Detemining the constants in a helix equation (Calculus of Variations question)

[hookie101]

This question is about variable end points in calculus of variations. I understand the basic principle of how you would find the various equations, but I embarrasingly keep getting stuck on when determining the constants.

Question: Find the equation of a frictionless wire between the point (0,1) and the line x=a such that a particle slides down under gravity in least time. (Question also wants the values of the constants that would come from the Euler equation and the integration part of the problem. This is the part I'm stuck at).

The equation I arrived to was

x=(1/A^2)[Ø - (1/2)sin2Ø] + B

where Ay^1/2 = sinØ

I'm not entirely sure how I would determine A and B by only knowing the point (0,1). Considering that this is an equation of a helix, my first assumption was that when x=0, Ø=0, so leaving me with B=0. However this still leaves me with A to deal with. If anyone knows how to find the constants, I would appreciate it. It would help me even more to show me how you would find the constants for any general equation you'll eventually get from the Calculus of Variations method, since I had this sort of problem numerous times.

Additional part: Is it possible to rewrite the above equation in terms of y and Ø too?

 

[lippyka]

For the additional part, the equation can be rewritten in terms of y and Ø as y=A^2[sinØ - (1/2)sin2Ø] + Bwhere Ay^1/2 = sinØ. In this case, A and B are still unknown constants and can be determined by setting y=1 when x=0, so that B=1. Additionally, when x=0, Ø=0 and thus Ay^1/2 = 0, so A=0.