These two questions come from my assignment and the final calculus baffles me a lot:
Given a Riemann metric on a region $D$:
- $D$= upper-half plane, $ds^2=vdu^2+vdv^2$;
- $D=\mathbb{R}^2$, $ds^2=(1+u^2+v^2)^{-2}(du^2+dv^2)$.
Find their geodesics.
For the first one, we have $$L=\frac{1}{2}(v\dot{u}^2+v\dot{v}^2),$$ and $$0=\frac{\partial L}{\partial u}=\frac{d}{dt}\frac{\partial L}{\partial \dot{u}},\quad L=C_2.$$ The first equation implies that $$\frac{\partial L}{\partial \dot{u}}=v\dot{u}=C_1.$$Hence, $$\frac{C_1^2}{2v}+\frac{1}{2}v \dot{v}^2=C_2,$$ or $\dot v^2=2C_2/v-C_1^2/v^2$. Taking square root of both sides and integrating, we have $$\frac{\sqrt{2C_2v-C_1^2}(C_2v+C_1^2)}{3C_2^2}=t+C_3.$$After this, do I need to do the further calculation to give an explicit formula for $(u(t),v(t))$?
For the second one, Euler-Lagrange's equation implies that $$-2u=\ddot{u}(1+v^2+u^2)-4\ddot{u}(\dot u+\dot v),$$which is far beyond my ability to solve. Apart from that, I noticed that this metric has a similar form as the stereographic projection. Will this be helpful to solving this question?