# How to interpret the following ODE's of Newton's movement laws expressed in polar coordinates?

The question relates to the movement of a spaceship which is at first hovering above the Earth at height $H$ with it's thrusters at max. The captain then proceeds to turn the spaceship $90^{\circ}$ so that it's thrusters act perpendicular to Earth's surface. The question asks whether the spaceship will crash or if it will make it to space and this is to be solved numerically in Matlab with help of either Runge-Kuttas method or the built in ODE45 function.

The problem is that I'm not that smooth when it comes to ODE's so I'm having difficulties interpreting the ODE's that are given in the assignment. According to the assignment the situation is described in Newton's laws of motion expressed in polar coordinates. So far so good. However some of the variables used are not explained, and when $r$ is used I don't understand if it's the explicit distance from Earth to the spaceship or a function or something else. The given ODE's are as follows:

$$\frac{d^2 r}{dt^2} - r(\frac{d\phi}{dt})^2 = G cos(\alpha) - g\frac{R²}{r²}$$ $$r\frac{d²\phi}{dt²} + 2\frac{dr}{dt} \frac{d\phi}{dt} = G sin(\alpha)$$

Where $\alpha$ was $0^{\circ}$ before the maneuver but $90^{\circ}$ after (at time $t = 0$). $R$ is Earth's radius, $g$ is gravity at Earth's surface and $G$ is the gravity at the height $H$ where the spaceship hovered at equilibrium between Earth's gravity and the force from the thrusters: $G = gR²/(R+H)²$.

The assignment then explains that the initial values are given by the fact that the spaceship was completely still before the cosmonauts maneuver.

So how do I interpret these ODE's? What does $\phi$ represent? And what are the actual coordinates?

EDIT 1: Is the following the correct interpretation of the ODE's?

$$r''(t) - r(\phi(t))² = Gcos(\alpha) - g \frac{R²}{r(t)}$$ $$r(\phi''(t)) + 2r(t)\phi(t) = Gsin(\alpha)$$

I have the solution now to the ODE's in MATLAB through the ODE45 solver. But I want to understand the ODE's better. For example I want to plot the curve in the polarplot that exists in MATLAB and I need the $x$ and $y$ coordinates of the spaceship to compute the angle and radius. Is there a way to extract that information from these ODE's or the resulting vectors obtained from ODE45?

EDIT 2: Finally starting to get a hang of this. The equations can be written as $$r''(t) - r(t)(\phi'(t))² = Gcos(\alpha) - g \frac{R²}{r(t)}$$ $$r(t)\phi''(t) + 2r'(t)\phi'(t) = Gsin(\alpha)$$

And as has been stated in the comments $\phi(t)$ is the angle between the $x$-axis and the line from Earth's center to the spaceship and $r(t)$ the distance between Earth's surface and the spaceship. Which is convenient since ODE45 returns vectors of both when the proper set of first order ODE's are provided.

• The left hand side of your two equations are the acceleration vector for a point constrainted to be in a plane and expressed in polar coordinates. It's not clear for me the actions the ship does or the idealisation for the rotation of the earth: is the earth rotating and the ship is hovering still wrt an intertial frame? and, later, the thrust is always in the same direction $\alpha$? etc. – Rafa Budría May 10 '17 at 16:26
• $\phi$ is the polar angle, i.e. the angle between the line from the ship to the center of the Earth and some other line from the center of the Earth. $r$ is the distance of the ship from the center of the Earth. – Rafa Budría May 10 '17 at 16:29
• I think that the $\phi$ is supposed to be a time derivative in the first equation after the edit. – Cuhrazatee May 10 '17 at 16:36
• @RafaBudria It's a very elementary task which doesn't regard the rotation of the Earth. It's only purpose is to know how to solve the problem in MATLAB. – ChristofferAB May 10 '17 at 20:10