its conjugate momen
tum p. Hamiltonian systems (sometimes mistakenly identified with the notion
of conservative systems) always have such pairs of variables, and so the pha
se space is even dimensional.
In the study of dissipative systems the term "degree of freedom" is often us
ed differently, to mean a single coordinate dimension of the phase space. Th
is can lead to confusion, and it is advisable to check which meaning of the
term is intended in a particular context.
Those with a physics background generally prefer to stick with the Hamiltoni
an definition of the term "degree of freedom." For a more general system the
proper term is "order" which is equal to the dimension of the phase space.
Note that a dynamical system with N d.o.f. Hamiltonian nominally moves in a
2N dimensional phase space. However, if H(q,p) is time independent, then ene
rgy is conserved, and therefore the motion is really on a 2N-1 dimensional e
nergy surface, H(q,p) = E. Thus e.g. the planar, circular restricted 3 body
problem is 2 d.o.f., and motion is on the 3D energy surface of constant "Jac
obi constant." It can be reduced to a 2D area preserving map by Poincare sec
tion (see [2.6]).
If the Hamiltonian is time dependent, then we generally say it has an additi
onal 1/2 degree of freedom, since this adds one dimension to the phase space
. (i.e. 1 1/2 d.o.f. means three variables, q, p and t, and energy is no lon
ger conserved).
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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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