Ex 2.1 Curves

Posted on December 10, 2022 by rpflee

Functional Differential Geometry

a. The rectangular coordinate equation for the Lemniscate of Bernoulli is

$\displaystyle{(x^2 + y^2)^2 = 2 a^2 (x^2 - y^2)}$ .

Find the expression in polar coordinates.

b. Describe a helix space curve in both rectangular and cylindrical coordinates.

~~~

(define-coordinates (up x y) R2-rect)

(define-coordinates (up r theta) R2-polar)

;

(define R2-rect-chi (chart R2-rect))

; R2-rect-chi
;     generates the rectangle coordinates of a point.

(define R2-rect-chi-inverse (point R2-rect))

; R2-rect-chi-inverse
;     gets the abstract representation.

(x (R2-rect-chi-inverse (up 'x0 'y0)))

; Function x
;     gets the x coordinate
;          of an (abstract-represented) point.

;

(define R2-polar-chi (chart R2-polar))

(define R2-polar-chi-inverse (point R2-polar))

(x (R2-polar-chi-inverse (up 'r0 'theta0)))

(r (R2-polar-chi-inverse (up 'r0 'theta0)))

(r (R2-rect-chi-inverse (up 'x0 'y0)))

(theta (R2-rect-chi-inverse (up 'x0 'y0)))

;

(define h (+ (* x (square r)) (cube y)))

(define R2-rect-point
  (R2-rect-chi-inverse (up 'x_0 'y_0)))

(show-expression
 (h R2-rect-point))

(show-expression
 (h (R2-polar-chi-inverse (up 'r_0 'theta_0))))

(show-expression
 ((- r (* 2 'a (+ 1 (cos theta))))
  ((point R2-rect) (up 'x 'y))))

; Ex 2.1 a

(show-expression
 ((- (square (+ (square x) (square y)))
     (* 2 (square 'a) (- (square x) (square y))))
  ((point R2-rect) (up 'x 'y))))

(show-expression
 ((- (square (+ (square x) (square y)))
     (* 2 (square 'a) (- (square x) (square y))))
  ((point R2-polar) (up 'r 'theta))))

; Ex 2.1 b

(define-coordinates (up r theta z) R3-cyl)

(define-coordinates (up x y z) R3-rect)

(show-expression
 ((- (up r z) (up 'R (* 'a theta)))
  ((point R3-cyl) (up 'r 'theta 'z))))

(show-expression
 ((- (up r z) (up 'R (* 'a theta)))
  ((point R3-rect) (up 'x 'y 'z))))

— Me@2022-12-10 10:29:59 AM

Ex 2.0

Posted on November 18, 2022 by rpflee

Functional Differential Geometry

~~~


(define R2 (make-manifold R^n 2))

(define U (patch 'origin R2))


(define R2-rect (coordinate-system 'rectangular U))

(define R2-polar (coordinate-system 'polar/cylindrical U))


(define R2-rect-chi (chart R2-rect))

(define R2-rect-chi-inverse (point R2-rect))

(define R2-polar-chi (chart R2-polar))

(define R2-polar-chi-inverse (point R2-polar))


(show-expression
 ((compose R2-polar-chi R2-rect-chi-inverse)
  (up 'x_0 'y_0)))

(show-expression
 ((D (compose R2-rect-chi R2-polar-chi-inverse))
  (up 'r_0 'theta_0)))

(define R2->R (-> (UP Real Real) Real))

(define f
  (compose (literal-function 'f-rect R2->R) R2-rect-chi))



(define R2-rect-point (R2-rect-chi-inverse (up 'x_0 'y_0)))

(define corresponding-polar-point
  (R2-polar-chi-inverse
   (up (sqrt (+ (square 'x_0) (square 'y_0)))
       (atan 'y_0 'x_0))))


(f R2-rect-point)

(f corresponding-polar-point)

(show-expression
 (f R2-rect-point))

(show-expression
 (f corresponding-polar-point))

