SICM Exercise 1.11

Solution to exercise 1.11 of Structure and Interpretation of Classical Mechanics by Gerald Jay Sussman and Jack Wisdom.

Kepler’s third law

🛈 Note

A Lagrangian suitable for studying the relative motion of two particles, of masses \(m_1\) and \(m_2\), with potential energy \(V\), is:

(define ((L-central-polar m V) local)
  (let ((q (coordinate local))
        (qdot (velocity local)))
    (let ((r (ref q 0)) (phi (ref q 1))
          (rdot (ref qdot 0)) (phidot (ref qdot 1)))
      (- (* 1/2 m
            (+ (square rdot) (square (* r phidot))) )
         (V r)))))

The argument \(m\) is the reduced mass of the system

\[ m = \frac{m_1 m_2}{m_1 + m_2} \]

(define ((gravitational-energy G m1 m2) r)
  (- (/ (* G m1 m2) r)))

where \(r\) is the distance between the two particles. Consider the simple situation of the particles in circular orbits around their common center of mass. Construct a circular orbit and plug it into the Lagrange equations. Show that the residual gives Kepler’s law:

\[n^2 a^3 = G(m_1 + m_2)\]

where \(n\) is the angular frequency of the orbit and a is the distance between the particles.

Since the Lagrangian does not depend on the coordinate system used, we can descibe the path using polar coordinates. In this case the solution is \( (a, nt) \).

(define ((Lagrange-equations Lagrangian) q)
  (- (D (compose ((partial 2) Lagrangian) (Gamma q)))
     (compose ((partial 1) Lagrangian) (Gamma q))))

(define ((L-central-polar m V) local)
  (let ((q (coordinate local))
        (qdot (velocity local)))
    (let ((r (ref q 0)) (phi (ref q 1))
          (rdot (ref qdot 0)) (phidot (ref qdot 1)))
      (- (* 1/2 m
            (+ (square rdot) (square (* r phidot))) )
         (V r)))))

(define ((gravitational-energy G m1 m2) r)
  (- (/ (* G m1 m2) r)))

(define (proposed-solution t)
  (up 'a (* 'n t)))

(show-expression
 (((Lagrange-equations (L-central-polar (/ (* 'm_1 'm_2) (+ 'm_1 'm_2))
                                        (gravitational-energy 'G 'm_1 'm_2)))
   proposed-solution) 't))

Running the code above, we get that

We can simplify and rearrange to get