Some dynamical and Bundle questions arising from certain map $P:TS^{n}\to S^{n}$ Define  the  map $$P:TS^{n}\to S^{n}  \;\;\;\text{by}\;\;  P((x,v))=\frac{x+v}{\parallel x+v \parallel}$$  where $$TS^{n}=\{(x,v)\in S^{n} \
\times \mathbb{R}^{n+1}\mid v \perp x \}$$
This  map is used in the  book of  Alain Hatcher, Algebraic topology,  to  give  a proof  for the fact that every vector field on even spheres must vanish  at some point of the sphere.
Question:

  
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*Does  $P$ define  a  (nontrivial)  fiber bundle?
  
*Define  the  Hamiltonian function $H:TS^{n} \to \mathbb{R}$  with $H(x,v)={\parallel P(x,v)-x \parallel}^{2} $  where the  latter norm is the standard Euclidean norm on $\mathbb{R}^{n+1}$. What can be said about the  dynamical  behavior of the  corresponding  Hamiltonian vector  field $X_{H}$? Are there  any periodic  orbit?
  
*Assume that $V$ is  a vector  field on the  sphere. To $V$, we  associate a self map $f(x)=P(x,V(x))$ on the  sphere. Are there  some relations  between the  continuous  dynamics  of  $V$  and the  discrete dynamics  of $f$. Note  that $V$  and  $f$ have  the same fixed points.
  

 A: There is a wonderful trick - I think promulgated by Moser - for viewing Hamiltonian flows on the cotangent bundle of the sphere as the reduction of a   Hamiltonian flow on the ambient phase space of (x,p)'s , i.e on  ${\mathbb R}^{n+1} \times{\mathbb R}^{n+1}$ and  which shows that your flow is in fact a geodesic flow running at twice the speed.
View the sphere as the space of rays through the origin, so orbits
of the ${\mathbb R}^+$ action $x \mapsto \lambda x$.  The cotangent lift of this action to the ambient phase space is $(x,p) \mapsto (\lambda x, p/\lambda)$.  The momentum map for this scaling action is $J(x,p) = \langle x, p \rangle$. The reduced space at $J =0$ is canonically the cotangent bundle of the sphere.  A Hamiltonian H for the cotangent bundle of the sphere becomes then a function on the ambient phase space which   is homogeneous of degree 0 -- i.e invariant --
with respect to this  scaling.  Example: the Hamiltonian for geodesic flow on the sphere  is $H_{geod} = (1/2)|x|^2 |p|^2$.  Your Hamiltonian, made homogeneous of degree 0 , is $H = |(x/|x|- |x|p)|^2 = (1 - 2<x,p> + |x|^2|p|^2)$. But we must fix $J = 0$.  So $H = 1 + 2 H_{geod}$ -- your
flow is geodesic flow, running at twice the speed. 
