We derive the gravitation law of Newton, given the three laws of Kepler. We avoid sophisticated vector analysis properties, and stick to basic calculus whenever possible.
The first law of Kepler states that the orbit of a planet revolving around the Sun, is an ellipse. We know the ellipse is confined within a plane, by the third law of Newton. Without loss of generality, let the length be scaled, so that the longer axis of ellipse along the \(x\) axis has length \(1\). We also let the shorter axis of ellipse along the \(y\) axis has length \(\beta\)\(\;\leq\;\)\(1\), so that the focal point lies on the \(x\) axis. The sun lies at \(\left\langle\vphantom{\gamma ,\, 0}\right.\)\(\gamma\)\(,\,\)\(0\)\(\left.\vphantom{\gamma ,\, 0}\right\rangle\) the focal point of ellipse. Introduce the focal distance:
\(\gamma\)\(\;\equiv\;\)\(\surd\!\)\(\left(\vphantom{1 \,-\, \beta \vphantom{ \beta}^{2}}\right.\)\(1\)\(\,-\,\)\(\beta \vphantom{ \beta}^{2}\)\(\left.\vphantom{1 \,-\, \beta \vphantom{ \beta}^{2}}\right)\)\(;\,\)
Define the position vector \(r\), velocity \(v\), acceleration \(a\). They are functions of \(\vartheta\)\(\;=\;\)\(\vartheta\)\(\left[\vphantom{t}\right.\)\(t\)\(\left.\vphantom{t}\right]\), the counterclockwise angle originating from the sun. With a prime denoting derivative of \(t\), they are found as thus:
Position:
\(r\)\(\;=\;\)\(\left\langle\vphantom{\,\mathrm{Cos}\, \vartheta \,-\, \gamma ,\, \beta \,\mathrm{Sin}\, \vartheta}\right.\)\(\,\mathrm{Cos}\,\)\(\vartheta\)\(\,-\,\)\(\gamma\)\(,\,\)\(\beta\)\(\,\mathrm{Sin}\,\)\(\vartheta\)\(\left.\vphantom{\,\mathrm{Cos}\, \vartheta \,-\, \gamma ,\, \beta \,\mathrm{Sin}\, \vartheta}\right\rangle\)\(;\,\)
Velocity:
\(v\)\(\;=\;\)\(\left\langle\vphantom{\,-\, \vartheta \vphantom{ \vartheta}^{\prime} \,\mathrm{Sin}\, \vartheta ,\, \beta \vartheta \vphantom{ \vartheta}^{\prime} \,\mathrm{Cos}\, \vartheta}\right.\)\(\,-\,\)\(\vartheta \vphantom{ \vartheta}^{\prime}\)\(\,\mathrm{Sin}\,\)\(\vartheta\)\(,\,\)\(\beta\)\(\vartheta \vphantom{ \vartheta}^{\prime}\)\(\,\mathrm{Cos}\,\)\(\vartheta\)\(\left.\vphantom{\,-\, \vartheta \vphantom{ \vartheta}^{\prime} \,\mathrm{Sin}\, \vartheta ,\, \beta \vartheta \vphantom{ \vartheta}^{\prime} \,\mathrm{Cos}\, \vartheta}\right\rangle\)\(;\,\)
Acceleration:
\(a\)\(\;=\;\)\(\left\langle\vphantom{\,-\, \vartheta \vphantom{ \vartheta}^{\prime\prime} \,\mathrm{Sin}\, \vartheta \,-\, \vartheta \vphantom{ \vartheta}^{2 \prime} \,\mathrm{Cos}\, \vartheta ,\, \beta \vartheta \vphantom{ \vartheta}^{\prime\prime} \,\mathrm{Cos}\, \vartheta \,-\, \beta \vartheta \vphantom{ \vartheta}^{2 \prime} \,\mathrm{Sin}\, \vartheta}\right.\)\(\,-\,\)\(\vartheta \vphantom{ \vartheta}^{\prime\prime}\)\(\,\mathrm{Sin}\,\)\(\vartheta\)\(\,-\,\)\(\vartheta \vphantom{ \vartheta}^{2 \prime}\)\(\,\mathrm{Cos}\,\)\(\vartheta\)\(,\,\)\(\beta\)\(\vartheta \vphantom{ \vartheta}^{\prime\prime}\)\(\,\mathrm{Cos}\,\)\(\vartheta\)\(\,-\,\)\(\beta\)\(\vartheta \vphantom{ \vartheta}^{2 \prime}\)\(\,\mathrm{Sin}\,\)\(\vartheta\)\(\left.\vphantom{\,-\, \vartheta \vphantom{ \vartheta}^{\prime\prime} \,\mathrm{Sin}\, \vartheta \,-\, \vartheta \vphantom{ \vartheta}^{2 \prime} \,\mathrm{Cos}\, \vartheta ,\, \beta \vartheta \vphantom{ \vartheta}^{\prime\prime} \,\mathrm{Cos}\, \vartheta \,-\, \beta \vartheta \vphantom{ \vartheta}^{2 \prime} \,\mathrm{Sin}\, \vartheta}\right\rangle\)\(;\,\)
