Daily Archives: September 15, 2023

Ex 1.30 Driven spherical pendulum, 2

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Structure and Interpretation of Classical Mechanics

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What symmetry(ies) can you have?

Find coordinates that express the symmetry.

What is conserved?

Give [the] analytic expression(s) for the conserved quantity(ies).

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[guess]

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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 (L-driven m R qs U)
  (compose
   (L-rect m R qs U)
   (F->C (sf->rf qs))))

(let* ((U (U-gravity 'g 'm))
       (xs (lambda (t) 0))
       (ys (lambda (t) 0))
       (zs (literal-function 'z_s))
       (qs (up xs ys zs))
       (L (L-driven 'm 'R qs U))
       (q (up (literal-function 'r)
              (literal-function 'theta)
              (literal-function 'phi)
              (literal-function 'lambda))))
  (show-expression
   ((compose L (Gamma q)) 't)))

(+ (* 1/2 m (expt (r t) 2)
            (expt (sin (theta t)) 2)
            (expt ((D phi) t) 2))
   (* -1 m ((D z_s) t)
           (r t)
           (sin (theta t))
           ((D theta) t))
   (* 1/2 m (expt (r t) 2)
            (expt ((D theta) t) 2))
   (* -1 g m (cos (theta t)) (r t))
   (* m ((D r) t)
        ((D z_s) t)
        (cos (theta t)))
   (* (expt R 2) (lambda t))
   (* -1 g m (z_s t))
   (* 1/2 m (expt ((D r) t) 2))
   (* 1/2 m (expt ((D z_s) t) 2))
   (* -1 (lambda t) (expt (r t) 2)))

\displaystyle{ \begin{aligned} L &= \frac{1}{2} m \left( \dot r^2 + r^2 \dot \theta^2 + r^2 (\sin \theta)^2 \dot \phi^2 + \dot z_s^2 \right) \\ & - m g \left( r \cos \theta + z_s(t) \right) - m \dot z_s \left( r \dot \theta \sin \theta - \dot r \cos \theta \right) \\ & - \lambda(t) (r^2 - R^2) \\ \end{aligned} }

(let* ((U (U-gravity 'g 'm))
       (xs (lambda (t) 0))
       (ys (lambda (t) 0))
       (zs (literal-function 'z_s))
       (qs (up xs ys zs))
       (L (L-driven 'm 'R qs U))
       (q (up (literal-function 'r)
              (literal-function 'theta)
              (literal-function 'phi)
              (literal-function 'lambda))))
  (show-expression
   (((Lagrange-equations L) q) 't)))


(let* ((U (U-gravity 'g 'm))
       (xs (lambda (t) 0))
       (ys (lambda (t) 0))
       (zs (literal-function 'z_s))
       (qs (up xs ys zs))
       (L (L-driven 'm 'R qs U))
       (q (up (literal-function 'r)
              (literal-function 'theta)
              (literal-function 'phi)
              (literal-function 'lambda))))
  (show-expression
   ((compose (Lagrangian->energy L) (Gamma q)) 't)))


(let* ((U (U-gravity 'g 'm))
       (xs (lambda (t) 0))
       (ys (lambda (t) 0))
       (zs (literal-function 'z_s))
       (qs (up xs ys zs))
       (L (L-driven 'm 'R qs U))
       (q (up (literal-function 'r)
              (literal-function 'theta)
              (literal-function 'phi)
              (literal-function 'lambda))))
  (show-expression
   ((compose ((partial 1) L) (Gamma q)) 't)))

The \displaystyle{\varphi} component of the force is zero because \displaystyle{\varphi} does not appear in the Lagrangian (it is a cyclic variable). The corresponding momentum component is conserved. Compute the momenta: 

(let* ((U (U-gravity 'g 'm))
       (xs (lambda (t) 0))
       (ys (lambda (t) 0))
       (zs (literal-function 'z_s))
       (qs (up xs ys zs))
       (L (L-driven 'm 'R qs U))
       (q (up (literal-function 'r)
              (literal-function 'theta)
              (literal-function 'phi)
              (literal-function 'lambda))))
  (show-expression
   ((compose ((partial 2) L) (Gamma q)) 't)))

[guess]

— Me@2023-08-20 05:02:09 PM

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