I have a map from positive reals to positive reals, of the form $$ f(d_1) = \frac{\int_0^\pi \int_0^\pi \int_0^{2\pi} 4\pi sin^2(\frac{\theta}{2})(1+(1-cos\theta )(v_1^2-1)) e^{k(d_1 + d_2 + d_3 + (1-cos\theta)(d_1(v_1^2 -1) + d_2(v_2^2 -1) + d_3(v_3^2 -1))}d\theta da db}{\int_0^\pi \int_0^\pi \int_0^{2\pi} 4\pi sin^2(\frac{\theta}{2}) e^{k(d_1 + d_2 + d_3 + (1-cos\theta)(d_1(v_1^2 -1) + d_2(v_2^2 -1) + d_3(v_3^2 -1))}d\theta da db} $$ where $$ v_1 = \sin a \cos b,\\ v_2 = \sin a \sin b,\\ v_3= \cos a$$. and $,d_2,d_3$ are positive constants. I am studying the dynamics of particles under a certain model on $SO(3)$. This above-mentioned function shows up while studying this, in fact, this function's non-zero fixed point is one of the coordinates of the center of mass (if more details are needed I will elaborate in comments). My guess is that there is a non-zero $x$ such that for $a<x, a<f(a)<x$ and for $x<b, x<f(b)<b$. I make this guess because this sort of behavior was observed in a different geometry. I need help showing this. How do I go about proving this?
Add a comment
|