Meanwhile, the second law of Kepler states that the area of the triangle swept by the line which passes through the planet and the sun, in unit time, is constant. This vector identity gives the square of area spanned by \(r\) and \(v\) as thus:
\(\left|r\right| \vphantom{ \left|r\right|}^{2}\)\(\left|v\right| \vphantom{ \left|v\right|}^{2}\)\(\,-\,\)\(\left(r \,\cdot\, v\right) \vphantom{ \left(r \,\cdot\, v\right)}^{2}\)\(\;=\;\)\(\,\mathsf{constant}\,\)\(;\,\)
Expressions Position and Velocity yield a constraint on \(\vartheta \vphantom{ \vartheta}^{\prime}\):
\(\left(\vphantom{\mathrm{Cos} \left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2} \,-\, 2 \gamma \,\mathrm{Cos}\, \vartheta \,+\, \gamma \vphantom{ \gamma}^{2} \,+\, \beta \vphantom{ \beta}^{2} \,\mathrm{Sin}\, \vartheta \vphantom{ \vartheta}^{2}}\right.\)\(\,\mathrm{Cos}\,\)\(\left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2}\)\(\,-\,\)\(2\)\(\gamma\)\(\,\mathrm{Cos}\,\)\(\vartheta\)\(\,+\,\)\(\gamma \vphantom{ \gamma}^{2}\)\(\,+\,\)\(\beta \vphantom{ \beta}^{2}\)\(\,\mathrm{Sin}\,\)\(\vartheta \vphantom{ \vartheta}^{2}\)\(\left.\vphantom{\mathrm{Cos} \left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2} \,-\, 2 \gamma \,\mathrm{Cos}\, \vartheta \,+\, \gamma \vphantom{ \gamma}^{2} \,+\, \beta \vphantom{ \beta}^{2} \,\mathrm{Sin}\, \vartheta \vphantom{ \vartheta}^{2}}\right)\)\(\,\cdot\,\)\(\vartheta \vphantom{ \vartheta}^{\prime}\)\(\left(\vphantom{\mathrm{Sin} \left[\vartheta\right] \vphantom{\mathrm{Sin} \left[\vartheta\right]}^{2} \,+\, \beta \vphantom{ \beta}^{2} \mathrm{Cos} \left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2}}\right.\)\(\,\mathrm{Sin}\,\)\(\left[\vartheta\right] \vphantom{\mathrm{Sin} \left[\vartheta\right]}^{2}\)\(\,+\,\)\(\beta \vphantom{ \beta}^{2}\)\(\,\mathrm{Cos}\,\)\(\left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2}\)\(\left.\vphantom{\mathrm{Sin} \left[\vartheta\right] \vphantom{\mathrm{Sin} \left[\vartheta\right]}^{2} \,+\, \beta \vphantom{ \beta}^{2} \mathrm{Cos} \left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2}}\right)\)\(\,-\,\)\(\vartheta \vphantom{ \vartheta}^{\prime 2}\)\(\left(\,-\, \,\mathrm{Cos}\, \vartheta \,\mathrm{Sin}\, \vartheta \,+\, \gamma \,\mathrm{Sin}\, \vartheta \,+\, \beta \vphantom{ \beta}^{2} \,\mathrm{Sin}\, \vartheta \,\mathrm{Cos}\, \vartheta\right) \vphantom{ \left(\,-\, \,\mathrm{Cos}\, \vartheta \,\mathrm{Sin}\, \vartheta \,+\, \gamma \,\mathrm{Sin}\, \vartheta \,+\, \beta \vphantom{ \beta}^{2} \,\mathrm{Sin}\, \vartheta \,\mathrm{Cos}\, \vartheta\right)}^{2}\)\(\;=\;\)\(\,\mathsf{constant}\,\)\(;\,\)
By arranging, by \(\,\mathrm{Sin}\,\)\(\left[\vartheta\right] \vphantom{\mathrm{Sin} \left[\vartheta\right]}^{2}\)\(\,+\,\)\(\,\mathrm{Cos}\,\)\(\left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2}\)\(\;=\;\)\(1\), by absorbing \(\beta \vphantom{ \beta}^{2}\) into the constant, and by pulling out \(\vartheta \vphantom{ \vartheta}^{\prime 2}\), we get:
