The transcendence necessary for computing the sphere and wave front in Martinet sub-Riemannian geometry (Q2771540)

The problem of minimization of energy, which is equivalent to the problem of sub-Riemannian geometry, is the optimal control problem: NEWLINE\[NEWLINE\begin{cases} \displaystyle{dq\over dt}(t)=u_1(t)G_1(q(t))+u_2(t)G_2(q(t)),\\ \displaystyle \min\limits_{u(\cdot)}\int_0^{T}(a(q(t))u_1^2(t)+c(q(t))u_2^2(t)) dt. \end{cases} NEWLINE\]NEWLINE From Pontryagin's maximum principle it follows that minimizing solutions satisfy the equations NEWLINE\[NEWLINE\dot q={\partial H_\nu\over\partial p},\;\dot p=-{\partial H_\nu\over\partial q},\;{\partial H_\nu\over\partial u}=0,NEWLINE\]NEWLINE where \(\displaystyle H_{nu}=\sum_{i=1}^2u_{i}\langle p,G(q)\rangle-\nu(au_1^2+cu_2^2)\) is pseudo-Hamiltonian. This paper deals with integrability by quadratures of geodesic equations for the normal form of degree 0: NEWLINE\[NEWLINEg=(1+\alpha y)^2dx^2+(1+\beta x+\gamma y)^2 dy^2,\qquad \alpha,\beta,\gamma\in {\mathbb R}.NEWLINE\]NEWLINE For the case of Martinet geometry the authors prove that sub-Riemannian sphere belongs to exp-log category if the geodesic equations is integrable by quadratures. The scale of asymptotic representations is presented.NEWLINENEWLINEFor the entire collection see [Zbl 0949.00042].