What is the path ?

Question

A drop of water falls onto a football and rolls down , following the path of steepest descent ; that is moves in the direction tangent to the football most nearly vertically downward.

Find the path the water drop follows if the surface of the football is ellipsoidal and given by the equation

4 x² + y² +4 z² =9

and the drop starts at the point ( 1, 1, 1 ).

Answer 1

This problem caught my attention a couple of years ago while reading a calculus book ( I don't remember the name of the author) and I copied it down , but I never gave it a second thought until my house went up to flames. I rediscovered it some months ago going over what was left after the fire.

My take is that the the key phrase here is " following the path of steepest descent " .

Therefore the path in question

r (t) = < x(t) , y(t) , z(t) > which must lie on the surface of the ellipsoid 4 x² + y² +4 z² =9

should be such that its projection on the xy-plane

σ (t) = < x(t) , y(t) > will be orthogonal to the level curves of the function z= f(x,y) = !1/2) √( 9-4x²-y²) in

which corresponds to the " northern " half of the ellipsoid, and also σ (t) = < x(t) , y(t) > will such an

orientation opposite of the direction of the gradient of f at every point

But the level curves of the function f is the family of the curves such that 4x^{2 +}y² = k , k constant. 0≤ k≤9

and they are ellipses. The differential equation whose set of solutions is the above described family of

curves is 8x +2y dy/dx = 0 , or equivalently dy/dx = -4x/y.

Now the family of curves that is orthogonal to the ellipses has differential equation dy/dx = y/4x

which is a separable one and its solution is y⁴ = x +c and since it must be passing through (1,1) then c= 0.

Hence

r (t) = < x(t) , y(t) , z(t) >= < t , t^1/4, (1/2) [ 9- 4t² -t^1/2 ]^1/2 , 1 ≤ t

Answer 2

Here is a path that stays on the ball, starting at (1,1,1) and ending at the ball z=0 plane

at the point (5/4,(√11)/2,0.

The projection on the xy plane is y=[(2√11)-4]x-(2√11)+5

for x=1 to x=5/4

this path was chosen based on the tangents dy/dz and dx/dz

If the path is p(t)=(x(t),y(t),z(t))

1≤t≤5/4

then x(t)=t

y(t)=[(2√11)-4]t-(2√11)+5

z(t)=+(1/2)√[9-4x²(t)-y²(t)]

z(t) is a bit messy but this has drop veering to the front a tad and down to (5/4,(√11)/2,0)

and my theory is the drop falls off at z=0.

Answer 3

Point 1,1,1 would be slightly down the side and forward from the top center of the football, disregarding the laces in the picture, and would continue to roll down that route back towards the underneath "center" bottom because gravity forces it towards that bottom center.