FRCTIONAL GRADIENT REPRESENTATION OF ARBITRATRY ORDER FOR MECHANICAL SYSTEM

Gradient system is an important type of mathematical system. When solving out its potential function and making it as Lyapunov function, one can use the characters of gradient system to study the equilibrium and stability of this system by Lyapunov function. Stability of constrained mechanical system is important and has broad applications. If constrained mechanical system can be transformed into gradient system, one can use the characters of gradient system to study stability of constrained mechanical system. However, not all the constrained mechanical system can be transformed into gradient system, but can be transformed into fractional gradient system under some conditions. The fractional gradient system can propose a new thinking for the stability research of these mechanical systems. Up to now the gradient representation of constrained mechanical system is limited into second order fractional gradient system (which fractional order \alpha = 2 ), to our best knowledge, there is no arbitrary order gradient representation for constrained mechanical system, which limits its application in constrained mechanical system. Different from former studies about only second order gradient representation of constrained mechanical system, new fractional gradient representation of holonomic mechanical system is studied. Two new type of propositions for holonomic mechanical system transforming into arbitrary order fractional gradient system are given, the results can degenerate into classical gradient system and second order fractional gradient system. Two examples are given to illustrate the results.