An Implementation Method for the Geodesics with Constraints on Heisenberg Manifolds

In this paper we address the implementation issue of the geodesics method with constraints on Heisenberg manifolds. First we present more details on the method in order to facilitate its implementation and second we consider Mathematica as a software tool for the simulation. This implementation is of great importance since it allows easy and direct determination of Ricci tensor, which plays a fundamental role in the Heisenberg manifold metric.


Introduction
Geodesics plays an important role in many applications, especially in nuclear physics, image processing, ••• Ovidiu Calin and Vittorio Mangione [1] considered the Heisenberg manifold structure to provide a qualitative characterization for geodesics under nonholonomic constraints.This method offers an excellent description or the solution of Euler-Lagrange equation associated to lagrangians with linear and quadratic constraints.Therefore it is highly desirable to consider the implementation of this interesting mathematical method.In this paper we investigate the implementation of the method presented in reference [1].Due to the fact that the mathematical work in reference [1] lacks some details that are necessary for implementation, we address this issue by including such required details with complete proofs, after presenting the method described elsewhere [1] in an appropriate manner we implement and simulate mathematical.This implementation approach can successfully well illustrate the variation of some parameters such as Christoffel coefficients, ••• and ••• that are required in the determination of tensors.Our approach is also of great importance especially in the case where the determination of the geodesics is carried out by minimizing a performance index.Therefore our implementation approach can be considered as an attractive complement for the work of [1].
The working hypotheses: We take the examples studied in [1] with the following: 1) Expressions of the vector fields , 2 2 2) The Heisenberg Laplacian operator 3) The sub-Riemannian geometry can be defined on by: is the trajectory of a particle of mass m = 1 its energy is given by       is a Riemannian metric which will be specified later.5) We consider successively the two expressions: where w is the 1-form such And  is a constant, which has the physical significance of a charge. is a 1-form which will be defined later.
6) We consider as in [2], the Heisenberg group This group is non commutative and the law of the group is polynomial and can be written in 3   The Lie algebra of H is spanned by the matrices 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 .
for which the following relations hold This relation is not so senseless even if it can be very easily proved with a little computation.
It is indeed coming from the Baker-Campb-Hausdorff formula which expresses the product of the exponential of two matrices as the exponential of some quantity.
To be more precise, for two matrices M and N:  where is a Lie series which depends on the iterated brackets of M and N: In this case of the Heisenberg group whose Lie algebra is nilpotent of order 2, this serie stops after the first bracket term.We prefer to work with the exponential coordinates are the coordinates in the Lie algebra.
We identify g H  then with the triple such that: The group law in these coordinates becomes: And the inverse element is: The expressions of the left invariant vector fields in these exponential coordinates are then Whereas the right-invariant vector fields is written: Reference [3] was the first to check easily that: , 4 X X T   , it follows that K is not involutive.The distribution K will be called the horizontal distribution.
A more detailed about sub-Riemannian structures see in [4].
A curve The vector fields (6) defines a unique Riemannian metric h such as For the coefficients calculus, we have used a little Mathematica program: 4 y h So, we have : As the coefficients  are symmetric in i and j the matrix of coefficients is For a detailed study of sub-Riemannian geodesics on Heisenbeg group see in [5].

Main Results [6] Heisenberg group case:
We shall construct the Euler-Lagrange equation for the Lagrangian (5) in the Levi-Civita connection form.

Lemma 1
If are the components of the Ricci tensor with respect to the metric on H then

Proof
We will calculate the coefficients of the Ricci tensor from the following relation: After, we will compare these results with those of theorem.
For the first calculation, we use the mathematica program: Clear [coord, metric, inversemetric, affine, riemann, ricci,scalar, x, y, t] n 3 affine , , affine s, j, k affine i, l,s , ToString R j, l , ricci j, l , j,1, n , l,1, j Likewise, Using (10) w e we swap her  with   , we get the Equation (11) on components: To show the values of the Riemann tensor, add the following command : If  is an horizontal curve ,then where We see does not depend on  .Using lemma1: The next proposition is a generalization of the previous corollary to any vector field.

Proposition 1
For any vector field V we have where Copyright © 2012 SciRes.AM

Corollary 2
The actions and In particular, the ex  trema will be geodesics in the metric with coefficients   ij R  .at, ev It is interestin non-holonomic c g th en if the Lagrangian (5) has a onstraint, the minimizers still behave as geodesics in a certain metric.This is given in the following result.

Theorem 1
The Euler-Lagrange equation for the Lagrangian ( 5) is where

A More General Case [6]
Heisenberg manifold case We have investigated the case when the vector fields are given by the formula (3).We shall deal in this section with the more general case of vector fields.

 
A computation shows the vector fields So, we have: As the Riemannian metric is symmetric, we obtain we get the Copyright © 2012 SciRes.AM Lagrangian in (5).In this case 0   .ne The following result is a ge ralization of lemma1.We shall denote by are the components e Ricci tensor with respect t the metric given in (20), t of th h is obtained by flipping the sign in(20) For this, we use the mathematica program And the Ricci scalar is : 2) we put : Table If UnsameQ affine i, j, k , 0 , ToString i, j, k , affine i, j, k Remarque: To show the values of the Riemann tensor ,add the following command : reach the extrema for the same functions.In particular, the extrema will be geodesics in the metric ij g and obe the equation y 0 The last equation can be written also as which com letes the proof.