Motion in Gravitational Field

PSarkar

I want to find out the general equations of motion for a particle with an initial velocity [itex]v_0[/itex] in a gravitational field by a point/spherical mass (assuming this is a large mass which doesnt move). Assume that the origin of the coordinate system is the point mass. If the vector equation of the particle's path is [itex]\mathbf{r}(t)[/itex], then the acceleration should be the second derivative,
[tex]\frac{d^2 \mathbf{r}}{dt^2}[/tex]
The acceleration is caused by the gravitaional field (acceleration field) given by,
[tex]A(\mathbf{r}) = GM\frac{\mathbf{r}}{|\mathbf{r}|^3}[/tex]
But we already have the acceleration of the particle,
[tex]\frac{d^2 \mathbf{r}}{dt^2} = GM\frac{\mathbf{r}}{|\mathbf{r}|^3}[/tex]
So the general solution for [itex]\mathbf{r}[/itex] can be found by solving the above differential equation but I couldn't do it. Can anyone show me how it is done?

PSarkar

Does anybody know how to solve it or is there no exact analytic solution?

chrisbaird

Yes, this equation is solvable analytically. It is best to approach it using Lagrangians. The solutions are elliptical orbits first described by Kepler's laws.

PSarkar

I have seen Lagrangians and Euler-Lagrange equation but I don't know them well. Would someone kindly show me the steps in the solution?

PSarkar

Does any one know how to do it?

tripleM

By Kepler:

[itex]\frac{d^{2}r}{dt^{2}} = GM\frac{r}{|r|^{2}}[/itex]

by Newton:
F=ma, P=mv

[itex]F=m*GM\frac{r}{|r|^{2}}[/itex]

Where m is the mass of the orbiting body and M is the point mass (center of orbit).
We also know that motion for a particle follows the path dictated by:

[itex] r=r(r,θ,t)=re_{r}[/itex]

[itex] v=e_{r}\frac{dr}{dt}+rωe_{θ}[/itex]

[itex] a=e_{r}(\frac{d^{2}r}{dt^{2}}-rω^{2})+(rα+2ω\frac{dr}{dt})e_{θ}[/itex]

This is using a polar coordinate system. If you substitute your acceleration into this equation and then solve for zero velocity, you will have a suitable equation for two dimensional orbit. For higher dimensions there is considerably more work, and I frankly have too much else to do!

By the way: This is taken directly from Newton's equations of motion, Under General Planar Motion.

Khashishi

You can find a detailed derivation in Landau and Lifshitz, Mechanics.