﻿ A Correspondence Between Statically and Dynamically Determined Force Fields
San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
U.S.A.

A Correspondence Between Statically and
Dynamically Determined Force Fields

## Background

Consider the following two situations in mathematical physics.

• A charge is distributed over some geometric region R and the charge density is known. The charge could be gravitational mass, electrical charge, magnetic charge or nucleonic charge (the charge associated with the nuclear strong force). The charge density is given by a function ρ(Z), where Z is a position vector. This is the static situation.
• A point particle of charge q traverses the same geometric region R. A probability density function is determined based on the proportion of the time the particle spends at various locations. The expected value of the force at any location can be computed based upon that probability density function. This is the dynamic situation.

Suppose the function giving the force dF on a unit charge at position Z due to a charge of dq located at position ζ is given by

#### dF = F(s)dq

where s is the distance between point Z and point ζ. This distance is given by |Z−ζ|. The increment of charge is given by ρ(ζ)dR.

The intensity vector G of the field at Z is then given by

#### G(Z) = ∫R F(|Z−ζ|)ρ(ζ)dR

The intensity at Z is constant so the expected value of G, E{ }, is just G; i.e.,

#### E{G(Z)} = G(Z) = ∫R F(|Z−ζ|)ρ(ζ)dR

On the other hand in the dynamic situation the field intensity at Z fluctuates. When the charge Q is at point ζ the force on a unit charge at point Z is given by

#### H(t) = F(|Z−ζ|)Q

Let P(ζ) be the probability density function representing the proportion of the time the charge spends at point ζ. For a particle traveling along a path at a velocity of v the probability density function is given by

#### P(ζ) = 1/(T|v(ζ|)

where T is the total time required to traverse the path.

The expected value of the vector of field intensity H is then given by

## The Correspondence

Note that if the charge density ρ(ζ) is given by P(ζ)Q then the dynamic case problem is mathematically identical to the static case problem. This correspondence points up the significance of the time-spent probability density function to mechanics. It is as significant as the spatial distribution of any charge.

From this correspondence it then follows that if the force law is strictly an inverse distance square law then the effect of a particle that uniformly traverses a spherical surface is the same, on average outside of the sphere as if it were located at the center of the sphere. Inside the sphere the effect is zero. This is relevant for the field due to the positive electrical charge of the protons in nuclei.

The so-called strong nuclear force is not a strictly inverse distance squared force because the particles carrying the force, the pi-mesons, decay over time and hence with distance. This makes the likely functional form inverse distance squared weighted by a negative exponential; i.e. H·exp(−s/s0)/s², where H and s0 are parameters.

## Extensions

The above derivation which was for a single particle applies equally as well to multiple particles. The multiple particles may be independent or interlinked. As a particular case it would apply to structures that rotate.

Thus for n particles with spent-time probability density functions and charges of PI and QI, respectively. for i=1...n the equivalent charge density ρ(ζ) is given by

## A Spherical Shape for Nuclei

It is not easy to find a static structure for charges that is spherical. It is however easy to specify a dynamic structure that produces the appearance of a sphere. For example, a circular ring that is flipping rapidly over and over like a flipped coin gives that spherical appearance. If the ring is a ring of linked particles and it is rotating about an axis through its center perpendicular to its plane then the particles appear at all different points of the sphere of rotation.

## Conclusions

A particle with a generic charge of Q which traverses a periodic path with a spent-time probability density function of P(ζ) has the same expected value effect as a static charge distribution density equal to P(ζ)Q.

This also applies to multiple particle systems whether they are independent particles or linked together in a structure.