chez-scmutils/scmutils/calculus/Lie.scm

#| -*-Scheme-*-

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    1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005,
    2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Massachusetts
    Institute of Technology

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it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or (at
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WITHOUT ANY WARRANTY; without even the implied warranty of
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USA.

|#

#|
;;; Generalize to rank k.
(define ((Lie-derivative X) Y)
  (vector-field? X)
  (assert (form-field? Y))
  (procedure->1form-field
   (lambda (Z)
     (- ((Lie-derivative X) (Y Z))
	(Y ((Lie-derivative X) Z))))
   'foo))
|#

(define (Lie-derivative X)
  (cond ((function? X)			;compatability with SICM Hamiltonian
	 (make-operator
	  (lambda (F)
	    ;(assert (function? F))
	    (Poisson-bracket F X))
	  `(Lie-derivative ,X)))
	((vector-field? X)
	 (make-operator
	  (lambda (Y)
	    (cond
	     ((function? Y) (X Y))
	     ((vector-field? Y) (commutator X Y))
	     ((form-field? Y)		; Hawking & Ellis p.29
	      (let ((k (get-rank Y)))
		(procedure->nform-field
		 (lambda vectors
		   (assert (= k (length vectors)))
		   (- ((Lie-derivative X) (apply Y vectors))
		      (sigma (lambda (i)
			       (apply Y
				      (list-with-substituted-coord vectors i
					  ((Lie-derivative X)
					   (list-ref vectors i)))))
			     0 (- k 1))))
		 k
		 `((Lie-derivative ,(diffop-name X))
		   ,(diffop-name Y)))))
	     ((structure? Y)
	      (s:map/r (Lie-derivative X) Y))
	     (else
	      (error "Bad argument -- Lie-derivative"))))
	  `(Lie-derivative ,X)))
	(else
	 (error "Bad vector field -- Lie-derivative"))))

#|
(install-coordinates R3-rect (up 'x 'y 'z))

(define R3-rect-point ((R3-rect '->point) (up 'x0 'y0 'z0)))

(install-coordinates R3-cyl (up 'r 'theta 'zeta))

(define R3-cyl-point ((R3-cyl '->point) (up 'r0 'theta0 'zeta0)))


(define w (literal-1form-field 'w R3-rect))
(define u (literal-1form-field 'u R3-rect))
(define v (literal-1form-field 'v R3-rect))

(define X (literal-vector-field 'X R3-rect))
(define Y (literal-vector-field 'Y R3-rect))
(define Z (literal-vector-field 'Z R3-rect))
(define W (literal-vector-field 'W R3-rect))

(define f (literal-scalar-field 'f R3-rect))

(clear-arguments)
(suppress-arguments (list '(up x0 y0 z0)))

(pec ((((Lie-derivative X) w) Y) R3-rect-point)
     (compose arg-suppressor simplify))
#| Result:
(+ (* ((partial 0) w_0) X^0 Y^0)
   (* ((partial 0) w_1) X^0 Y^1)
   (* ((partial 0) w_2) X^0 Y^2)
   (* ((partial 1) w_0) X^1 Y^0)
   (* ((partial 1) w_1) X^1 Y^1)
   (* ((partial 1) w_2) X^1 Y^2)
   (* ((partial 2) w_0) X^2 Y^0)
   (* ((partial 2) w_1) X^2 Y^1)
   (* ((partial 2) w_2) X^2 Y^2)
   (* ((partial 0) X^0) w_0 Y^0)
   (* ((partial 1) X^0) w_0 Y^1)
   (* ((partial 2) X^0) w_0 Y^2)
   (* ((partial 0) X^1) w_1 Y^0)
   (* ((partial 1) X^1) w_1 Y^1)
   (* ((partial 2) X^1) w_1 Y^2)
   (* ((partial 0) X^2) Y^0 w_2)
   (* ((partial 1) X^2) Y^1 w_2)
   (* ((partial 2) X^2) w_2 Y^2))
|#


(pec ((- ((d ((Lie-derivative X) f)) Y)
	 (((Lie-derivative X) (d f)) Y) )
      R3-rect-point)
     (compose arg-suppressor simplify))
#| Result:
0
|#

(pec ((- ((d ((Lie-derivative X) w)) Y Z)
	 (((Lie-derivative X) (d w)) Y Z) )
      ((R3-rect '->point) (up 'x^0 'y^0 'z^0)))
     (compose arg-suppressor simplify))
#| Result:
0
|#
|#

#|
(install-coordinates R2-rect (up 'x 'y))

