Ex 2.2 Stereographic Projection

Functional Differential Geometry

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The points on the plane can also be specified with polar coordinates $\displaystyle{(\rho, \theta)}$ and the points on the sphere are specified both by Riemann coordinates and the traditional colatitude and longitude $\displaystyle{(\phi, \lambda)}$.

(show-expression
((compose
(chart S2-spherical)
(point S2-Riemann)
(chart R2-rect)
(point R2-polar))
(up 'rho 'theta)))


~~~

1. The code


(up 'rho 'theta)


represents the polar coordinates of a point.

2. The function


(point R2-polar)


generates an abstract point from a point in R2-polar coordinates.

3. The function


(chart R2-rect)


gives the rect coordinates given an abstract point on the plane R2.

(show-expression
((compose
(chart R2-rect)
(point R2-polar))
(up 'rho 'theta)))

4.

The procedure (point S2-Riemann) gives the point on the sphere given rectangular coordinates on the plane.

In other words, the function


(point S2-Riemann)


generates an abstract point-on-the-sphere (S2) from a point-on-the-plane (R2) in rect coordinates. In other words,


S2-Riemann

means


S2-rect

.

5.

Perform an analogous computation to get the polar coordinates of the point on the plane corresponding to a point on the sphere given by its colatitude and longitude.

(show-expression
((compose
(chart R2-polar)
(point R2-rect)
(chart S2-Riemann)
(point S2-spherical))
(up 'phi 'lambda)))

— Me@2023-04-22 10:42:50 PM

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Posted in FDG

Ex 2.1 Curves

Functional Differential Geometry

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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

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Posted in FDG

Ex 2.0

Functional Differential Geometry

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~~~


(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

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Posted in FDG

Ex 1.1: Motion on a Sphere, 2

Functional Differential Geometry

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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)


.

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

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

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— Me@2022-10-27 10:30:50 AM

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Posted in FDG

Ex 1.1: Motion on a Sphere

Functional Differential Geometry

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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)

(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}}

.

So

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

for some constant $C$.

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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}}

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— Me@2022-10-08 04:56:27 PM

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Posted in FDG

Functional Differential Geometry

Chapter 1: Introduction

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(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)

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

(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)))


— Me@2022.09.18 04:25:12 PM

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2022.09.18 Sunday ACHK

Posted in FDG