# Interactive Scheme Mechanics Scheme Mechanics, also known as SCMUTILS, has been used in two books, Structure and Interpretation of Classical Mechanics (), and Functional Differential Geometry (, ).

## Exercise 1.9a ()

We derive the Lagrange equations for the following system. A particle of mass $$m$$ moves in a two-dimensional potential $$V(x, y) = (x^2 + y^2)/2 + x^2 y - y^3/3$$, where $$x$$ and $$y$$ are rectangular coordinates of the particle. A Lagrangian is $$L(t; x, y; v_x, v_y) = (1/2) m (v_x^2 + v_y^2) - V(x, y)$$.

(define (Lagrange-eq L q) (- (D (compose ((partial 2) L) (Gamma q))) (compose ((partial 1) L) (Gamma q)))) (define (V x y) (+ (/ (+ (square x) (square y)) 2) (* (square x) y) (- (/ (cube y) 3)))) (define ((L m) local) (let ((vx (ref (velocity local) 0)) (vy (ref (velocity local) 1)) ( x (ref (coordinate local) 0)) ( y (ref (coordinate local) 1))) (- (/ (* m (+ (square vx) (square vy))) 2) (V x y)))) (define (q t) (up ((literal-function 'x) t) ((literal-function 'y) t))) (print-expression ((Lagrange-eq (L 'm) q) 't)) )