21 Canonical Transformation 1

Ashok Goyal

1. Introduction

A particle in a central–force field may execute bounded or open noncircular motion depending on the nature of the force. If the particle in executing one complete revolution does not return to its original position it signals a deviation from the inverse – square law force however slight. An irregularity in the motion of the planet mercury was observed. It was observed that the perihelion of mercury that the semi-major axis advances at the rate of roughly 574 sec. of arc per century. Calculation of the influence of other planets predicted an advance of approximately 531 sec of arc per century leaving a deficit of about 40 sec of arc per century. Einstein’s General Theory of Relativity in post – Newtonian approximation was able to account for this difference of 43 sec of arc per century and became one of the greatest triumphs of General Relativity. In this unit we will discuss the conditions for closed orbits, stability of the orbits and an estimate of the advance of the perihelion of mercury in the presence of deviations from the inverse-square law.

2. Open and Closed Orbits:

We saw earlier that the radial velocity of a particle in a central field is given by

4. Stability of Orbits.

If  the total energy of the system is equal to the minimum of the effective potential i.e.

Substituting for P  from equation (11.15)

Substituting for U from (10.24) and writing

Thus the perihelion advance or the shift in the semi-major axis is proportional to the strength of the inverse cubic repulsive force and the orbit is a ‘precessing ellipse’ shown in Fig. 1.

6. Summary

•  A particle moving in a force field where the force is either inverse-square or a linear harmonic force has a closed orbit.
• The inverse cubic force law results in an open precessing elliptical orbit.
• The circular orbit of a particle moving in a general central force field is stable provided

where ρ is the radius of the circular orbit.

• The perihelion of mercury arises because of the presence of a small repulsive inverse cubic force.