How would I convert kinetic energy for one particle in six dimensions (x, y, z, and their time derivatives) from Cartesian to spherical coordinates? Note: #dotq = (delq)/(delt)#, and this is from a 1962 Statistical Mechanics textbook by Norman Davidson.
In Cartesian, #K = K(x,y,z,(delx)/(delt), (dely)/(delt), (delz)/(delt))# , and in spherical, #K = K(r,theta,phi, (delr)/(delt), (deltheta)/(delt), (delphi)/(delt))# .
For convenience, we set #dotq = (delq)/(delt)# .
Not sure if anyone actually needs this, but this was quite crazy to do, and I'm willing to share how I did it so that if anyone ever needs to do this... they can? There is a lot of nested chain rule and product rule!
In Cartesian,
For convenience, we set
Not sure if anyone actually needs this, but this was quite crazy to do, and I'm willing to share how I did it so that if anyone ever needs to do this... they can? There is a lot of nested chain rule and product rule!
1 Answer
For spherical coordinates (where
#x = rsinthetacosphi# ,
#y = rsinthetasinphi# ,
#z = rcostheta# .
To express
#K = 1/2m(dotx^2 + doty^2 + dotz^2)#
Let's focus on
#dotx = (delx)/(delt) = r(dotthetacosthetacosphi - dotphisinthetasinphi) + dotrsinthetacosphi#
#doty = (dely)/(delt) = r(dotphisinthetacosphi + dotthetacosthetasinphi) + dotrsinthetasinphi#
#dotz = (delz)/(delt) = -rdotthetasintheta + dotrcostheta#
Note that
#dotx^2 = [r(dotthetacosthetacosphi - dotphisinthetasinphi) + dotrsinthetacosphi]^2#
#= r^2(dotthetacosthetacosphi - dotphisinthetasinphi)^2 + dotr^2sin^2thetacos^2phi + 2rdotrsinthetacosphi(dotthetacosthetacosphi - dotphisinthetasinphi)#
#= r^2dottheta^2cos^2thetacos^2phi - 2r^2dotthetadotphisinthetacosthetasinphicosphi + r^2dotphi^2sin^2thetasin^2phi + dotr^2sin^2thetacos^2phi + 2rdotrdotthetasinthetacosthetacos^2phi - 2rdotrdotphisin^2thetasinphicosphi#
#= (r^2dottheta^2costheta + 2rdotrdotthetasintheta)costhetacos^2phi - (2r^2dotthetadotphisinthetacostheta + 2rdotrdotphisin^2theta) sinphicosphi + (r^2dotphi^2sin^2phi + dotr^2cos^2phi)sin^2theta#
Now, we repeat for
#doty^2 = r^2(dotphisinthetacosphi + dotthetacosthetasinphi)^2 + dotr^2sin^2thetasin^2phi + 2rdotrsinthetasinphi(dotphisinthetacosphi + dotthetacosthetasinphi)#
#= r^2dotphi^2sin^2thetacos^2phi + r^2dottheta^2cos^2thetasin^2phi + 2r^2dotthetadotphisinthetacosthetasinphicosphi + dotr^2sin^2thetasin^2phi + 2rdotrdotphisin^2thetasinphicosphi + 2rdotrdotthetasin^2phisinthetacostheta#
#= (r^2dottheta^2costheta + 2rdotrdotthetasintheta)costhetasin^2phi + (2r^2dotthetadotphisinthetacostheta + 2rdotrdotphisin^2theta)sinphicosphi + (r^2dotphi^2cos^2phi + dotr^2sin^2phi)sin^2theta#
There is hope, as some
#dotz^2 = (-rdotthetasintheta + dotrcostheta)^2#
#= dotr^2cos^2theta - 2rdotrdotthetacosthetasintheta + r^2dottheta^2sin^2theta#
And now, adding these together gives us
#(2K)/m = dotx^2 + doty^2 + dotz^2#
#= (r^2dottheta^2costheta + 2rdotrdotthetasintheta)costhetacos^2phi - cancel((2r^2dotthetadotphisinthetacostheta + 2rdotrdotphisin^2theta) sinphicosphi) + (r^2dotphi^2sin^2phi + dotr^2cos^2phi)sin^2theta + (r^2dottheta^2costheta + 2rdotrdotthetasintheta)costhetasin^2phi + cancel((2r^2dotthetadotphisinthetacostheta + 2rdotrdotphisin^2theta)sinphicosphi) + (r^2dotphi^2cos^2phi + dotr^2sin^2phi)sin^2theta + dotr^2cos^2theta - 2rdotrdotthetacosthetasintheta + r^2dottheta^2sin^2theta#
Regroup the
#=> (r^2dottheta^2costheta + 2rdotrdotthetasintheta)costhetacos^2phi + (r^2dotphi^2sin^2phi + dotr^2cos^2phi + r^2dotphi^2cos^2phi + dotr^2sin^2phi)sin^2theta + (r^2dottheta^2costheta + 2rdotrdotthetasintheta)costhetasin^2phi + dotr^2cos^2theta - 2rdotrdotthetacosthetasintheta + r^2dottheta^2sin^2theta#
Getting some
#=> r^2dottheta^2cos^2theta + 2rdotrdotthetasinthetacostheta+ (r^2dotphi^2 + dotr^2)sin^2theta + dotr^2cos^2theta - 2rdotrdotthetacosthetasintheta + r^2dottheta^2sin^2theta#
And some more happening with the first and last terms, and the third and fourth terms (upon multiplying the
#=> r^2dottheta^2sin^2theta + r^2dottheta^2cos^2theta + cancel(2rdotrdotthetasinthetacostheta) + r^2dotphi^2sin^2theta + dotr^2sin^2theta + dotr^2cos^2theta - cancel(2rdotrdotthetasinthetacostheta)#
#= dotr^2 + r^2dottheta^2 + r^2dotphi^2sin^2theta#
Finally, multiply by
#color(blue)(K = 1/2mdotr^2 + 1/2mr^2dottheta^2 + 1/2mr^2dotphi^2sin^2theta)#