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chapter1.scm
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(define
( (L-free-particle mass) local)
(let
( (v (velocity local)))
(* 1/2 mass (dot-product v v)) ) )
(define q
(up
(literal-function 'x)
(literal-function 'y)
(literal-function 'z) ) )
(define demo1a (q 't))
(define demo1b ((D q) 't))
(define demo1c ((Gamma q) 't))
(define demo1d
( (compose
(L-free-particle 'm)
(Gamma q) )
't ) )
;(show-expression demo1d)
(define
(Lagrangian-action L q t1 t2)
(definite-integral (compose L (Gamma q)) t1 t2) )
(define
(test-path t)
(up
(+ (* 4 t) 7)
(+ (* 3 t) 5)
(+ (* 2 t) 1) ) )
(define demo1e (Lagrangian-action (L-free-particle 3.0) test-path 0.0 10.0))
(define
( (make-eta nu t1 t2) t)
(*
(- t t1)
(- t t2)
(nu t) ) )
(define
( (varied-free-particle-action mass q nu t1 t2) eps )
(let
( (eta (make-eta nu t1 t2)))
(Lagrangian-action
(L-free-particle mass)
(+ q (* eps eta))
t1
t2 ) ) )
(define demo1f
( (varied-free-particle-action 3.0 test-path (up sin cos square) 0.0 10.0) 0.001 ) )
(define demo1g
(minimize (varied-free-particle-action 3.0 test-path (up sin cos square) 0.0 10.0) -2.0 1.0) )
(define
( (parametric-path-action Lagrangian t0 q0 t1 q1) qs)
(let
( (path (make-path t0 q0 t1 q1 qs)))
(Lagrangian-action Lagrangian path t0 t1) ) )
(define
(find-path Lagrangian t0 q0 t1 q1 n)
(let
( (initial-qs (linear-interpolants q0 q1 n)))
(let
( (minimizing-qs (multidimensional-minimize (parametric-path-action Lagrangian t0 q0 t1 q1) initial-qs)))
(make-path t0 q0 t1 q1 minimizing-qs) ) ) )
(define
( (L-harmonic m k) local)
(let
( (q (coordinate local))
(v (velocity local)) )
(- (* 1/2 m (square v))
(* 1/2 k (square q)) ) ) )
(define q (find-path (L-harmonic 1.0 1.0) 0. 1. :pi/2 0. 3))
;(define win2 (frame 0. :pi/2 0. 1.2))
(define win2 #f)
(define
( (parametric-path-action Lagrangian t0 q0 t1 q1) intermediate-qs)
(let
( (path (make-path t0 q0 t1 q1 intermediate-qs)))
(if win2
(begin
(graphics-clear win2)
(plot-function win2 path t0 t1 (/ (- t1 t0) 100)) ) )
(Lagrangian-action Lagrangian path t0 t1) ) )
(define demo1h
(find-path (L-harmonic 1. 1.) 0. 1. :pi/2 0. 2) )
(define
((Lagrange-equations Lagrangian) q)
(- (D (compose ((partial 2) Lagrangian) (Gamma q)))
(compose ((partial 1) Lagrangian) (Gamma q)) ) )
(define
(test-path t)
(up
(+ (* 'a t) 'a0)
(+ (* 'b t) 'b0)
(+ (* 'c t) 'c0) ) )
(define demo1i (((Lagrange-equations (L-free-particle 'm)) test-path) 't))
;(print-expression demo1i)
(define demo1j
( ( (Lagrange-equations (L-free-particle 'm))
(literal-function 'x) ) 't ) )
;(show-expression demo1j)
(define
(proposed-solution t)
(* 'A (cos (+ (* 'omega t) 'phi))) )
(define demo1k
(((Lagrange-equations (L-harmonic 'm 'k)) proposed-solution) 't) )
;(show-expression demo1k)
(define
((L-central-polar m V) local)
(let
( (q (coordinate local))
(qdot (velocity local)) )
(let
( (r (ref q 0))
(phi (ref q 1))
(rdot (ref qdot 0))
(phidot (ref qdot 1)) )
(- (* 1/2 m
(+ (square rdot)
(square (* r phidot)) ) )
(V r) ) ) ) )
(define
((gravitational-energy G m1 m2) r)
(- (/ (* G m1 m2) r)) )
(define
((L-uniform-acceleration m g) local)
(let
( (q (coordinate local))
(v (velocity local)) )
(let
( (y (ref q 1)))
(- (* 1/2 m (square v))
(* m g y) ) ) ) )
(define demo1l
( ((Lagrange-equations (L-uniform-acceleration 'm 'g))
(up
(literal-function 'x)
(literal-function 'y) ) ) 't ) )
