Ex 1.18 Bead on a triaxial surface, 2

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

\displaystyle{\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1}

for some constants a, b, and c. Identify suitable generalized coordinates, formulate a Lagrangian, and find Lagrange’s equations.

~~~

[guess]

The generalized coordinates:

\displaystyle{\begin{aligned}  x &= a \sin(\theta )\cos(\varphi),\\  y &= b \sin(\theta )\sin(\varphi),\\  z &= c \cos(\theta),  \end{aligned}}

where

{\displaystyle 0\leq \theta \leq \pi ,\qquad 0\leq \varphi <2\pi .}

.

{\displaystyle {\begin{aligned} \dot x &= a \dot \theta \cos(\theta) \cos(\varphi) - a \dot \varphi\sin(\theta ) \sin(\varphi) \\ \dot y &= b \dot \theta \cos(\theta) \sin(\varphi) + b \dot \varphi \sin(\theta) \cos(\varphi) \\ \dot z &= - c \dot \theta \sin(\theta) \end{aligned}}}


(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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2021.03.02 Tuesday (c) All rights reserved by ACHK