— Me@2022-11-18 11:22:36 AM

Ex 1.1: Motion on a Sphere, 2

Posted on October 28, 2022 by rpflee

Functional Differential Geometry

The metric for a unit sphere, expressed in colatitude $\displaystyle{\theta}$ and longitude $\displaystyle{\phi}$ , is

$\displaystyle{g(u,v) = d\theta(u) d \theta(v) + (\sin \theta)^2 d \phi (u) d \phi (v)}$

Compute the Lagrange equations for motion of a free particle on the sphere …

~~~

(define ((L2 mass metric) place velocity)
  (* 1/2
     mass
     ((metric velocity velocity) place)))

(define ((Lc mass metric coordsys) state)
  (let ((x (coordinates state))
        (v (velocities state))
        (e (coordinate-system->vector-basis
            coordsys)))
    ((L2 mass metric)
     ((point coordsys) x) (* e v))))

(define the-metric
  (literal-metric 'g R2-rect))

(define L
  (Lc 'm the-metric R2-rect))

(L (up 't (up 'x 'y) (up 'vx 'vy)))

(show-expression
 (L (up 't (up 'x 'y) (up 'v_x 'v_y))))

$\displaystyle{g(u,v) = d\theta(u) d \theta(v) + (\sin \theta)^2 d \phi (u) d \phi (v)}$

(show-expression
 (L (up 't
    (up 'theta 'phi)
    (up 'thetadot 'phidot))))

When $\displaystyle{R = 1}$ and $\displaystyle{ [g] = \begin{bmatrix} 1 & 0 \\ 0 & (\sin \theta)^2 \end{bmatrix}}$ ,

$\displaystyle{ \frac{1}{2} m (\sin \theta)^2 \dot \phi^2 + 0 + \frac{1}{2} m \dot \theta^2}$

— Me@2022-10-27 10:30:50 AM

Ex 1.1: Motion on a Sphere

Posted on October 9, 2022 by rpflee

Functional Differential Geometry

The metric for a unit sphere, expressed in colatitude $\displaystyle{\theta}$ and longitude $\displaystyle{\phi}$ , is

$\displaystyle{g(u,v) = d\theta(u) d \theta(v) + (\sin \theta)^2 d \phi (u) d \phi (v)}$

Compute the Lagrange equations for motion of a free particle on the sphere and convince yourself that they describe great circles. For example, consider the motion on the equator $\displaystyle{\theta = \pi/2}$ and motion on a line of longitude ( $\displaystyle{\phi}$ is constant).

~~~

(define ((Lfree mass) state)
  (* 1/2 mass (square (velocity state))))

(define ((sphere->R3 R) state)
  (let ((q (coordinate state)))
    (let ((theta (ref q 0)) (phi (ref q 1)))
      (up (* R (sin theta) (cos phi))
          (* R (sin theta) (sin phi))
          (* R (cos theta))))))

(define ((F->C F) local)
  (up (time local)
      (F local)
      (+ (((partial 0) F) local)
         (* (((partial 1) F) local)
            (velocity local)))))

(define (Lsphere m R)
  (compose (Lfree m) (F->C (sphere->R3 R))))

(show-expression
 ((Lsphere 'm 'R)
  (up 't
      (up 'theta 'phi)
      (up 'thetadot 'phidot))))

(show-expression
 (((Lagrange-equations
    (Lsphere 'm 'R))
   (up
    (literal-function 'theta)
    (literal-function 'phi)))
  't))

$\displaystyle{ \begin{aligned} - \sin \theta (D \phi)^2 \cos \theta + D^2 \theta &= 0 \\ 2 \sin \theta D \theta D \phi \cos \theta + D^2 \phi (\sin \theta)^2 &= 0 \\ \\ D^2 \theta &= (D \phi)^2 \cos \theta \sin \theta \\ D( D \phi (\sin \theta)^2) &= 0 \\ \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} D \phi (\sin \theta)^2 &\equiv C \\ \\ \end{aligned}}$

for some constant $C$ .