\(\eta\)\(\;\equiv\;\)\(1\)\(\,-\,\)\(\gamma\)\(\,\mathrm{Cos}\,\)\(\vartheta\)\(;\,\)
\(\vartheta \vphantom{ \vartheta}^{\prime 2}\)\(\eta \vphantom{ \eta}^{2}\)\(\;=\;\)\(\,\mathsf{constant}\,\)\(;\,\)
Here, notice \(1\)\(\,-\,\)\(\gamma\)\(\,\mathrm{Cos}\,\)\(\vartheta\)\(\;>\;\)\(0\). Besides, if we agree that \(\vartheta\) revolves counterclockwise, \(\vartheta \vphantom{ \vartheta}^{\prime}\)\(\;>\;\)\(0\) also does, hence:
Equal_areas:
\(\vartheta \vphantom{ \vartheta}^{\prime}\)\(\eta\)\(\;=\;\)\(\,\mathsf{constant}\,\)
\(\;\equiv\;\)\(K\)\(;\,\)
\(\;>\;\)\(0\)\(;\,\)
Differentiation on both sides of Equal_areas gives:
Angle_derivative:
\(\vartheta \vphantom{ \vartheta}^{\prime\prime}\)\(\eta\)\(\,+\,\)\(\vartheta \vphantom{ \vartheta}^{\prime 2}\)\(\gamma\)\(\,\mathrm{Sin}\,\)\(\vartheta\)\(\;=\;\)\(0\)\(;\,\)
After further manipulation, we have (except perhaps when both sides are singular):
\(\dfrac{-\, \vartheta \vphantom{ \vartheta}^{\prime\prime} \,\mathrm{Sin}\, \vartheta \,-\, \vartheta \vphantom{ \vartheta}^{\prime 2} \,\mathrm{Cos}\, \vartheta}{\mathrm{Cos}\, \vartheta \,-\, \gamma}\)\(\;=\;\)\(\dfrac{\vartheta \vphantom{ \vartheta}^{\prime\prime} \,\mathrm{Cos}\, \vartheta \,-\, \vartheta \vphantom{ \vartheta}^{\prime 2} \,\mathrm{Sin}\, \vartheta}{\mathrm{Sin}\, \vartheta}\)\(;\,\)
This indicates that the force exerted by the Sun to the planet is on the same line which connects them, but along opposite direction.
It remains to verify the magnitude \(\left|a\right| \vphantom{ \left|a\right|}^{2}\). We start from expression Acceleration:
\(\left|r\right| \vphantom{ \left|r\right|}^{4}\)\(\left|a\right| \vphantom{ \left|a\right|}^{2}\)\(\;=\;\)\(\eta \vphantom{ \eta}^{4}\)\(\,\cdot\,\)\(\vartheta \vphantom{ \vartheta}^{\prime\prime 2}\)\(\left(\vphantom{\beta \vphantom{ \beta}^{2} \mathrm{Cos} \left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2} \,+\, \mathrm{Sin} \left[\vartheta\right] \vphantom{\mathrm{Sin} \left[\vartheta\right]}^{2}}\right.\)\(\beta \vphantom{ \beta}^{2}\)\(\,\mathrm{Cos}\,\)\(\left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2}\)\(\,+\,\)\(\,\mathrm{Sin}\,\)\(\left[\vartheta\right] \vphantom{\mathrm{Sin} \left[\vartheta\right]}^{2}\)\(\left.\vphantom{\beta \vphantom{ \beta}^{2} \mathrm{Cos} \left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2} \,+\, \mathrm{Sin} \left[\vartheta\right] \vphantom{\mathrm{Sin} \left[\vartheta\right]}^{2}}\right)\)\(\,+\,\)\(\eta \vphantom{ \eta}^{4}\)\(\,\cdot\,\)\(2\)\(\vartheta \vphantom{ \vartheta}^{\prime\prime}\)\(\vartheta \vphantom{ \vartheta}^{\prime 2}\)\(\gamma \vphantom{ \gamma}^{2}\)\(\,\mathrm{Cos}\,\)\(\vartheta\)\(\,\mathrm{Sin}\,\)\(\vartheta\)\(\,+\,\)\(\eta \vphantom{ \eta}^{4}\)\(\,\cdot\,\)\(\vartheta \vphantom{ \vartheta}^{\prime 4}\)\(\left(\vphantom{\mathrm{Cos} \left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2} \,+\, \beta \vphantom{ \beta}^{2} \mathrm{Sin} \left[\vartheta\right] \vphantom{\mathrm{Sin} \left[\vartheta\right]}^{2}}\right.\)\(\,\mathrm{Cos}\,\)\(\left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2}\)\(\,+\,\)\(\beta \vphantom{ \beta}^{2}\)\(\,\mathrm{Sin}\,\)\(\left[\vartheta\right] \vphantom{\mathrm{Sin} \left[\vartheta\right]}^{2}\)\(\left.\vphantom{\mathrm{Cos} \left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2} \,+\, \beta \vphantom{ \beta}^{2} \mathrm{Sin} \left[\vartheta\right] \vphantom{\mathrm{Sin} \left[\vartheta\right]}^{2}}\right)\)\(;\,\)
