Transforming Changes in Position and Gradients from Rectangular to Spherical Coordinates.

  Small changes in position transform nearly linearly from rectangular to spherical coordinates and we can represent this transformation by the matrix L. The inverse of this matrix can be used to transform the gradient matrices in a similar manner. Starting with δr̄ in the spherical coordinate system and using the chain rule again we can evaluate δr, δθ, and δφ in terms of δx, δy, and δz and then substitute the coefficients of δθ and δφ for the partial derivatives to get the components of the transformation matrix L.

Finding the coefficients of δθ and δφ is a little tricky since we have ranges 0≤θ≤π and 0≤φ≤2π in spherical coordinates while the inverse tangent is defined only for -π/2≤θ≤π/2. It helps to rewrite the inverse tangent using expressions that are continuous for the segments of a circle with positive and negative values of y and using δ[tan^-1u]=δu/(1+u²) to evaluate δθ and δφ to get the results above.

The determinant of L is equal to 1 which confirms the magnitude of the δr̄ is the same in both coordinate systems.

Multiplying L on the right by the transpose of L verifies that it is the right inverse of L.

One can show that the transpose of L is also the left inverse of L.

So we arrive at the following summary of results.

Variations of the Radial Unit Vector

  We can use the chain rule to find an expression for the variation of a function, f, in terms of its variables and then factor this into two parts, one involving only positions and the second containing changes in position. This works for both spherical coordinates and rectangular coordinates. The first factor is known as the gradient and is commonly written as ∇f.

For future reference we collect some formulas for spherical coordinates here.

We can use the same trick to factor the changes in the direction cosines of the radial unit vector but in this case the coefficients of the changes in position are vectors. As expected both sets of gradients can be represented by the same formula in both coordinate systems. The matrices G' and G are the coefficients of the changes in position in spherical and rectangular coordinates respectively.

The Mirror Focal Points

  In Part I, section II.[10] of his Theory of Systems of Rays Hamilton connects the signs in the sum of the action with the shape of the mirrors and the nature of their focal points. An elliptical mirror has a real focus to which the rays converge and the sign of the length of the ray is taken as positive. For a hyperbolic mirror the rays diverge from the focus and the sign used is negative. We can use ℓ and ϑ to see what happens to the rays if these variables are changed slightly. The positions of the foci and the point of reflection allow us to determine the relations between the focal distances a and b and parameters ℓ and ϑ.

Independent adjustments to ℓ and ϑ allow us to find two nearby points on the elliptical and hyperbolic mirrors through the original point of reflection. Each pair of the new rays have new values of a and b which we can use to find a new reflection point (x,y). Placing the focal points on the x-axis simplifies the solution for the reflection point.

The values derived using the formulas starting with the parameters ℓ and ϑ check with the values obtained in the previous results. Plotting the rays for these reflections shows neighboring rays converge for the focus of the elliptical mirror and diverge for the hyperbolic mirror.

Supplemental (Jun 19): The parameters ℓ and ϑ form an auxiliary coordinate system derived from the biradial coordinates (a,b) for the focal points of the ellipse. For every pair of Cartesian coordinates (x,y) there is a unique pair of the auxiliary coordinates (ℓ, ϑ) in the plane of the rays. More generally the surfaces of reflection are ellipsoids and hyperboloids of revolution about the axis through the focal points. The auxiliary coordinates are related to prolate spheroidal coordinates. For both the ellipsoidal and hyperboloidal mirrors the angle of incidence equals the angle of reflection but while the reflected rays of the ellipsoidal mirror pass through the second focus those of the hyperboloid do not.

A Justification of Hamilton's Hat Trick

 Hamilton noted that the direction cosines do not change for variations along a ray. But getting from here to his three equations is a bit of a leap. One can show that the variations in the direction cosines, δα_i, are equal to a dot product and therefore proportional to the projection of the variation in position, δr̅ onto directed line segments A̅_i.

This shouldn't depend on what coordinate system we use so we can try to see what happens in spherical coordinates. We don't need to know the exact values of the components of the gradients A̅_i so we can represent them by a set of functions g. Since the direction cosines don't change for variations along the ray we conclude that g_ir=0 and the gradients are perpendicular to the direction of the ray.

Hamilton's methods appear to be essentially geometrical which employs projections and so he would not need a complete understanding of vector analysis to make some of the deductions.

Hamilton's Hat Trick

  Hamilton performed a bit of a hat trick when deriving the conditions for the integrability of a first order differential. In subsection [8] he makes an observation about the unit vector for the direction of a ray and then states three conditions that must be satisfied. Where do they come from? We can start with deriving expressions for the direction cosines α, β, & γ and see what happens to the magnitude of the unit vector as we vary the point of intersection.

The result is a condition involving the direction cosines and their gradients but it isn't quite what we're looking for. We can next try evaluating the gradient of one of the direction cosines and see what happens.

We find that the gradient can be expressed as the sum of two vectors. The same can be done for the other gradients and taking the dot product of each of the gradients with the direction unit vector we get Hamiton's three equations.

Supplemental (Jun 9): We can verify the first condition found by using the formulas for the direction cosine gradients.

Hamilton appears to have arrived at his set of equations by noting that the direction of change of the direction cosines is perpendicular to the direction of the ray so their projections onto the direction of the ray would be zero.