Tuesday, June 19, 2018

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.

Sunday, June 10, 2018

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, δαi, are equal to a dot product and therefore proportional to the projection of the variation in position, δr̅ onto directed line segments A̅i.

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 A̅i 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 gir=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.

Friday, June 8, 2018

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.

Wednesday, May 30, 2018

Hamilton - Theory of Systems of Rays, Reflection from the Hyperbola

  Hamilton discusses the reflection of a ray from a hyperbolic mirror in Theory of Systems of Rays, Section 1, item 5. This appears to be what he is talking about. Light from focal point P travels to the intersection point R and is reflected by the hyperbola beyond the elliptical mirror. The tangent and normal directions are reversed for the hyperbola. The path of the light reflecting from the other side of the hyperbola is along the extension of the initial ray from P to R.

One can find more information about ellipses and hyperbolas in Casey, A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections.

Saturday, May 26, 2018

A First Look Into Hamilton's Theory of Systems of Rays

  Following the argument in Hamilton's Theory of Systems of Rays can be somewhat challenging. It starts out with the reflection of light from a mirror and he gives an analytic representation the law of reflection in equation (A). Hamilton appears to be using biradial coordinates with ρ and ρ' being the distances from the foci or poles R and R' below. Let the unit vector m̂ represent the direction of a line in the plane of the mirror and n̂ the direction normal to it. Since the angle of incidence equals the angle of reflection the magnitudes of the components of the directions, ê and ê', of the rays from the point reflection to the two points R and R' are equal except for the sign the projection onto the mirror. This allows us to derive the vector equivalent of equation (A) which is the projection of these vectors onto the direction of an arbitrary line, l.

One can show that the angle of incidence being equal to that of reflection is equivalent to the path length ℓ=a+b being a minimum at the point of reflection as follows. Here the two radii are a and b and the reference points P and Q.

One proceeds by deriving the condition for the direction ê of the line along the mirror which results from changing the point of reflection by a infinitesimal amount. The first two terms of Taylor's series are used to estimate the changes for the radii a and b.

Next we can find the direction the equation for ê satisfies by doing a simple search for the components of the unknown vector.

So the direction we need is that of â-b̂ corresponding to the unit vector ê=(â-b̂)/| â-b̂|.

The biradial coordinates are not as useful as they might be since we can't vary the two radii independently. We know however that changes in position along the mirror don't change the value of the pathlength ℓ locally so we can try using this as one coordinate. Replacing the direction in the differential dℓ with that perpendicular to it and integrating we get another function of the position R, ϑ=a-b.

Both dℓ and dϑ are exact differentials which means that the result of the integration from one point to another depends just on the endpoints and not the path between. One can derive the condition for an exact differential by varying the path of integration.

Using the expressions for da and db as a functions of x and y above we can now show that dℓ and dϑ satisfy this condition in the plane of the reflection.

The curves of constant ℓ and ϑ can be shown to be ellipses and hyperbolas respectively by substituting the functions of a(x,y) and b(x,y) above into their definitions.

Here's a specific example to help tie everything together.

Hopefully this will help illuminate some of Hamilton's 1824 paper on rays.

Monday, April 2, 2018

Poggendorff's Apparatus

  Poggendorff in 1853 designed an apparatus to demonstrate the changes in the tensions in Atwood's Machine that result from the acceleration of the masses. Mach gives a brief description in his Science of Mechanics. A compound Atwood's Machine is easier to analyze since the suspension points from the upper pulley don't change while the ends of the beam in Poggendorff's apparatus do. Only two parameters are required to specify the configuration of the compound machine, the rotation angles of the two pulleys or the arcs s ant t.

The positions of the masses are specified by the vertical coordinates, xi. From the balance of forces for each mass we obtain expressions relating the tensions and torques to the two accelerations, s̈ and ẗ.

One can then find the formulas for the individual tensions and by using the difference in tensions acting on the two pulleys we can derive a pair of linear equations for the unknown accelerations.

If we have cylindrical pulleys we can relate their moment of inertia to their masses as follows.

We are now in a position to compute the accelerations and tensions for the compound pulley by solving the system of linear equations.

All the tensions are reduced except for that of the first mass when the masses are accelerating.

Thursday, March 29, 2018

Torqued Pulley

  Consider a variation of Atwood's machine with a mass hanging from a string wrapped around a pulley.

The expressions for the height of the mass and balance of forces are the same as before. For the pulley we need to balance the torque, the product of tension and its distance from the center of the pulley, with the inertial force due to the acceleration of the pulley. It is assumed that the quantity of motion for the pulley is proportional to the rate of change of the angle θ or P=Iω=Iθ̇ where I is the moment of inertia. Note that the resistance to angular acceleration is directed upwards. This time the inertial factor, m+I/r2, includes the inertia of the pulley.

And the tension in the string is again less than the gravitational force acting on the mass m. Barton's Analytic Mechanics derives the formulas for Atwood's machine including the inertia of the pulley.