It may require patience to verify, but by \(\,\mathrm{Sin}\,\)\(\left[\vartheta\right] \vphantom{\mathrm{Sin} \left[\vartheta\right]}^{2}\)\(\,+\,\)\(\,\mathrm{Cos}\,\)\(\left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2}\)\(\;=\;\)\(1\), \(\beta \vphantom{ \beta}^{2}\)\(\,+\,\)\(\gamma \vphantom{ \gamma}^{2}\)\(\;=\;\)\(1\), and by repeatedly using AngleDer, we may obtain:
\(\left|r\right| \vphantom{ \left|r\right|}^{4}\)\(\left|a\right| \vphantom{ \left|a\right|}^{2}\)\(\;=\;\)\(\eta \vphantom{ \eta}^{2}\)\(\,\cdot\,\)\(\vartheta \vphantom{ \vartheta}^{\prime 4}\)\(\gamma \vphantom{ \gamma}^{2}\)\(\,\mathrm{Sin}\,\)\(\left[\vartheta\right] \vphantom{\mathrm{Sin} \left[\vartheta\right]}^{2}\)\(\,\cdot\,\)\(\left(\vphantom{\beta \vphantom{ \beta}^{2} \mathrm{Cos} \left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2} \,+\, \mathrm{Sin} \left[\vartheta\right] \vphantom{\mathrm{Sin} \left[\vartheta\right]}^{2}}\right.\)\(\beta \vphantom{ \beta}^{2}\)\(\,\mathrm{Cos}\,\)\(\left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2}\)\(\,+\,\)\(\,\mathrm{Sin}\,\)\(\left[\vartheta\right] \vphantom{\mathrm{Sin} \left[\vartheta\right]}^{2}\)\(\left.\vphantom{\beta \vphantom{ \beta}^{2} \mathrm{Cos} \left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2} \,+\, \mathrm{Sin} \left[\vartheta\right] \vphantom{\mathrm{Sin} \left[\vartheta\right]}^{2}}\right)\)\(\,+\,\)\(\eta \vphantom{ \eta}^{3}\)\(\,\cdot\,\)\(\vartheta \vphantom{ \vartheta}^{\prime 2}\)\(\left(\vphantom{\,-\, \gamma \,\mathrm{Sin}\, \vartheta}\right.\)\(\,-\,\)\(\gamma\)\(\,\mathrm{Sin}\,\)\(\vartheta\)\(\left.\vphantom{\,-\, \gamma \,\mathrm{Sin}\, \vartheta}\right)\)\(\,\cdot\,\)\(2\)\(\vartheta \vphantom{ \vartheta}^{\prime 2}\)\(\gamma \vphantom{ \gamma}^{2}\)\(\,\mathrm{Cos}\,\)\(\vartheta\)\(\,\mathrm{Sin}\,\)\(\vartheta\)\(\,+\,\)\(\eta \vphantom{ \eta}^{4}\)\(\,\cdot\,\)\(\vartheta \vphantom{ \vartheta}^{\prime 4}\)\(\,\cdot\,\)\(\left(\vphantom{\mathrm{Cos} \left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2} \,+\, \beta \vphantom{ \beta}^{2} \mathrm{Sin} \left[\vartheta\right] \vphantom{\mathrm{Sin} \left[\vartheta\right]}^{2}}\right.\)\(\,\mathrm{Cos}\,\)\(\left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2}\)\(\,+\,\)\(\beta \vphantom{ \beta}^{2}\)\(\,\mathrm{Sin}\,\)\(\left[\vartheta\right] \vphantom{\mathrm{Sin} \left[\vartheta\right]}^{2}\)\(\left.\vphantom{\mathrm{Cos} \left[\vartheta\right] \vphantom{\mathrm{Cos} \left[\vartheta\right]}^{2} \,+\, \beta \vphantom{ \beta}^{2} \mathrm{Sin} \left[\vartheta\right] \vphantom{\mathrm{Sin} \left[\vartheta\right]}^{2}}\right)\)
\(\;=\;\)\(K \vphantom{ K}^{4}\)\(;\,\)
This agrees the Newton law of gravitation.
❧ November 15, 2018; revised August 8, 2021