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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
+