Can Newton's Theory of Gravity Explain Planetary Elliptical Orbits?

[kent davidge]

Is it difficult to use Newton's theory of gravity for showing that planet's orbits must be elliptical?

 

[Charles Link]

Not tremendously difficult, but not nearly as simple as assuming circular orbits. The mechanics book by Kleppner and Kolenkow has a good derivation of this.

 

[hilbert2]

They can also be parabolic or hyperbolic, if the planet has a sufficient escape velocity. Finding the allowed orbits from Lagrange's equations of motion may require some work, unless you already know the solution (in which case you can test it by substitution).

 

[kent davidge]

I imagined the following approach. The equation of an ellipse is $x^2/a^2 + y^2/b^2 = 1$. If we define $\bar{x} = x / a, \bar{y} = y / b$ we have $\bar{x}^2+ \bar{y}^2 = 1$ which is the equation of a unit sphere.

Of course that satisfy Newton's equation of gravity and so do $x^2 / a^2$ and $y^2 / b^2$.

 

[hilbert2]

The trajectory of an object is not just a set of points, like a circle on the plane. The object has both a position and velocity at every instant.