Since $\displaystyle{ \begin{aligned} D \phi (\sin \theta)^2 &= 0 \\ \\ \end{aligned}}$ for some $\theta$ ,

$\displaystyle{ \begin{aligned} D \phi (\sin \theta)^2 &\equiv 0 \\ \\ \end{aligned}}$

This is equivalent to setting up the coordinate system such that the initial value of $\theta$ equals zero.

Also, since $\displaystyle{ \begin{aligned} (\sin \theta)^2 &\ne 0 \\ \\ \end{aligned}}$ for some $\theta$ ,

$\displaystyle{ \begin{aligned} D \phi &\equiv 0 \\ \\ \end{aligned}}$

— Me@2022-10-08 04:56:27 PM

Functional Differential Geometry

Posted on September 18, 2022 by rpflee

Chapter 1: Introduction

(define ((Gamma w) t)
  (up t (w t) ((D w) t)))

(define q-rect
  (up (literal-function 'x_0)
      (literal-function 'y_0)
      (literal-function 'x_1)
      (literal-function 'y_1)
      (literal-function 'F)))

(show-expression (q-rect 't))

(show-expression ((Gamma q-rect) 't))





(define ((Lfree mass) state)
  (* 1/2 mass (square (velocity state))))

(define ((sphere->R3 R) state)
  (let ((q (coordinate state)))
    (let ((theta (ref q 0)) (phi (ref q 1)))
      (up (* R (sin theta) (cos phi))
          (* R (sin theta) (sin phi))
          (* R (cos theta))))))

(define ((F->C F) local)
  (up (time local)
      (F local)
      (+ (((partial 0) F) local)
         (* (((partial 1) F) local)
            (velocity local)))))

(define (Lsphere m R)
  (compose (Lfree m) (F->C (sphere->R3 R))))

((Lsphere 'm 'R)
 (up 't
     (up 'theta 'phi)
     (up 'thetadot 'phidot)))

(show-expression
 ((Lsphere 'm 'R)
  (up 't
      (up 'theta 'phi)
      (up 'thetadot 'phidot))))



(define ((L2 mass metric) place velocity)
  (* 1/2
     mass
     ((metric velocity velocity) place)))

(define ((Lc mass metric coordsys) state)
  (let ((x (coordinates state))
        (v (velocities state))
        (e (coordinate-system->vector-basis
            coordsys)))
    ((L2 mass metric)
     ((point coordsys) x) (* e v))))

(define the-metric
  (literal-metric 'g R2-rect))

(define L
  (Lc 'm the-metric R2-rect))

(L (up 't (up 'x 'y) (up 'vx 'vy)))

(show-expression
 (L (up 't (up 'x 'y) (up 'v_x 'v_y))))



(define gamma
  (literal-manifold-map 'q R1-rect R2-rect))

((chart R2-rect)
 (gamma ((point R1-rect) 't)))

(show-expression
 ((chart R2-rect)
  (gamma ((point R1-rect) 't))))



(define coordinate-path
  (compose
   (chart R2-rect) gamma (point R1-rect)))

(coordinate-path 't)

(define Lagrange-residuals
  (((Lagrange-equations L)
    coordinate-path) 't))

(show-expression
 Lagrange-residuals)





(define-coordinates t R1-rect)

(define Cartan
  (Christoffel->Cartan
   (metric->Christoffel-2
    the-metric
    (coordinate-system->basis R2-rect))))

(define geodesic-equation-residuals
  (((((covariant-derivative Cartan gamma) d/dt)
     ((differential gamma) d/dt))
    (chart R2-rect))
   ((point R1-rect) 't)))

(define metric-components
  (metric->components
   the-metric
   (coordinate-system->basis R2-rect)))

(- Lagrange-residuals
   (* (* 'm (metric-components
             (gamma ((point R1-rect) 't))))
      geodesic-equation-residuals))

(show-expression
 (- Lagrange-residuals
    (* (* 'm (metric-components
              (gamma ((point R1-rect) 't))))
       geodesic-equation-residuals)))