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Международная молодежная конференция «Геометрия и управление»
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Cut Locus in the Riemannian Problem on Alexey Podobryaev Program Systems Institute RAS, Pereslavl-Zalesskiy, Russia |
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Аннотация: The parametrization of Riemannian geodesics on We represent the Maxwell strata, cut locus and give the equation for the cut time. When one eigenvalue of Riemannian metric moves to infinity, the parametrization of geodesics, conjugate time and locus, cut time and locus in the Riemannian problem converge to the sub-Riemannian ones that were considered by U. Boscain and F. Rossi [2]. Let Let Theorem. Let $$ \cos \tau \cos(\tau \eta \bar{p}_3) - \bar{p}_3 \sin \tau \sin(\tau \eta \bar{p}_3) = 0 $$ (1) If (2) If $$ \left\{ \begin{array}{lll} \frac{2 \pi I_1}{|p|}, & \text{if} & \frac{1}{2 \eta} \leqslant |\bar{p}_3| < 1, \\ \frac{2 I_1 \tau_{cut}(\eta, \bar{p}_3)}{|p|}, & \text{if} & |\bar{p}_3| < \frac{1}{2 \eta}. \end{array} \right. $$ Theorem. (1) If (2) If $$ J_{\eta} = \{\exp(\pm \varphi e_3) \ | \ \varphi \in [2 \pi (1 + \eta), \pi] \}. $$ The proof of these theorems is based on considering the symmetry group of Hamiltonian vector field of Pontryagin maximum principle, finding its fixed points (Maxwell strata), considering some open sets bounded by Maxwell strata, which are diffeomorphic by exponential map. This method was presented by Yu. L. Sachkov for the Euler elastic problem [3]. Proposition. The geodesics parametrization, conjugate time and locus, cut time and locus in sub-Riemannian problem on Example. Cut locus in the sub-Riemannin problem has two components: $$ S^1 \setminus \{\mathrm{id}\} = \{\exp(\varphi e_3) \ | \ \varphi \in (0, 2 \pi) \}. $$ The stratum Язык доклада: английский Список литературы
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