Conjugate Time in the Sub-Riemannian Problem on the Cartan Group

Abstract

The Cartan group is the free nilpotent Lie group of rank 2 and step 3. We consider the left-invariant sub-Riemannian problem on the Cartan group defined by an inner product in the first layer of its Lie algebra. This problem gives a nilpotent approximation of an arbitrary sub-Riemannian problem with the growth vector (2,3,5). In previous works, we described a group of symmetries of the sub-Riemannian problem on the Cartan group, and the corresponding Maxwell time — the first time when symmetric geodesics intersect one another. It is known that geodesics are not globally optimal after the Maxwell time. In this work, we study local optimality of geodesics on the Cartan group. We prove that the first conjugate time along a geodesic is not less than the Maxwell time corresponding to the group of symmetries. We characterize geodesics for which the first conjugate time is equal to the first Maxwell time. Moreover, we describe continuity of the first conjugate time near infinite values.

Appendix: . Explicit formulas for coefficients of Jacobian

In this appendix we present explicit formulas for the coefficients a₀₁, a₁₂ of the Jacobian J₁ (5.2) in the domain C₂.

$$ \begin{array}{@{}rcl@{}} &&a_{01} = \sum\limits_{j=0}^{3} \sum\limits_{i=0}^{j} c_{i, j-i} F^{i}(u_{1}) E^{j-i}(u_{1}), \\ &&c_{00} = - 2 k^{2} \sin^{2} u_{1} \cos^{2} u_{1} \sqrt{1-k^{2} \sin^{2} u_{1}}, \\ &&c_{01} = \cos u_{1} \sin u_{1} (4+k^{2}(1-6\sin^{2} u_{1})), \\ &&c_{02} = - 3 \sqrt{1-k^{2} \sin^{2} u_{1}} (1-2\sin^{2} u_{1}), \\ &&c_{11} = (2-k^{2}) \sqrt{1-k^{2} \sin^{2} u_{1}}, \\ &&c_{20} = (1-k^{2}) \sqrt{1-k^{2} \sin^{2} u_{1}} (1-2\sin^{2} u_{1}), \end{array} $$

$$ \begin{array}{@{}rcl@{}} &&c_{03} = - 2 \cos u_{1} \sin u_{1}, \\ &&c_{12} = (2-k^{2}) \cos u_{1} \sin u_{1}, \\ &&c_{21} = 2(1-k^{2}) \cos u_{1} \sin u_{1}, \\ &&c_{30} = -(1-k^{2})(2-k^{2}) \cos u_{1} \sin u_{1}, \end{array} $$

(A.1)

$$ \begin{array}{@{}rcl@{}} &&a_{21} = \sum\limits_{j=0}^{5} \sum\limits_{i=0}^{j} d_{i,j-i} F^{i}(u_{1}) E^{j-i}(u_{1}), \\ &&d_{00} = - 6 k^{7} \sin^{3} u_{1} \cos^{3} u_{1}, \\ &&d_{01} = 20 k^{5} \sin^{2} u_{1} \cos^{2} u_{1} \sqrt{1-k^{2} \sin^{2} u_{1}}, \\ &&d_{10} = - 6 k^{5} (2-k^{2}) \sin^{2} u_{1} \cos^{2} u_{1} \sqrt{1-k^{2} \sin^{2} u_{1}}, \\ &&d_{02} = - 2 k^{3} \sin u_{1} \cos u_{1}(12-k^{2}(1 + 10 \sin^{2} u_{1})), \\ &&d_{11} = \frac{k^{3}}{2} \sin u_{1} \cos u_{1}(32 - 8 k^{2}(1+6\sin^{2} u_{1}) + 3 k^{4}(1+8\sin^{2} u_{1})), \\ &&d_{20} = \frac{k^{3}}{2} \sin u_{1} \cos u_{1}(16 + 3 k^{6} \sin^{2} u_{1} + k^{4} (9-8 \sin^{2} u_{1})-4k^{2}(7-2 \sin^{2} u_{1})), \\ &&d_{03} = 8 k (2-k^{2}) \sqrt{1-k^{2} \sin^{2} u_{1}}, \\ &&d_{12} = - \frac{k}{2}(32 - 32 k^{2} + 15 k^{4}) \sqrt{1-k^{2} \sin^{2} u_{1}}, \\ &&d_{21} = - \frac{k}{2}(32 - 48 k^{2} + 10k^{4}+3k^{6}) \sqrt{1-k^{2} \sin^{2} u_{1}}, \\ &&d_{30} = \frac k2 (32 - 64 k^{2} + 41 k^{4} - 9 k^{6}) \sqrt{1-k^{2} \sin^{2} u_{1}}, \\ &&d_{04} = -10k^{3} \sin u_{1} \cos u_{1}, \\ &&d_{13} = 12 k^{3}(2-k^{2}) \sin u_{1} \cos u_{1}, \\ &&d_{22} = - \frac 32 k^{3}(8-8k^{2}+3k^{4}) \sin u_{1} \cos u_{1}, \\ &&d_{31} = - \frac{k^{3}}{2}(16-24k^{2}+6k^{4}+k^{6}) \sin u_{1} \cos u_{1}, \\ &&d_{40} = \frac 32 k^{3}(1-k^{2})(2-k^{2})^{2} \sin u_{1} \cos u_{1}, \\ &&d_{05} = 4 k \sqrt{1-k^{2} \sin^{2} u_{1}}, \\ &&d_{14} = - 6 k (2-k^{2}) \sqrt{1-k^{2} \sin^{2} u_{1}}, \\ &&d_{23} = k (8-8k^{2} +3k^{4}) \sqrt{1-k^{2} \sin^{2} u_{1}}, \\ &&d_{32} = \frac k2 (16 -24k^{2}+6k^{4}+k^{6})\sqrt{1-k^{2} \sin^{2} u_{1}}, \\ &&d_{41} = -3k(1-k^{2})(2-k^{2})^{2} \sqrt{1-k^{2} \sin^{2} u_{1}}, \\ &&d_{50} = \frac k2 (1-k^{2})(2-k^{2})^{3} \sqrt{1-k^{2} \sin^{2} u_{1}}. \end{array} $$

(A.2)

Here F(u₁) and E(u₁) are elliptic integrals of the first and second kinds [42].