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LSpice
LSpice
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The most spectacular application is the theory of orbits under a central force field. This is basically what Kepler and Newton did (not, of course, using this this notation). One of Kepler's key observations on the orbit of Mars was the constancy of a certain quantity associated with points on the orbit. In modern terms he had noticed that $r(1+ e\cos \theta)$ was constant along the orbit. We now recognise this instantly as the equation of an ellipse with the origin (the sun) at a focus. I suggest that you look up the theorem of Binet for more information on this theme. There are other applications of the use of forms $r=f(\theta)$ (even better $rf(\theta)=1$). Space is too short to give a detailed list but there are many such, e.g., unifying the basic explicit solutions of the motivating problems of the elementary calculus of variations.

The most spectacular application is the theory of orbits under a central force field. This is basically what Kepler and Newton did (not, of course, using this this notation). One of Kepler's key observations on the orbit of Mars was the constancy of a certain quantity associated with points on the orbit. In modern terms he had noticed that $r(1+ e\cos \theta)$ was constant along the orbit. We now recognise this instantly as the equation of an ellipse with the origin (the sun) at a focus. I suggest that you look up the theorem of Binet for more information on this theme. There are other applications of the use of forms $r=f(\theta)$ (even better $rf(\theta)=1$). Space is too short to give a detailed list but there are many such, e.g., unifying the basic explicit solutions of the motivating problems of the elementary calculus of variations.

The most spectacular application is the theory of orbits under a central force field. This is basically what Kepler and Newton did (not, of course, using this notation). One of Kepler's key observations on the orbit of Mars was the constancy of a certain quantity associated with points on the orbit. In modern terms he had noticed that $r(1+ e\cos \theta)$ was constant along the orbit. We now recognise this instantly as the equation of an ellipse with the origin (the sun) at a focus. I suggest that you look up the theorem of Binet for more information on this theme. There are other applications of the use of forms $r=f(\theta)$ (even better $rf(\theta)=1$). Space is too short to give a detailed list but there are many such, e.g., unifying the basic explicit solutions of the motivating problems of the elementary calculus of variations.

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user6891
user6891
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The most spectacular application is the theory of orbits under a central force field. This is basically what Kepler and Newton did (not, of course, using this this notation). One of Kepler's key observations on the orbit of Mars was the constancy of a certain quantity associated with points on the orbit. In modern terms he had noticed that $r(1+ e\cos \theta)$ was constant along the orbit. We now recognise this instantly as the equation of an ellipse with the origin (the sun) at a focus. I suggest that you look up the theorem of Binet for more information on this theme. There are other applications of the use of forms $r=f(\theta)$ (even better $rf(\theta)=1$). Space is too short to give a detailed list but there are many such, e.g., unifying the basic explicit solutions of the motivating problems of the elementary calculus of variations.