Minimum for a Horizontal Line

  What happens if the path is limited to a horizontal line instead of an ellipse?

In this case y equals a constant value and there is only one independent variable which we can take to be x. We can easily determine a formula for the potential energy and set its derivative equal to zero
to get an equation that can be solve for x.

On simplifying this equation we get a simple expression for the square of x and we have to be careful about the sign when taking the square root. All the factors are positive except for y so we need a minus sign to make the right side of the equation positive and then we can solve for x. The equation for x allows us to determine that the triangles involving the angles are similar and we see that angles are also equal at equilibrium for this problem. A check using the value of y_min for the ellipse shows that we get the same x_min.

What's remarkable is that for both problems the angles are equal at the equilibrium position. We get the same results whether we use the ellipse or the horizontal line which is tangent at the equilibrium point.