Friday, August 9, 2013
Illuminating Hamilton on the Path of Light & Planets
In 1833 Hamilton wrote a paper on the path of light and the planets which dealt with the use of a characteristic or generating function to determine the path of motion. He included two examples which showed that a straight path was the minimum path for some alternative paths. The first example was that of a straight line of length V and the set of circular arcs, Vε, for which V is the chord. The two are shown as bold lines in the figure below. The parameter ε is determined by the ratio of the height of the arc, a, to 1/2 V.
The formula for the length of the arc, Vε, given in Eqn 6 can be derived as follows.
For small ε, tan-1ε is approximately ε and the factor on the right hand side of the bottom line involving ε reduces to ε2+1 and therefore Vε=V when ε=0.
Calculation shows that Hamilton's expression in Eqn 7 for the length of the arc is not equal to that obtained from the formula above. One can derive a similar formula that works but first one needs to derive an integral expression for θ=tan-1ε.
This expression can be integrated by parts and substituted into the formula in Eqn 6 to get a new formula for the length of the arc.
Once again Vε=V when ε=0 and the product of the two factors in the second term on the right increases with increasing positive ε. Since the formula involves functions of ε2 and their averages it is symmetric, that is, it has the same value when ε is replaced by -ε.
The second example involves a straight line and elliptic arcs with V, Vε and ε given in the figure below.
One can derive Hamilton's formula in Eqn 10 as follows allowing for differences in notation.
Again one sees that Vε=V when ε=0 and since the second term on the right hand side is a sum of positive quantities it increases with ε. The integral in Eqn 9 is the length of the elliptic arc and is smallest when ε=0 since only positive quantities are involved in the summation. One can also show that Vε is symmetric and that if 0<ε<η then, for I(ε)=Vε/V, 1<I(ε)<I(η) and so I(ε) increases with positive ε.
Hamilton was not the first to use the minimum path approach in mechanics. Over one hundred years earlier the Bernouli brothers showed that the brachistochrone is the path of minimum time for a descent between any two points.
Supplemental (Aug 20): The figure shown is not correct. Eqn. (9) in Hamilton's paper for the arc length indicates that the φ is the polar angle corresponding to the "parametric latitude." It doesn't indicate the direction of a point on the ellipse but its use simplifies the necessary equations.