(define R2-rect-point ((R2-rect '->point) (up 'x0 'y0)))

(define X (literal-vector-field 'X R2-rect))
(define Y (literal-vector-field 'Y R2-rect))

(define f (literal-scalar-field 'f R2-rect))

(clear-arguments)
(suppress-arguments (list '(up x0 y0)))

(pec ((((Lie-derivative X) Y) f) R2-rect-point)
    (compose arg-suppressor simplify))
#| Result:
(+ (* ((partial 0) Y^0) X^0 ((partial 0) f))
   (* ((partial 0) Y^1) X^0 ((partial 1) f))
   (* ((partial 1) Y^0) X^1 ((partial 0) f))
   (* ((partial 1) Y^1) X^1 ((partial 1) f))
   (* -1 ((partial 0) X^0) Y^0 ((partial 0) f))
   (* -1 ((partial 0) X^1) Y^0 ((partial 1) f))
   (* -1 ((partial 1) X^0) ((partial 0) f) Y^1)
   (* -1 ((partial 1) X^1) Y^1 ((partial 1) f)))
|#


Let phi_t(x) be the integral curve of V from x for interval t

L_V Y (f) (x) = lim_t->0 ( Y(f) (phi_t (x)) - (d phi_t)(Y)(f)(x))/t
              = D (lambda (t) 
                       ( Y(f) (phi_t (x)) - (d phi_t)(Y)(f)(x)))
                (t=0)
so let g(t) = ( Y(f) (phi_t (x)) - (d phi_t)(Y)(f)(x))
            = ( Y(f) (phi_t (x)) - Y(f circ phi_t)(x))

we only need linear terms in phi_t(x)


Perhaps

       phi_t(x) = (I + t v(I))(x)

Is this correct?  No!, cannot add to a manifold point. ***********

       g(t) = ( Y(f) ((I + t v(I))(x)) - Y(f circ (I + t v(I)))(x))


(define ((((Lie-test V) Y) f) x)
  (define (g t)
    (- ((Y f) ((+ identity (* t (V identity))) x))
       ((Y (compose f (+ identity (* t (V identity))))) x)))
  ((D g) 0))


(pec (- ((((Lie-test X) Y) f) R2-rect-point)
	((((Lie-derivative X) Y) f) R2-rect-point)))
#| Result:
0
|#

;;; this result is a consequence of confusing manifold points
;;; with tuples of coordinates in the embedding space.

(clear-arguments)
|#

#|
;;; Lie derivative satisfies extended Leibnitz rule

(define V (literal-vector-field 'V R2-rect))

(define Y (literal-vector-field 'Y R2-rect))

(define q_0 (up 'q_x 'q_y))

(define m ((R2-rect '->point) q_0))
;Value: m


(define f (literal-manifold-function 'f R2-rect))


(define e_0 (literal-vector-field 'e_0 R2-rect))
(define e_1 (literal-vector-field 'e_1 R2-rect))

(define vector-basis (down e_0 e_1))
(define 1form-basis (vector-basis->dual (down e_0 e_1) R2-rect))
(define basis (make-basis vector-basis 1form-basis))

(define Y^i (1form-basis Y))

(pe ((- (((Lie-derivative V) Y) f)
	(+ (* (s:map/r (Lie-derivative V) Y^i) (vector-basis f))
	   (* Y^i ((s:map/r (Lie-derivative V) vector-basis) f))))
     m))
0
|#

#|
;;; Computation of Lie derivatives by difference quotient.

(define X (literal-vector-field 'X R2-rect))

(define Y (literal-vector-field 'Y R2-rect))

(define q_0 (up 'q_x 'q_y))

(define m_0
  ((R2-rect '->point) q_0))

(define ((q coords)  t)
  (+ coords
     (* t
	((X (R2-rect '->coords))
	 ((R2-rect '->point) coords)))))

(define (gamma initial-point)
  (compose (R2-rect '->point)
	   (q ((R2-rect '->coords) initial-point))))

(define ((phi^X t) point)
  ((gamma point) t))

(define f (literal-manifold-function 'f R2-rect))


