# 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