Mazhitova A. D.

 

Mechanical-Mathematical Department, Al-Farabi Kazakh National University,

Al-Farabi ave. 71, Almaty 050038, Kazakhstan;

e-mail: Akmaral.Mazhitova@kaznu.kz

 

Sub-Riemannian problem on the three-dimensional

Lie group SOLV –

 

In this paper we study geodesics of a left-invariant sub-Riemannian metric on the three-dimensional solvable Lie group. A system of differential equations for geodesics fined by Pontryagin maximum principle and by using Hamiltonian structure. In a generic case the normal geodesics are described by elliptic functions.

Let us consider the three-dimensional Lie group SOLV− formed by all matrices

of the form

,  x, y, z ∈ R.                  (1)

Its Lie algebra is spanned by the vectors

 

 

meeting the following commutation relations: . We take a new basis  in which the commutation relations take the form  . Let us consider a left-invariant metric on  SOLV−, which is defined by its values in the unit of the group: .  The Lie group  SOLV− is diffeomorphic to the space .  Indeed,   are the global coordinates on  SOLV− and they also may be considered as global coordinates on . The tangent space at each point of  SOLV− is spanned by matrices of the form

 

which are the left translations of the basic vectors:

 

For the basis  we have

 .

 

The inner product takes the form

=                            (2)

 

In this paper we study the sub-Riemannian problem on the three-dimensional Lie group SOLV− defined by the two-dimensional left-invariant distribution   with left-invariant Riemannian metric (2). Geodesics of sub-Riemannian metrics satisfy the Pontryagin maximum  principle (see, for instance, [5])

 

                   .         (3)

 

The Hamiltonian equations take the form

 

                  (4)

        

The system (3) has three first integrals:

 

 

which are functionally independent almost everywhere, and therefore the system is completely integrable. Since the flow is left-invariant as well as the distribution  ∆  and the metric, without loss of generality we assume, that all geodesics originate at

the unit of group, that is, we have the following initial conditions for the

system (3):

 

                           (5)

           

In the sequel, we put

 

.

 

By substituting these expressions into (3), we fine a expression for  pz  and by this from third equation of  (4) we obtain

 

,   where                        (6)

 

The last expression is not integrated in terms of elementary functions and defines an elliptic integrals, except of special cases, when this elliptic integral degenerates.

 

Theorem.  In a generic case the normal geodesics (with the initial condition (5)) are described by the formulas (for pz > 0):

 

 

In the degenerated cases the normal geodesics (with the initial condition (5)) are described in terms of elementary functions by the formulas

 

               

                                       

 where  is discriminant  of subradical expression in (6)

 

;                               

.     

The qualitative behavior of generic normal geodesic is quite complicated.

 

 

References

 

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