On a Riemannian manifold (M, g) we consider the k + 1 functions F1, ..., F k , G and construct the vector fields that conserve F1, ..., F k and dissipate G with a prescribed rate. We study the geometry of these vector fields and prove that they are of gradient type on regular leaves corresponding to F1, ..., F k . By using these constructions we show that the cubic Morrison dissipation and the Landau-Lifschitz equation can be formulated in a unitary form.MSC: 37C10, 58A10, 53B21, 70E20.

The stability for all generic equilibria of the Lie-Poisson dynamics of the so(4) rigid body dynamics is completely determined. It is shown that for the generalized rigid body certain Cartan subalgebras (called of coordinate type) of so(n) are equilibrium points for the rigid body dynamics. In the case of so(4) there are three coordinate type Cartan subalgebras whose intersection with a regular adjoint orbit gives three Weyl group orbits of equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical rigid body in Communicated by A. Bloch. P. Birtea · I. Caşu ( )

On a constraint manifold we give an explicit formula for the Hessian matrix of a cost function that involves the Hessian matrix of a prolonged function and the Hessian matrices of the constraint functions. We give an explicit formula for the case of the orthogonal group O(n) by using only Euclidean coordinates on R n 2 . An optimization problem on SO(3) is completely carried out. Its applications to nonlinear stability problems are also analyzed.MSC: 53Bxx, 58A05, 58E50, 51F25, 34K20.

Considering orthogonal Stiefel manifolds as constraint manifolds, we give an explicit description of a set of local coordinates that also generate a basis for the tangent space in any point of the orthogonal Stiefel manifolds. We show how this construction depends on the choice of a submatrix of full rank. Embedding a gradient vector field on an orthogonal Stiefel manifold in the ambient space, we give explicit necessary and sufficient conditions for a critical point of a cost function defined on such manifolds. We explicitly describe the steepest descent algorithm on the orthogonal Stiefel manifold using the ambient coordinates and not the local coordinates of the manifold. We point out the dependence of the recurrence sequence that defines the algorithm on the choice of a full rank submatrix. We illustrate the algorithm in the case of Brockett cost functions.MSC: 53Bxx, 65Kxx, 90Cxx

We give a description of numerical Newton algorithm on a constraint manifold using only the ambient coordinates (usually Euclidean coordinates) and the geometry of the constraint manifold. We apply the numerical Newton algorithm on a sphere in order to find the critical configurations of the 5-electron Thomson problem. As a result, we find a new critical configuration of a regular pentagonal type. We also make an analytical study of the critical configurations found previously and determine their nature using Morse-Bott theory. Last section contains an analytical study of critical configurations for Riesz s-energy of 5-electron on a sphere and their bifurcation behavior is pointed out.MSC: 49M15, 53-XX