7 Hamiltonian Formalism

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

The motion is therefore confined in the region  and these points are the turning points of the motion. For certain value of the potential , there may be only one root in which case the motion is circular. If the motion of the particle is such that the orbit is closed after finite number of round travel between  and , the central potential satisfies certain conditions. If the orbit after finite number of oscillations does not come back to the initial orbit, the orbit is said to be open as shown below:

To find the condition for closed orbit, we need to find the change in angle  after the particle has made one complete oscillation from  and  and back; thus

Example: Show that if the potential  varies as some integer power of radial distance a closed non-circular path will result only if  or +1. In terms of the variable , equation (11.4) can be written as

The integral (11.5) can be expressed in terms of circular functions provided the radical in the integral (11.5) can be expressed in the form

For the stability of the orbit, it is required that  is imaginary  will increase exponentially making the  orbit to be unstable. Therefore the stability of the orbit is ensured provided

5. Perihelion of Mercury

We mentioned in the introduction that the perihelion of Mercury that is semi-major axis was found to advance at the rate of roughly 574 sec. of arc per century. This would happen if the law of force deviated to the inverse square law. We will assume the presence of a small  force in addition to the inverse square law to understand the advance in the perihelion. This small  force could for example arise due to the pressure of other gravitating bodies i.e. other planets, comets, etc. apart from the sun which provides the major inverse square force. We take the potential  to be

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 providedwhere ρ is the radius of the circular orbit.
• The perihelion of mercury arises because of the presence of a small repulsive inverse cubic force.