The arbitrary order Gauss principle of relative motion dynamics is studied.Firstly, the concept of arbitrary order derivative space of relative acceleration is proposed, and the variational rules of generalized Gauss variation in this space are given.On this basis, the arbitrary order Gauss principle for dynamics systems in relative motion with two-sided ideal constraints is given from the kinetic equation of relative motion of particles.By defining the arbitrary order generalized compulsion function, the arbitrary order Gauss least compulsion principle for dynamics systems in relative motion with two-sided ideal constraints is established.Secondly, the arbitrary order Gauss principle of relative motion dynamics in generalized coordinates is studied, and its Appell form, Lagrange form and Nielsen form are given.Thirdly, the application of the obtained principle to nonholonomic mechanics is studied.The Gauss least compulsion principle and its generalized coordinate form for nonholonomic systems of arbitrary order in relative motion are given.Finally, we introduce how to establish the motion equations of higher order nonholonomic systems in relative motion by using the Gauss least compulsion principle of arbitrary order.