# How to solve for first integrals of motion

Tags:
1. Feb 21, 2015

### mat

1. The problem statement, all variables and given/known data
A point particle is moving in a field, where its potential energy is U=-α/r. Find first motion integrals.

2. Relevant equations
How to derive it

3. The attempt at a solution
I only figured out that all of this is related to the conservation of energy, but i dont know even the basic steps how to approach such problem..

2. Feb 21, 2015

### Orodruin

Staff Emeritus
Hi and welcome to Physics Forums!

In order for us to help you properly, you will need to provide more information on what you have tried and what conclusions you have come to by applying what you have learnt in class or in a textbook. For example, what are first integrals of motion? How do they relate to conserved quantities? What are the equations governing the behaviour of a particle in a potential? Do you know what Noether's theorem is?

3. Feb 22, 2015

### mat

Oh thanks for the reply. So i do have a vague understanding of what first integrals are and how they differ from constants of motion. First integral is some property of motion which does not change with respect to time, is symmetrical. So for the problem at hand we have motion in a central force field (potential energy depends only on position vector) and we can choose spherical coordinate system here to take advantage of the fact that the angular momentum does not change and we have motion in one plane. Now i read that since we have non changing angular momentum (dL/dt=0) we have three independant constants of motion (don't really know the reason why..) and we need to analyze those to get first integrals (i know that all first integrals are constants of motion, but not all constants of motion are first integrals). We construct lagrangian in polar coordinates L=T-V=1/2m((dr/dt)^2+(rdφ/dt)^2)-U(r) (still, don't actually know now what it is) and from this equation we can extract first integrals (?). I read about Noather's a little... As you can see, i have little understanding about this problem, but i must do it until 2015-02-25 and i am becoming desprate :D. Any clarification/solution/explanation is most welcome, we dont have worked examples on such problems and so it is immensly hard for me to just simply apply the theory.

4. Feb 22, 2015

### mat

I posted what i know above. Can i solve this problem just with Noether's theorem? Thank you

5. Feb 24, 2015

### Orodruin

Staff Emeritus
Noether's theorem relates symmetries of the Lagrangian to conserved quantities. All of the conservation laws you are familiar with are in one way or another corresponding to such a symmetry. For example, energy conservation is a direct implication of time translation invariance of the Lagrangian. Your Lagrangian has more symmetries than that and therefore more conserved quantities.

When we talk about first integrals in physics, this is normally equivalent to talking about constants of motion.