Determination of the apsidal angles and Bertrand's theorem

Area of Science:

• Classical Mechanics
• Celestial Mechanics
• Mathematical Physics

Background:

• The study of central force motion is fundamental in physics.
• Bertrand's theorem is a key result concerning stable orbits in central potentials.
• Calculating apsidal angles and precession rates is crucial for understanding orbital dynamics.

Purpose of the Study:

• To derive a general expression for determining apsidal angles for any central potential.
• To investigate conditions under which apsidal angles are independent of energy and angular momentum.
• To provide an alternative, nonperturbative proof of Bertrand's theorem.

Main Methods:

• Derivation of a general formula for apsidal angles.
• Analysis of the conditions for energy and angular momentum independence.
• Application of the derived formula to prove Bertrand's theorem nonperturbatively.

Main Results:

• An expression for calculating apsidal angles applicable to arbitrary central potentials.
• Identification of conditions for apsidal angle independence from mechanical energy and angular momentum.
• A novel, nonperturbative proof of Bertrand's theorem.

Conclusions:

• The derived formula offers a versatile tool for analyzing orbital precession.
• The findings provide deeper insights into the conditions governing stable orbits in central potentials.
• The nonperturbative proof of Bertrand's theorem simplifies and generalizes its understanding.