Here's how the plotter works

Newton's second law is$$

\vec{F}=m\vec{a}

$$Newton's law of gravity is$$

\vec{F}=\frac{GMm}{r^2}

$$They give you two differential equations of motion in polar coordinates$$

\frac{d^2r}{dt^2}-r\left(\frac{d\theta}{dt}\right)^2=-\frac{GM}{r^2}

$$$$

2\frac{dr}{dt}\frac{d\theta}{dt}+r\frac{d^2\theta}{dt^2}=0

$$Kepler's laws are

\vec{F}=m\vec{a}

$$Newton's law of gravity is$$

\vec{F}=\frac{GMm}{r^2}

$$They give you two differential equations of motion in polar coordinates$$

\frac{d^2r}{dt^2}-r\left(\frac{d\theta}{dt}\right)^2=-\frac{GM}{r^2}

$$$$

2\frac{dr}{dt}\frac{d\theta}{dt}+r\frac{d^2\theta}{dt^2}=0

$$Kepler's laws are

- The orbit of a planet is an ellipse with the Sun at one of the two foci.
- A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
- The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit.

The second differential equation quickly gives you Kepler's second law. It is more difficult to get$$

r=\frac{P}{1+e\cos{\left(\theta-\phi\right)}}

$$That is a circle when ##e=0##, an ellipse when ##\left|e\right|<1##, a parabola when ##\left|e\right|=1## and a hyperbola when ##\left|e\right|>1##. ##\phi## is the angle of the axis of symmetry of the curve. ##P## is a magic constant. The equation gives you Kepler's first law.

r=\frac{P}{1+e\cos{\left(\theta-\phi\right)}}

$$That is a circle when ##e=0##, an ellipse when ##\left|e\right|<1##, a parabola when ##\left|e\right|=1## and a hyperbola when ##\left|e\right|>1##. ##\phi## is the angle of the axis of symmetry of the curve. ##P## is a magic constant. The equation gives you Kepler's first law.

Newton must have been very pleased when he did those. Part of the job was inventing calculus!

The initial conditions determine ##P,e,\phi##. That requires more work if you only know the initial position and velocity.

Full details (except inventing calculus) in Commentary 5.4 Orbital toypdf. (9 pages)

The spreadsheets for plotting the curves are at

Commentary 5.4 Orbital toy.xlsm (with animation macros)

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