Ex 3.1 State Derivatives

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Functional Differential Geometry

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Newton’s equations:

\displaystyle{ \begin{aligned} D^2 X(t) &= A_x(X(t), Y(t)) \\ D^2 Y(t) &= A_y(X(t), Y(t)) \end{aligned} }

Coordinate path \displaystyle{\sigma}:

\displaystyle{ \begin{aligned} \sigma &= \chi \circ \gamma \\ \chi &= (\text{t}, \text{x}, \text{y}, \text{v}_x, \text{v}_y) \\ \end{aligned} }

What is \chi?

It is a coordinate system on the manifold \mathbf{M}.

What is \gamma?

It is a parametric path.

(p.29) The integral curve of \mathbf{v}

\displaystyle{ \begin{aligned} \gamma_\mathbf{m}^\mathbf{v}&: \mathbf{R} \to \mathbf{M} \\ \end{aligned} }

is a parametric path on \mathbf{M} satisfying

\displaystyle{ \begin{aligned} D(\text{f} \circ \gamma_\mathbf{m}^\mathbf{v})(t) &= \mathbf{v}(\text{f})(\gamma_\mathbf{m}^\mathbf{v}(t)) = (\mathbf{v}(\text{f}) \circ \gamma_\mathbf{m}^\mathbf{v})(t) \\ \gamma_\mathbf{m}^\mathbf{v}(0) &= \mathbf{m} \\ \end{aligned} }

What is \mathbf{m}?

(p.22) Let \mathbf{m} be a point on a manifold, \mathbf{v} be a vector on the manifold, and \mathbf{f} be a real-valued function on the manifold.

(define R5 (make-manifold R^n 5))
 
(define U5 (patch 'origin R5))
 
(define R5-rect
  (coordinate-system 'rectangular U5))

(define R5-rect-chi-inverse
  (point R5-rect))

(define R5-rect-point
  (R5-rect-chi-inverse
   (up 't 'x 'y 'v_x 'v_y)))
 
(define-coordinates
  (up t x y v_x v_y) R5-rect)
 
(define v5
  (literal-vector-field 'b R5-rect))
 
(show-expression
 ((v5 (literal-manifold-function
       'f_rect
       R5-rect))
  R5-rect-point))


(show-expression
 ((v5 (chart R5-rect)) R5-rect-point))

(p.29) We can recover the differential equations satisfied by a coordinate representation of the integral curve by letting \mathbf{f}=\chi.

\displaystyle{\begin{aligned} D(f \circ \gamma_\mathbf{m}^\mathbf{v})(t) &= \mathbf{v}(f)(\gamma_\mathbf{m}^\mathbf{v}(t)) = (\mathbf{v}(f) \circ \gamma_\mathbf{m}^\mathbf{v})(t) \\ \gamma_\mathbf{m}^\mathbf{v}(0) &= \mathbf{m} \\ \end{aligned}}

(p.29)

\displaystyle{\begin{aligned} D \sigma(t) &= D(\chi \circ \gamma) (t) \\ &= ... \\ &= (b \circ \sigma)(t), \\ \end{aligned}}

where \displaystyle{ b = \mathbf{v}(\chi) \circ \chi^{-1}} is the coefficient function for the vector field \displaystyle{ \mathbf{v}} for the coordinates \displaystyle{ \chi} (see equation 3.7).

(define R5->R
  (-> (UP Real Real Real Real Real) Real))

(print-expression
 ((literal-function 'f (-> (X Real Real) Real))
  'x 'y))

(define v5
  (components->vector-field
   (up (lambda (b) 1)
       (lambda (b) (ref b 3))
       (lambda (b) (ref b 4))
       (lambda (b)
         ((literal-function 'A_x
                            (-> (UP Real Real) Real))
          (up (ref b 1) (ref b 2))))
       (lambda (b)
         ((literal-function 'A_y
                            (-> (UP Real Real) Real))
          (up (ref b 1) (ref b 2)))))
   R5-rect))

(show-expression
 ((v5 (chart R5-rect)) R5-rect-point))


(series:for-each
 print-expression
 (((exp (* 't v5)) (chart R5-rect))
  ((point R5-rect) (up 't 'x 'y 'v_x 'v_y)))
 3)

(series:for-each
 show-expression
 (((exp (* 't v5)) (chart R5-rect))
  ((point R5-rect) (up 't 'x 'y 'v_x 'v_y)))
 4)

(define ((((evolution order) delta-t v) f) m)
  (series:sum
   (((exp (* delta-t v)) f) m)
   order))

(show-expression
 ((((evolution 1) 'Delta_t v5) (chart R5-rect))
  ((point R5-rect)
   (up 't_0 'x_0 'y_0 'v_x0 'v_y0))))


(show-expression
 ((((evolution 2) 'Delta_t v5) (chart R5-rect))
  ((point R5-rect) 
   (up 't_0 'x_0 'y_0 'v_x0 'v_y0))))

— Me@2023-07-21 08:12:30 PM

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2023.07.23 Sunday (c) All rights reserved by ACHK