1.) The Soap Bubble Problem: Consider a curve ð‘¦(ð‘¥) from (ð‘¥0, ð‘¦0) = (0,ð‘…) to (ð‘¥1, ð‘¦1) = (ð», ð‘…) Revolve this curve around the x-axis to produce a surface of revolution. Recall that the surface area of a surface of revolution is given by ð´ = 2ðœ‹ âˆ« yds. Find the function ð‘¦(ð‘¥) that minimizes the surface area.
[This is called the soap bubble problem because if you take 2 wire rings of radius ð‘… that are parallel to each other, and dip them in soapy water, the bubble that forms between the rings minimizes the potential energy associated with surface tension forces. This energy is proportional to the surface area.]

2.) Fermat's Principle of Least Time: Consider a medium in which the index of refraction is a function of only radius: ð‘›(ð‘Ÿ) = ð‘›0(ð‘Ÿ0/ð‘Ÿ)Â² Show that light rays that pass through the origin make circular trajectories. Hints: Light rays will travel along the trajectory ð‘Ÿ(âˆ…) that minimizes the travel time. Use polar coordinates in two dimensions. Use Euler's second equation.

3.) Hamilton's Principle: Lagrange & Hamilton showed that the trajectory of a particle minimizes the "action". Let's demonstrate this principle with a very simple example. Consider throwing a ball straight upward with initial velocity ð‘£0 Ignore air resistance. The ball reaches a maximum height and returns to the initial height. Let's take ð‘” = 10 m/sÂ² and ð‘£0 = 20 m/s. As you work through the problem, use variables and constants and only substitute numerical values at the end of each part.
(a) Apply Lagrange's equations to find ð‘§(ð‘¡).
(b) Substitute ð‘§(ð‘¡) into the action functional and explicitly integrate to find the action for the actual trajectory from the time the ball is released until the time the ball returns to its initial height.
(c) Now let's test a wrong trajectory. The first half of a sine function sort of looks like the actual trajectory: ð‘§wrong,1 = ð´ sin(ðœ”ð‘¡). Find ð´ and ðœ” so that this wrong trajectory matches the correct initial velocity and the correct travel time. Graph the correct trajectory and this wrong trajectory. How do they differ? Substitute this wrong trajectory into the action functional and compute the action. How does it compare to the action for the correct trajectory?
(d) Let's try one more wrong path. Let's use a sine function again, but this time, find A and to so that the wrong trajectory matches the correct travel time and the maximum height. Add this to the same graph in part
(c). Substitute this wrong trajectory into the action functional and compute the action. How does it compare to the action for the correct trajectory?

1.) The Soap Bubble Problem: Consider a curve ð‘¦(ð‘¥) from (ð‘¥...

1.) The Soap Bubble Problem: Consider a curve ð‘¦(ð‘¥) from (ð‘¥...