A gradient representation of holonomic system in the event space

The dynamics research in the event space has important geometric and mechanical meanings, and great progress has been made in this field. A gradient system is a kind of important systems in differential equations and dynamical systems, and is receiving more and more attention. In this paper, a gradient representation and a fractional gradient representation of a holonomic system in the event space are studied. First, the differential equations of motion for the system are established and expressed in the first order form. Second, we have obtained the condition under which the system can be considered as a gradient system and also the condition under which the system can be considered as a fractional gradient system. When a constrained mechanical system is transformed into a gradient system or a fractional gradient system, one can use the properties of the gradient system or the fractional gradient system to study the integration and the stability of a constrained mechanical system. Finally, two examples are given to illustrate the application of the results. The event space is known as more extensive than the configuration space, therefore, the result in the configuration space is a special case of this paper.