How do I correctly introduce a time parameter into this equation?

Question

So, for the past few years it's been my goal to create an equation that would give me the position of an object in a gravitational field at time $t$, given it's initial position and velocity. At first the problem was that I didn't know enough to do the math. Now that I can do multivariable calculus I thought that problem would be solved, but I've just ended up running into a new problem. Please don't tell me how to solve it, but if you can give me a hint that would be great. Here's the set up for the problem:

A planet of mass M (and radius = 0) is situated at the origin. I know that the magnitude of acceleration due to gravity is $$\frac{GM}{r^2}$$ so an object at $(x,y)$ will have acceleration $$a(x,y)= \frac{GM}{x^2+y^2},$$ or, as a vector, $$\overrightarrow{a}(x,y)= \langle \frac{GM}{x^2+y^2}cos\theta, \frac{GM}{x^2+y^2}sin\theta\rangle$$

$$= \langle \frac{GM}{x^2+y^2}\frac{x}{\sqrt{x^2+y^2}}, \frac{GM}{x^2+y^2}\frac{y}{\sqrt{x^2+y^2}}\rangle$$ $$= \langle \frac{GMx}{(x^2+y^2)^{3/2}}, \frac{GMy}{(x^2+y^2)^{3/2}}\rangle$$

So, here's where I'm stuck. I can integrate with respect to distance and get

$$ W(x,y) = \langle -\frac{GM}{\sqrt{x^2+y^2}}, -\frac{GM}{\sqrt {x^2+y^2}}\rangle$$

which I think is a vector who's magnitude is the work done, but that doesn't tell me anything about time. I can integrate with respect to time, but that would give

$$f(x,y)= \langle \frac{GMx}{(x^2+y^2)^{3/2}}t, \frac{GMy}{(x^2+y^2)^{3/2}}t\rangle$$

which... I mean is naïve at best. It doesn't take into account the change in position that happens over time. The only thing that I can think of to do is somehow find parametric equations where x and y are functions of t, but that's basically what I'm trying to do anyway.

Any ideas? I want to find an equation such that I can put in a location and velocity and the equation will tell me what path the object will take. Is that even possible?

Answer 1

The path of the object will be an ellipse, a parabola, or a hyperbola. (An ellipse may also be a circle.)

If you draw line segments to the central point from any two points the moving mass passes through, the area enclosed by those two segments and the path will be proportional to the time taken to travel between those two points. The reason for this is that the angular momentum of the object around the central point never changes.

Mathematically, you can satisfy conservation of energy and conservation of momentum for a given initial position and velocity only within a certain range of distances from the central point. That range of distances tells you the shape of the orbit.

There's then a closed-form solution that gives you the exact position at any future time.

That's a few hints. If you need more, look up "Euler-Lagrange equation of planetary motion" on a good search engine.

This was a topic in an upper-level undergraduate physics course I took; I forget whether we were able to get through it in one 80-minute lecture, so expect to do quite a bit of discovery if you want to figure it out on your own. I don't think it relies on things you would normally learn in a course on multivariate calculus.