(pe ((D (lambda (t)
	  (- ((Y f) ((phi^X t) m_0))
	     ((Y (compose f (phi^X t))) m_0))))
     0))
(+ (* -1 (((partial 1) X^0) (up q_x q_y)) (Y^1 (up q_x q_y)) (((partial 0) f) (up q_x q_y)))
   (* -1 (Y^1 (up q_x q_y)) (((partial 1) X^1) (up q_x q_y)) (((partial 1) f) (up q_x q_y)))
   (* (((partial 1) Y^0) (up q_x q_y)) (((partial 0) f) (up q_x q_y)) (X^1 (up q_x q_y)))
   (* (((partial 0) f) (up q_x q_y)) (((partial 0) Y^0) (up q_x q_y)) (X^0 (up q_x q_y)))
   (* -1 (((partial 0) f) (up q_x q_y)) (((partial 0) X^0) (up q_x q_y)) (Y^0 (up q_x q_y)))
   (* (((partial 1) Y^1) (up q_x q_y)) (((partial 1) f) (up q_x q_y)) (X^1 (up q_x q_y)))
   (* (((partial 1) f) (up q_x q_y)) (((partial 0) Y^1) (up q_x q_y)) (X^0 (up q_x q_y)))
   (* -1 (((partial 1) f) (up q_x q_y)) (((partial 0) X^1) (up q_x q_y)) (Y^0 (up q_x q_y))))


(pe ((((Lie-derivative X) Y) f) m_0))
(+ (* -1 (((partial 1) X^0) (up q_x q_y)) (Y^1 (up q_x q_y)) (((partial 0) f) (up q_x q_y)))
   (* -1 (Y^1 (up q_x q_y)) (((partial 1) X^1) (up q_x q_y)) (((partial 1) f) (up q_x q_y)))
   (* (((partial 1) Y^0) (up q_x q_y)) (((partial 0) f) (up q_x q_y)) (X^1 (up q_x q_y)))
   (* (((partial 0) f) (up q_x q_y)) (((partial 0) Y^0) (up q_x q_y)) (X^0 (up q_x q_y)))
   (* -1 (((partial 0) f) (up q_x q_y)) (((partial 0) X^0) (up q_x q_y)) (Y^0 (up q_x q_y)))
   (* (((partial 1) Y^1) (up q_x q_y)) (((partial 1) f) (up q_x q_y)) (X^1 (up q_x q_y)))
   (* (((partial 1) f) (up q_x q_y)) (((partial 0) Y^1) (up q_x q_y)) (X^0 (up q_x q_y)))
   (* -1 (((partial 1) f) (up q_x q_y)) (((partial 0) X^1) (up q_x q_y)) (Y^0 (up q_x q_y))))


(pe (- ((D (lambda (t)
	  (- ((Y f) ((phi^X t) m_0))
	     ((Y (compose f (phi^X t))) m_0))))
	0)
       ((((Lie-derivative X) Y) f) m_0)))
0

(pe (- ((D (lambda (t)
	     (- ((Y f) ((phi^X t) m_0)) 
		((((pushforward-vector (phi^X t) (phi^X (- t)))
		   Y)
		  f)
		 ((phi^X t) m_0)))))
	0)
       ((((Lie-derivative X) Y) f) m_0)))
0

(pe (- ((D (lambda (t)
	     ((((pushforward-vector (phi^X (- t)) (phi^X t))
		Y)
	       f)
	      m_0)))
	0)
       ((((Lie-derivative X) Y) f) m_0)))
0
|#

#|

(define m ((R2-rect '->point) (up 'x 'y)))

(define V (literal-vector-field 'V R2-rect))

(define Y (literal-vector-field 'Y R2-rect))

(define f (literal-manifold-function  'f R2-rect))


(define e_0 (literal-vector-field 'e_0 R2-rect))
(define e_1 (literal-vector-field 'e_1 R2-rect))

(define vector-basis (down e_0 e_1))
(define 1form-basis (vector-basis->dual (down e_0 e_1) R2-rect))

(define e^0 (ref 1form-basis 0))
(define e^1 (ref 1form-basis 1))

(define basis (make-basis vector-basis 1form-basis))

(define (Delta^i_j v)
  (1form-basis (s:map/r (Lie-derivative v) vector-basis)))

;;; Verifying equation 0.184

(pec ((- (((Lie-derivative V) Y) f)
	 (* (vector-basis f)
	    (+ (V (1form-basis Y))
	       (* (1form-basis Y) (Delta^i_j V)))))
      m))

#| Result:
0
|#

;;; Indeed, a painful detail:

(pec ((- (* (1form-basis Y) ((s:map/r (Lie-derivative V) vector-basis) f))
	 (* (1form-basis Y) (Delta^i_j V) (vector-basis f)))
      m))

#| Result:
0
|#

;;; Even simpler

(pec ((- (* ((s:map/r (Lie-derivative V) vector-basis) f))
	 (* (Delta^i_j V) (vector-basis f)))
      m))

#| Result:
(down 0 0)
|#

|#