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(define ((Lfree mass) state)
(* 1/2 mass (square (velocity state))))
; geometric transform of spherical coordinates to 3D coordinates
; 'R' is radius
(define ((sphere->R3 R) state)
(let ((q (coordinate state)))
(let ((theta (ref q 0))
(phi (ref q 1)))
(up (* R (sin theta) (cos phi)) ; x
(* R (sin theta) (sin phi)) ; y
(* R (cos theta)))))) ; z
; apply a coordinate transform to a function over states (?)
(define ((F->C F) state)
(up (time state)
(F state)
(+ (((partial 0) F) state)
(* (((partial 1) F) state)
(velocity state)))))
(define (Lsphere m R)
(compose (Lfree m) (F->C (sphere->R3 R))))
((Lsphere 'm 'R)
(up 't (up 'theta 'phi) (up 'thetadot 'phidot)))
; (+ (* 1/2 (expt R 2) m (expt phidot 2) (expt (sin theta) 2)) (* 1/2 (expt R 2) m (expt thetadot 2)))
; -> 1/2 m R^2 phidot^2 sin(theta)^2 + 1/2 R^2 m thetadot^2
; -> 1/2 m R^2 ( phidot^2 sin(theta)^2 + thetadot^2 )
; this is kinetic energy of a particle on a sphere in spherical coordinates, as expected
(define ((L2 mass metric) place velocity)
(* 1/2 mass ((metric velocity velocity) place)))
; (metric velocity velocity) here is the magniture of velocity ("weighted sum of squares")
; 'Lc' is "Lagrangian in coordinate representation"
(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)))
; (+ (* 1/2 m (expt vx 2) (g_00 (up x y)))
; (* m vx vy (g_01 (up x y)))
; (* 1/2 m (expt vy 2) (g_11 (up x y))))
; same as 1.3, assuming g_01 and g_10 are the same (?)
; going to define a path on a 2D manifold
(define gamma (literal-manifold-map 'q R1-rect R2-rect))
; ahhh!
; ``((point R1-rect 't))` creates a point on the real like (R1-rect), based on coordinate 't.
; `((chart R2-rect) m)` takes a point m and returns coordinates
((chart R2-rect) (gamma ((point R1-rect) 't)))
; (up (q^0 t) (q^1 t))
(define coordinate-path
(compose (chart R2-rect) gamma (point R1-rect)))
(coordinate-path 't)
; (up (q^0 t) (q^1 t))
(define Lagrange-residuals
(((Lagrange-equations L) coordinate-path) 't))
Lagrange-residuals
; big mess
(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))
; (down 0 0)
; so the Lagrange equations correspond to the Christoffel coefficients
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