刘维尔定理 Liouville Theorem: The phase space density for a Hamiltonian system is an invariant of the motion Or equivalently, the phase space volume occupied by the system is conserved Ietp≡f0(x,p)and=(,p}常数:加速器物理:相空间密度不变,或相 空间体积守恒。 of2D+y·f a2b,(1),∂(i,f2)_O2o,02 2D文 i D f at at at a 02H 02H 对于哈密顿系统f2D∞+f2D=2 D 连续性方程 +V·p=0 at +V·f2D=0 f2p o2p Op:. O2D x十 0 at a 2D刘维尔定理 Let   f 2D (x, px ) and v  x  , p  x  ( ) ( ) x x f p p p f p f x x f t f p p f x x f t f f v t f D x x x D x D D D x D D x D D D   +   +   +   +   =   +   +   +  =         2 2 2 2 2 2 2 2 2 2 0 2 2 2 2 2 2 =    −    =   +   x D x D x x D D x p H f x p H f p p f x x f   2 2 2 2 0 D D D D x x df f f f x p d t t x p    = + + =    0 2 = d t df D 对于哈密顿系统 连续性方程 +  = 0   v t   2 2 0 D D f f v t  +   =  Liouville Theorem: The phase space density for a Hamiltonian system is an invariant of the motion. Or equivalently, the phase space volume occupied by the system is conserved. 刘维尔定理：复变函数：有界的调和函数是 常数；加速器物理：相空间密度不变，或相 空间体积守恒