- #1

lalbatros

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- 2

Is the Coulomb potential, 1/r, the only one that produces a periodic motion?

If no, is there a condition for periodicity to occur?

Thanks,

Michel

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In summary, the conversation discusses the conditions for periodic motion of a test particle in a central field. The Coulomb potential, 1/r, is not the only potential that can produce periodic motion, but there are certain conditions, such as an effective potential and closed orbits, that must be met for true periodicity to occur. While Goldstein's "Classical Mechanics" provides a list of force-laws that result in closed orbits, Landau-Lifchitz states that there are only two potentials that can produce closed trajectories: 1/r and r². The reason for this is not entirely clear and further investigation may be needed.

- #1

lalbatros

- 1,256

- 2

Is the Coulomb potential, 1/r, the only one that produces a periodic motion?

If no, is there a condition for periodicity to occur?

Thanks,

Michel

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- #2

- 10,334

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There's a list in Goldstein "Classical Mechanics" of the force-laws that give closed orbits, IIRC.

- #3

lalbatros

- 1,256

- 2

You gave me the idea to check in Landau-Lifchitz. (I don't have Goldstein unfortunately).

He states that there are only two potentials that result in closed trajectories: 1/r and r² . That's already good to know. However, I don't see where this magic comes from. The algebra is simple and clear, but it does not indicate some more "fundamental" reason.

Michel

PS: In other words, can a property of an integral be understood in another way?

- #4

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I've never had any deeper insight than the mathematical demonstration you've already seen, sorry.

A periodic orbit is a type of motion in which a body or particle moves in a repeated pattern around a fixed point or axis. It is characterized by a constant period, meaning that the time it takes for the body to complete one full cycle is always the same.

The Coulomb potential, also known as the electrostatic potential, is a fundamental force that describes the attraction or repulsion between charged particles. In the context of periodic orbits, the Coulomb potential is responsible for creating a stable equilibrium point around which the particle can orbit in a repeated pattern.

The shape of a periodic orbit is directly influenced by the strength of the Coulomb potential. If the potential is weak, the orbit will be more circular, while a stronger potential will result in a more elliptical orbit. Additionally, the orientation of the orbit can also be affected by the direction of the Coulomb force.

Yes, periodic orbits can be observed in many real-life systems, such as the motion of planets around the sun, electrons orbiting the nucleus of an atom, and even the motion of a pendulum. These systems exhibit periodic motion due to the presence of a Coulomb potential.

Yes, there are other forces and potentials that can produce periodic orbits. Some examples include the gravitational potential, magnetic potential, and harmonic potential. However, the Coulomb potential is a particularly important and commonly studied factor in producing periodic motion.

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