Orbital equations in polar coordinates

[TimK]

TL;DR Summary

I'm trying to develop a launch trajectory calculator using polar coordinates. When I integrate in polar coordinates I just get circles.

The equations of motion are:
$\ddot{r}-r{\dot{\theta}} ^{2} = -\frac{1}{r^{2}}$
for the radial acceleration and
$r\ddot{\theta} + 2\dot{r}\dot{\theta}= 0$
for the transverse acceleration

When I integrate these equations I get only circles. The energy of the system is constant and the angular momentum is constant but the trajectory is not an ellipse when it should be. What is going on here? Is there more then one solution?

 

[PeroK]

Summary:: I'm trying to develop a launch trajectory calculator using polar coordinates. When I integrate in polar coordinates I just get circles.

The equations of motion are:
$\ddot{r}-r{\dot{\theta}} ^{2} = -\frac{1}{r^{2}}$
for the radial acceleration and
$r\ddot{\theta} + 2\dot{r}\dot{\theta}= 0$
for the transverse acceleration

When I integrate these equations I get only circles. The energy of the system is constant and the angular momentum is constant but the trajectory is not an ellipse when it should be. What is going on here? Is there more then one solution?

A circle at constant speed is one solution, but the general solution is conic sections generally and an ellipse for closed orbits.

Without seeing your work it's not possible to see where you are going wrong. The derivation of elliptic orbits, however, is far from simple.

 

[TimK]

I should've mentioned that the circles are not centred on the origin. So the angular velocity is not constant for what should be an elliptical orbit.

 

[PeroK]

I should've mentioned that the circles are not centred on the origin. So the angular velocity is not constant for what should be an elliptical orbit.

That doesn't sound good!

Conservation of angular momentum is implied by a central force.

 

[TimK]

I just tried it again with a much larger eccentricity and it works. All I need now is a better integrator.

 

[mpresic3]

I have had some experience with these types of models of the inverse square problem and I have seen two different approaches. Rand Corporation in 1962 put together a rocket model, and that model integrates in polar coordinates in a manner similar to the way you are doing.

Another approach is to use cartesian coordinates x, y, . The computer does all the work anyway so you do not have elegant solutions. For example gravity in the x direction would just be :
GM x / r^3/2 or GMx / (x*x+y*y+z*z)^3/2. Similar for y directions, and z directions.

Often this is done by Runge Kutta, but Euler Integration is also possible.

When you get x(t), y(t)), you can always transform to polar coordinates later if you like plane polar coordinates better.

You do all the calculation in an inertial frame rather than an Earth fixed frame so you have no Coriolis or Centrifugal forces.

You account for rotation of the Earth or other rotation if necessary by using rotation matrices.