chez-scmutils/scmutils/mechanics/Hamiltonian-evolution.scm

#| -*-Scheme-*-

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    Institute of Technology

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it under the terms of the GNU General Public License as published by
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WITHOUT ANY WARRANTY; without even the implied warranty of
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#|
(define ((T-pend m l g ys) local)
  (let ((t (time local))
        (theta (coordinate local))
        (thetadot (velocity local)))
    (let ((ysdot (D ys)))
      (* 1/2 m
         (+ (square (* l thetadot))
            (square (ysdot t))
            (* 2 (ysdot t) l (sin theta) thetadot))))))

(define ((V-pend m l g ys) local)
  (let ((t (time local))
        (theta (coordinate local)))
    (* m g (- (ys t) (* l (cos theta))))))

(define L-pend (- T-pend V-pend))

(define ((periodic-drive amplitude frequency phase) t)
  (* amplitude (cos (+ (* frequency t) phase))))

(define (L-periodically-driven-pendulum m l g a omega)
  (let ((ys (periodic-drive a omega 0)))
    (L-pend m l g ys)))

(show-expression
 ((Lagrangian->Hamiltonian
    (L-periodically-driven-pendulum 'm 'l 'g 'a 'omega))
  (->H-state 't 'theta 'p_theta)))
(+
 (* -1/2
    (expt a 2)
    m
    (expt omega 2)
    (expt (cos theta) 2)
    (expt (sin (* omega t)) 2))
 (* a g m (cos (* omega t)))
 (/ (* a omega p_theta (sin theta) (sin (* omega t))) l)
 (* -1 g l m (cos theta))
 (/ (* 1/2 (expt p_theta 2)) (* (expt l 2) m)))

(define (H-pend-sysder m l g a omega)
  (Hamiltonian->state-derivative
    (Lagrangian->Hamiltonian
      (L-periodically-driven-pendulum m l g a omega))))

;;; for driven-pendulum-phase-space

(define window (frame -pi pi -10.0 10.0))

(start-gnuplot "pendulum-2x")

(define ((monitor-p-theta win) state)
  (let ((q ((principal-value pi) (coordinate state)))
	(p (momentum state)))
    (plot-point win q p)))
  
(let ((m 1.)		                ;m=1kg
      (l 1.)				;l=1m
      (g 9.8)				;g=9.8m/s$^2$
      (A 0.1)				;A=1/10 m
      (omega (* 2 (sqrt 9.8))))
  ((evolve H-pend-sysder m l g A omega)
   (up 0.0				;t$_0$=0
       1.0				;theta$_0$=1 radian
       0.0)				;thetadot$_0$=0 radians/s
   (monitor-p-theta window)
   0.01					;step between plotted points
   100.0				;final time
   1.0e-12))

(stop-gnuplot)

(graphics-clear window)

;;; for driven-pend-nuniq1 and 2
(let ((m 1.)		                ;m=1kg
      (l 1.)				;l=1m
      (g 9.8)				;g=9.8m/s$^2$
      (A 0.1)				;A=1/10 m
      (omega (* 2 (sqrt 9.8))))
  ((evolve H-pend-sysder m l g A omega)
   (up 0.0				;t$_0$=0
       1.0				;theta$_0$=1 radian
       0.0)				;thetadot$_0$=0 radians/s
   (monitor-p-theta window)
   0.01					;step between plotted points
   10.0					;final time
   1.0e-12))

(define ((monitor-pprime-theta mlA omega win) state)
  (let ((t (time state))
	(q ((principal-value pi) (coordinate state)))
	(p (momentum state)))
    (plot-point 
     win
     q 
     (+ p (* mlA omega (sin q) (sin (* omega t)))))))

(let ((m 1.)		                ;m=1kg
      (l 1.)				;l=1m
      (g 9.8)				;g=9.8m/s$^2$
      (A 0.1)				;A=1/10 m
      (omega (* 2 (sqrt 9.8))))
  ((evolve H-pend-sysder m l g A omega)
   (up 0.0				;t$_0$=0
       1.0				;theta$_0$=1 radian
       0.0)				;thetadot$_0$=0 radians/s
   (monitor-pprime-theta (* m l A) omega window)
   0.01					;step between plotted points
   10.0					;final time
   1.0e-12))

(graphics-close window)
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