Ex 1.18 Bead on a triaxial surface

Structure and Interpretation of Classical Mechanics


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.



The generalized coordinates:

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


{\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 theta (literal-function 'theta))

(define phi (literal-function 'phi))


(define (x t) (* 'a (sin (theta t)) (cos (phi t))))

((D x) 't)

(show-expression ((D x) 't))


(define (y t) (* 'b (sin (theta t)) (sin (phi t))))

(show-expression (y 't))

((D y) 't)

(show-expression ((D y) 't))


— Me@2021-02-16 07:20:25 AM



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