commit work from yesterday (thursday)

2 files changed, 170 insertions, 0 deletions

diff --git a/chapters/4-Basis-Fields.scm b/chapters/4-Basis-Fields.scm
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+++ b/chapters/4-Basis-Fields.scm
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+
+; pre-reqs
+(define R2->R (-> (UP Real Real) Real))
+(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))
+(define R2-rect-point (R2-rect-chi-inverse (up 'x0 'y0)))
+(define R2-polar-point (R2-polar-chi-inverse (up 'r0 'theta0)))
+
+(define-coordinates (up x y) R2-rect)
+(define-coordinates (up r theta) R2-polar)
+
+(define e0
+  (+ (* (literal-manifold-function 'e0x R2-rect) d/dx)
+     (* (literal-manifold-function 'e0y R2-rect) d/dy)))
+
+(define e1
+  (+ (* (literal-manifold-function 'e1x R2-rect) d/dx)
+     (* (literal-manifold-function 'e1y R2-rect) d/dy)))
+
+(define e-vector-basis (down e0 e1))
+(define e-dual-basis
+  (vector-basis->dual e-vector-basis R2-polar))
+
+((e-dual-basis e-vector-basis) R2-rect-point)
+; (up (down 1 0) (down 0 1))
+
+((e-dual-basis e-vector-basis) R2-polar-point)
+; (up (down 1 0) (down 0 1))
+
+; remember, e-vector-basis is "down"
+(define v
+  (* (up (literal-manifold-function 'b^0 R2-rect)
+         (literal-manifold-function 'b^1 R2-rect))
+     e-vector-basis))
+
+((e-dual-basis v) R2-rect-point)
+; (up (b^0 (up x0 y0)) (b^1 (up x0 y0)))
+
+(define (Jacobian to-basis from-basis)
+  (s:map/r (basis->1form-basis to-basis)
+           (basis->vector-basis from-basis)))
+
+(define b-rect
+  ((coordinate-system->1form-basis R2-rect)
+   (literal-vector-field 'b R2-rect)))
+
+(define b-polar
+  (* (Jacobian (coordinate-system->basis R2-polar)
+               (coordinate-system->basis R2-rect))
+     b-rect))
+
+(b-rect ((point R2-rect) (up 'x0 'y0)))
+; (up (b^0 (up x0 y0)) (b^1 (up x0 y0)))
+
+(b-polar ((point R2-rect) (up 'x0 'y0)))
+; (up (/ (+ (* x0 (b^0 (up x0 y0))) (* y0 (b^1 (up x0 y0))))
+;        (sqrt (+ (expt x0 2) (expt y0 2))))
+;     (/ (+ (* x0 (b^1 (up x0 y0))) (* -1 y0 (b^0 (up x0 y0))))
+;        (+ (expt x0 2) (expt y0 2))))
+; same as...
+; (x0 b0(m) + y0 b1(m)) / r
+; (x0 b1(m) - y0 b0(m)) / r^2
+
+; Commutators
+
+(let* ((polar-basis (coordinate-system->basis R2-polar))
+       (polar-vector-basis (basis->vector-basis polar-basis))
+       (polar-dual-basis (basis->1form-basis polar-basis))
+       (f (literal-manifold-function 'f-rect R2-rect)))
+  ((- ((commutator e0 e1) f)
+      (* (- (e0 (polar-dual-basis e1))
+            (e1 (polar-dual-basis e0)))
+         (polar-vector-basis f)))
+   R2-rect-point))
+; 0
+
+(define-coordinates (up x y z) R3-rect)
+
+(define R3->R (-> (UP Real Real Real) Real))
+(define R3-rect-chi (chart R3-rect))
+(define R3-rect-chi-inverse (point R3-rect))
+(define R3-rect-point (R3-rect-chi-inverse (up 'x0 'y0 'z0)))
+
+(define Jz (- (* x d/dy) (* y d/dx)))
+(define Jx (- (* y d/dz) (* z d/dy)))
+(define Jy (- (* z d/dx) (* x d/dz)))
+
+; huh, 'g' here was not defined. is it supposed to be a metric? probably not. a function?
+;(((+ (commutator Jx Jy) Jz) g) R3-rect-point)
+(define g (compose (literal-function 'g-rect R3->R) R3-rect-chi))
+
+; oh, I see, there was a footnote. I defined things a little differently, could have been:
+
+; (define R3-rect (coordinate-system-at 'rectangular 'origin R3))
+; (define-coordinates (up x y z) R3-rect)
+; (define R3-rect-point ((point R3-rect) (up 'x0 'y0 'z0)))
+; (define g (literal-manifold-function 'g-rect R3-rect))
+
+
+(((+ (commutator Jx Jy) Jz) g) R3-rect-point)
+; 0
+(((+ (commutator Jy Jz) Jx) g) R3-rect-point)
+; 0
+(((+ (commutator Jz Jx) Jy) g) R3-rect-point)
+; 0
+
+(define Euler-angles (coordinate-system-at 'Euler 'Euler-patch SO3))
+(define Euler-angles-chi-inverse (point Euler-angles))
+(define-coordinates (up theta phi psi) Euler-angles)
+(define SO3-point ((point Euler-angles) (up 'theta 'phi' psi)))
+(define f (literal-manifold-function 'f-Euler Euler-angles))
+
+; from p48
+(define e_x (+ (* (cos phi) d/dtheta)
+               (* -1 (sin phi) (/ (cos theta) (sin theta)) d/dphi)
+               (* (/ (sin phi) (sin theta)) d/dpsi)))
+(define e_y (+ (* (cos phi) (/ (cos theta) (sin theta)) d/dphi)
+               (* (sin phi) d/dtheta)
+               (* -1 (/ (cos phi) (sin theta)) d/dpsi)))
+(define e_z d/dphi)
+
+(((+ (commutator e_x e_y) e_z) f) SO3-point)
+; 0
+
+(((+ (commutator e_y e_z) e_x) f) SO3-point)
+; 0
+
+(((+ (commutator e_z e_x) e_y) f) SO3-point)
+; 0
+
+; that is a fair amount of computer algebra! verified by fiddling with the
+; definition of e_x, and the subtraction no longer comes out to zero
+
+;;; Exercise 4.1 Alternate Angles
+
+; 4.1a
+
+
+;;; Exersize 4.2 General Commutators
+
+; verify equation (4.38) (not done)
+
+;;; Exersize 4.3 SO(3) Basis and Angular Momentum Basis
+
+
+((Jx g) R3-rect-point)
+
+((e_x f) SO3-point)
diff --git a/log/2022-09-22.md b/log/2022-09-22.md
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--- /dev/null
+++ b/log/2022-09-22.md
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+
+Two days ago, after running through Func Diff Geometry chapter, spent a while
+reading Carroll Spacetime+Geometry, in early chapters. Particularly relevant
+was the discussion of 1-forms ("dual vectors") and vectors and tensors. Tensor
+language isn't really even used in this book, except in Appendix C. Should
+probably work through that appendix in another week or two.
+
+---
+
+Starting on chapter 4, about basis fields (basic vector fields and basis
+one-form fields), both coordinate and non-coordinate.
+
+What is an example of a non-coordinate basis? Rotations! This will probably
+come back up as... Lorentz transforms? Rotations of spacetime, as a change of
+frame.
+
+Commutators!
+
+The commutator of two coordinate basis fields is zero. This is one way to tell
+if basis fields are coordinate or non-coordinate. Is this sufficient proof?