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  • $\begingroup$ Are you sure about this "Using the second time derivative of g we have $\dot{x}^2+\dot{y}^2+\dot{z}^2=−(\ddot{x}x+\ddot{y}y+\ddot{z}z)$? $\endgroup$
    – basics
    Commented Jun 22 at 10:15
  • $\begingroup$ well after calling myself completely into question, yes. The first derivative being $2\dot xx + 2\dot yy + 2\dot zz$ $\endgroup$
    – Not a chance
    Commented Jun 22 at 10:40
  • $\begingroup$ Wouldn't that require $\ddot{g}=0$? Is that condition true? Also, why solve it in Cartesian instead of the more straight-forward spherical coordinates? $\endgroup$
    – Kyle Kanos
    Commented Jun 22 at 13:04
  • $\begingroup$ FWIW, I suspect there's something screwy in your derivation as your EOM (your last equation) does not appear to line up with expected equations (cf. this Physics.SE post). $\endgroup$
    – Kyle Kanos
    Commented Jun 22 at 13:15