| negative Jacobian -
07-29-2007, 03:20 PM
When canonical transformation is applied to the motion of a particle along a fixed circular path of radius R, the equation of constraint is R=sqrt(x+y) where x and y are coordinates of a Cartesian plane. Since there are two coordinates and one constraint, then 2-1=1, gives one degree of freedom. Similarly, this is also true for motion on the x-z and the y-z planes.
The one degree of freedom becomes the variable angular coordinate, q, of a generalized polar plane. R becomes the coordinate constant. Then the transformations are x=Rcosq and y=Rsinq. The Jacobian is the determinant of the matrix whose elements are partial derivatives of the source with respect to the destination coordinates such that by Liouville’s theorem the multi-dimensional volume integrals are invariance in phase space: J=¶(x,y)/¶(q,R). If R=1 then the Jacobian is -1. In theory only the absolute value validates Liouville’s theorem. In practice negative Jacobian implies orthogonality. Moreover in any generalized coordinate system of any dimension, orthogonality implies the existence of tangent bundles. On the other hand, it also indicates the existence of clockwise (CW) and counterclockwise (CCW) rotations. For a right-handed system positive Jacobian gives CW and negative Jacobian gives CCW rotations.
Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: ¶a(t)·¶r(t)=c²
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Re: negative Jacobian -
08-03-2007, 06:22 PM
Linear Mode
