In trying to understand Laplace-Runge-Lenz vector, I read in Wikipedia that the Kepler problem is mathematically equivalent to a particle moving freely on the surface of a four dimensional hypersphere.
The Lagrangian of the free particle on the sphere is
$$\mathcal L=\frac m2R^2(\dot\phi_1^2+\dot\phi_2^2\sin^2\phi_1+\dot\phi_3^2\sin^2\phi_1\sin^2\phi_2)$$
where the spherical coördinates are
$x_1=r\cos\phi_1,\ x_2=r\sin\phi_1\cos\phi_2,\ x_3=r\sin\phi_1\sin\phi_2\cos\phi_3,\ x_4=r\sin\phi_1\sin\phi_2\sin\phi_3$
The equation of motion in Kepler problem is $$m\ddot r-mr\omega^2+\frac{2k}{r^3}=0$$
How can I show they are equivalent?
Another small question, they say, "This higher symmetry results from two properties of the Kepler problem: the velocity vector always moves in a perfect circle..."
What do they mean by velocity vector moves in a perfect circle? Velocity vecctor is always perpendicular to the path wich can also be ellipse, parabola and hyperbola.
