Generalizations of the Brachistochrone Problem

1. Generalizations of the Brachistochrone Problem John Gemmer Co-Directed by Drs. Ron Umble and Michael Nolan

2. The Problem • Consider the following problem: Given two points A and B on some frictionless surface, what curve is traced on the surface by a particle falling in a gravitational field that starts at A and reaches B in the shortest time? • Using rudimentary techniques from calculus of variations, Newton showed in 1696 that the solution is a cycloid, provided the gravitational field is uniform and the surface is a vertical plane. • Johann Bernoulli was also able to show that the solution is a cycloid by using basic ideas from geometrical optics.

3. Plot of Newton’s Solution g ↓

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5. How Do We Find These Curves? • From conservation of total mechanical energy, we have • Therefore, the total elapsed time T, along an arbitrary path is given by • Finally, to minimize T we need to solve the Euler-Lagrange equation:

6. Arc-Length on a Surface Since surfaces are two-dimensional objects, it is advantageous to describe them using two coordinates. → Let The element of arc-length on the surface is: and

7. Examples of Surfaces

8. General Solution • Theorem 1: Let x(u,v) be a parameterization of a surface S with metric If V, E, and G are independent of u, then the solution to the Brachistochrone problem on S is given by the curve x(u,v(u)) where,

9. Surfaces of Revolution • One important application of this theorem is for particles falling on a surface of revolution with a gravitational field that is uniform along the axis of revolution. g ↓

10. Solution Curves on the Cone g ↓

11. Hyperboloid of One Sheet • The hyperboloid of one sheet is another interesting surface of revolution. What is interesting is that there is a bifurcation of the solutions at C ≈ .687. g ↓ g ↓

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13. Inverse-Square Fields We can also apply the theorem to the more general case of a particle falling in inverse square gravitational field but confined to the plane z=0. In this case V is proportional to -1/r.

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15. Geometrical Optics • Fermat’s Principle states that light propagates in a medium in such a way as to minimize the total time of travel between two points A and B. The time taken by the light to travel along the curve connecting A to B is: A B

16. The Eikonal Equation • A wave-front propagating through a medium with index of refraction n is a family of level surfaces or curves of some function L. • The gradient of L is normal to this surface and the flow lines of this vector field are thought of as light rays. So,

17. Alternative Approach • Theorem 2: Let x(u,v) be a parameterization of a surface S with metric If E, G, and n are independent of u, then the light rays on S are given by the curve x(u,v(u)) where, x g ↓ y

18. Alternative Approach

19. Inverse-Square Field

20. Special Relativity • If the particle’s velocity is near the speed of light, the Newtonian mechanical energy equation is replaced with its relativistic counterpart where • Therefore, the index of refraction is given by the equation:

21. Vertical Plane Revisited • Let us again consider a particle falling in a uniform gravitational field but confined to the vertical plane. But, we will now allow the particle to fall at relativistic velocities. In this case, we can apply Theorem 2 with the modified index of refraction given by:

22. Local Theory of Curves • Theorem 3: If α(s) is a unit speed curve with nonzero curvature κ(s), then • Definition: The function is called the torsion of α(s) at s. • Note that a curve is planar if and only if

23. Curvature of Light Rays In geometrical optics, light rays obey the equation

24. Inverse-Square Solutions Revisited • Using the light-ray curvature equation we can show that when falling in an inverse square field the torsion vanishes.

25. Future Directions • The problem still remains open for surfaces and fields that do not satisfy the conditions in Theorem 1. It may be easier to use the Eikonal equations to solve these more general problems. • We would also like to solve more problems for particles falling in gravitational fields that are not confined to a surface. The next step beyond an inverse-square field would be a dipole field.