Structure and Interpretation of Classical Mechanics
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A bead of mass m moves without friction on a triaxial ellipsoidal surface. In rectangular coordinates the surface satisfies
for some constants a, b, and c. Identify suitable generalized coordinates, formulate a Lagrangian, and find Lagrange’s equations.
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[guess]
The generalized coordinates:
where
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(define ((F->C F) local) (->local (time local) (F local) (+ (((partial 0) F) local) (* (((partial 1) F) local) (velocity local))))) ; (define ((e->r a b c) local) (let ((q (coordinate local))) (let ((theta (ref q 0)) (phi (ref q 1))) (let ((x (* a (sin theta) (cos phi))) (y (* b (sin theta) (sin phi))) (z (* c (cos theta)))) (up x y z))))) ; (define ((L-rect m) local) (let ((q (coordinate local)) (v (velocity local))) (* 1/2 m (square v)))) (define (L-e m a b c) (compose (L-rect m) (F->C (e->r a b c)))) ; (show-expression ((L-e 'm 'a 'b 'c) (->local 't (up 'theta 'phi) (up 'thetadot 'phidot)))) ; (show-expression (((Lagrange-equations (L-e 'm 'a 'b 'c)) (up (literal-function 'theta) (literal-function 'phi))) 't))
[guess]
— Me@2021-03-01 06:22:50 PM
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