;(show-expression demo1l)
(define
((L-central-rectangular m U) local)
(let
( (q (coordinate local))
(v (velocity local)) )
(- (* 1/2 m (square v))
(U (sqrt (square q))) ) ) )
(define demo1m
( ( (Lagrange-equations (L-central-rectangular 'm (literal-function 'U)))
(up
(literal-function 'x)
(literal-function 'y) ) )
't ) )
;(show-expression demo1m)
(define
((L-central-polar m U) local)
(let
( (q (coordinate local))
(qdot (velocity local)) )
(let
( (r (ref q 0))
(phi (ref q 1))
(rdot (ref qdot 0))
(phidot (ref qdot 1)) )
(- (* 1/2 m
(+ (square rdot)
(square (* r phidot)) ) )
(U r) ) ) ) )
(define demo1n
( ((Lagrange-equations (L-central-polar 'm (literal-function 'U)))
(up
(literal-function 'r)
(literal-function 'phi) ) ) 't ) )
;(show-expression demo1n)
(define
( (F->C F) local)
(up
(time local)
(F local)
(+ (((partial 0) F) local)
(* (((partial 1) F) local)
(velocity local) ) ) ) )
(define
(p->r local)
(let
( (polar-tuple (coordinate local)))
(let
( (r (ref polar-tuple 0))
(phi (ref polar-tuple 0)) )
(let
( (x (* r (cos phi)))
(y (* r (sin phi))) )
(up x y) ) ) ) )
(define demo1o
(velocity
( (F->C p->r)
(up 't
(up 'r 'phi)
(up 'rdot 'phidot) ) ) ) )
;(show-expression demo1o)
(define
(L-central-polar m U)
(compose
(L-central-rectangular m U)
(F->C p->r) ) )
(define demo1p
( (L-central-polar 'm (literal-function 'U))
(up 't
(up 'r 'phi)
(up 'rdot 'phidot) ) ) )
;(show-expression demo1p)
(define
((L-free-rectangular m) local)
(let
( (vx (ref (velocities local) 0))
(vy (ref (velocities local) 1)) )
(* 1/2 m (+ (square vx) (square vy))) ) )
(define
(L-free-polar m)
(compose
(L-free-rectangular m)
(F->C p->r) ) )
(define
((F Omega) local)
(let
( (t (time local))
(r (ref (coordinates local) 0))
(theta (ref (coordinates local) 1)) )
(up r (+ theta (* Omega t))) ) )
(define
(L-rotating-polar m Omega)
(compose
(L-free-polar m)
(F->C (F Omega)) ) )
(define
(L-rotating-rectangular m Omega)
(compose
(L-rotating-polar m Omega)
(F->C r->p) ) )
(define demo1q
( (L-rotating-rectangular 'm 'Omega)
(up 't
(up 'x_r 'y_r)
(up 'xdot_r 'ydot_r) ) ) )
(define demo1r
( ( (Lagrange-equations (L-rotating-rectangular 'm 'Omega))
(up
(literal-function 'x_r)
(literal-function 'y_r) ) )
't ) )
(define
( (T-pend m l g ys) local)
(let*
( (t (time local))
(theta (coordinate local))
(thetadot (velocity local))
(vys (D ys))
(vyst (vys t)) )
(* 1/2 m
(+ (square (* l thetadot))
(square vyst)
(* 2 l vyst thetadot (sin theta)) ) ) ) )
(define
( (V-pend m l g ys) local)
(let*
( (t (time local))
(theta (coordinate local))
(yst (ys t)) )
(* m g (- yst (* l (cos theta)))) ) )
(define L-pend (- T-pend V-pend))
(define demo1s
( ( (Lagrange-equations (L-pend 'm 'l 'g (literal-function 'y_s)))
(literal-function 'theta) ) 't ) )
;(show-expression demo1s)
; already defined this above in demo1l
(define
( (L-uniform-acceleration m g) local)
(let
( (q (coordinate local))
(v (velocity local)) )
(let
( (y (ref q 1)))
(- (* 1/2 m (square v))
(* m g y) ) ) ) )
(define
( (dp-coordinates l y_s) local)
(let
( (t (time local))
(theta (coordinate local)) )
(let
( (x (* l (sin theta)))
(y (- (y_s t)
(* l (cos theta)) ) ) )
(up x y) ) ) )
(define
(L-pend m l g y_s)
(compose
(L-uniform-acceleration m g)
(F->C (dp-coordinates l y_s)) ) )
(define demo1t
( (L-pend 'm 'l 'g (literal-function 'y_s))
(up 't 'theta 'thetadot) ) )
;(show-expression demo1t)