This paper considers control systems defined on Lie algebroids. After deriving basic controllability tests for general control systems, we specialize our discussion to the class of mechanical control systems on Lie algebroids. This class of systems includes mechanical systems subject to holonomic and nonholonomic constraints, mechanical systems with symmetry and mechanical systems evolving on s...

We apply nonlocal symmetry-like operators to systems of two first and two second-order ordinary differential equations to seek reduction to quadratures. The reduction of order of such systems is carried out with the help of analytic continuation of scalar equations in the complex plane. Examples are taken from the literature. Precisely it is shown how the reduction to quadratures of a system of...

We propose a new approach for detecting ellipses. The approach is based on the geometrical property that five points on an ellipse can determine the parameters of the ellipse, and the symmetry of ellipses is used to obtain these points. Using symmetry, we classify the edge points in an input image into several subimages. Ellipses with different symmetric axes will lie in different subimages. In...

The Lie theorem for Birkhoffian systems with time-scale nonshifted variational problems are studied, including free system, generalized system and constrained system. First, the Pfaff-Birkhoff principle is established, dynamical equations three under deduced. Afterwards, in problems, by introducing infinitesimal transformations, symmetry defined respectively. Then theorems kinds of deduced prov...

: Singular systems play an important role in many fields, and some new fractional operators, which are general, have been proposed recently. Therefore, singular on the basis of mixed derivatives including integer order derivative generalized operators studied. Firstly, Lagrange equations within established, primary constraints presented for systems. Then constrained Hamilton constructed by intr...

In the present paper, seventh order Caudrey-Dodd-Gibbon (CDG) equation is solved by Lie symmetry analysis. All geometry vector fields of KdV equations are presented. Using transformation CDG reduced into ordinary differential equations. These ODEs power series method to obtain exact solution. The convergence also discussed.