# Motion Integrals of a Particle in a Force Field

I am trying to wrap my head around the following problem:

A point particle is moving in a field, where its potential energy is U=-α/r. Find first motion integrals.

In our university we have no examples solved or explained on these types of problems whatsoever, therefore it is immensely hard for me to grasp the intuition of finding the answer... From the books I get that here we are talking about the motion in central force field, we choose spherical coordinate system with the origin where the center of force is, we are analyzing the angular momentum because of the symmetry, we somehow extract so called first motion integrals from there, but i have little idea of what they are or how to apply any of this... I need help to understand the principle of how to deal with such problems, thank you for any tips!

• If you don't understand what they are (supposing you've read Wikipedia), you can have a look at: physics.stackexchange.com/q/55861 – jinawee Feb 19 '15 at 22:28
• Helpful. But i need an example of how to solve the problem given, any help appreciated.. – FringeEvent Feb 20 '15 at 8:46
• I figured out, that i don't need to use lagrangians there and i can use conservation of energy to get the first integral: m*d^2*r/(dt)^2+-α/r=E. I need some help, if you could answer some of my questions, that would be great. 1) Can i get more first integrals from the given conditions? 2) Is it enough to solve the equation that i wrote for r to get the answer? 3) How do i deal with E in the equation, do i assume it is a function of r or what? – FringeEvent Feb 23 '15 at 18:35