MathsIsFun: that sounds really interesting...if you get it working id really like to hear about it..
my approach it a bit more primitive, i came to think the following -
suppose the cables center of mass is located in y1. applying a downward force F to the midpoint i equivalent to placing a weight at that point of the chain. the new systems (cable + weight ) center of mass will be located somewhere under y1, if i.e. the weight weighs w [N] and the m_cable=20kg the new center of mass will lie at
y2 = ( y1*m_cable + (y1-d)w/g ) / ( 2 ), where d is the distance between y1 and the weight.
A cable is attached at two (nonemoving)fixpoints. Now the cables midpoint is pulled downward, thereby destroying the catenary curve.
Question: How will the cables center of mass (CM) move: op, down or stay fixed?
I find it intuituitive that the CM will assume the lowest possible position in the catenary (cosh y) curve, but how can i "proove" this? and is it possible to describe the hight of CM as function of the downward midpoint-displacement?
I also wonder if the catenary curve really is "destroyed" - can one not argue, that catenary curve is just altered to fit the new lowered midpoint?