Exercise 1.11: Kepler’s third law

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

$$ L = \frac{1}{2} m \left(\dot{r}^2 + (r\dot{\phi})^2\right) + V(r) $$

The argument $m$ is the reduced mass of the system:

$$ m = \frac{m_1 m_2}{m_1 + m_2} $$

For gravity, the potential energy function is:

$$ V(r) = -\frac{G m_1 m_2}{r} $$

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 particles.

;; Define Lagrangian and derive EoMs
(defn L-central-polar [m V]
    (fn [[_ [r phi] [rdot phidot]]] 
      (- (* 1/2 m
            (+ (square rdot) (square (* r phidot))) )
         (V r))
        ))

(def m (/ (* 'm_1 'm_2) (+ 'm_1 'm_2) ))
(def V (fn [r] (- (/ (* 'G 'm_1 'm_2) r))))

(def eom-kepler 
        (Lagrange-equations (L-central-polar m V)))

(rendertexvec (let [state (up (literal-function 'r) (literal-function 'phi))]
            ((eom-kepler state) 't)))

;; Circular orbit
; r(t) = a
; phi(t) = n * t  
(def circ-solution
  (up (fn [t] 'a) (fn [t] (* 'n t))))
(rendertex (circ-solution 't))

(rendertexvec ((eom-kepler circ-solution) 't))
\begin{pmatrix}\displaystyle{\frac{- {a}^{3}\,m_1\,m_2\,{n}^{2} + G\,{m_1}^{2}\,m_2 + G\,m_1\,{m_2}^{2}}{{a}^{2}\,m_1 + {a}^{2}\,m_2}} \cr \cr \displaystyle{0}\end{pmatrix}

The above equation simplifies to:

$$ \begin{align*} -a^3 m_1 m_2 n^2 + G m_1 m_2 (m_1 + m_2) &= 0 \\ => G (m_1 + m_2) &= a^3 n^2\\ \end{align*} $$

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