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) (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)
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(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
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