<p>As a universal principle in analytical mechanics, Gauss principle is characterized by its extremal property, which differs from other differential variational principles. Because of its universality and extreme properties, the Gauss principle is not only theoretically important, but also has great practical value, such as in robot dynamics, multi-body systems, approximate solutions to dynamics equations, etc. In this paper, the arbitrary-order Gauss principle is proposed and its application in nonholonomic mechanics is studied. Firstly, the concept of the space spanned by arbitrary-order derivative of acceleration is proposed, and Gauss principle of mechanical system with two-sided ideal constraints is established in this space. By defining the generalized compulsion function, it is proved that in the arbitrary-order derivative space of acceleration this function yields a stationary value along the path of real motion. Secondly, three kinds of arbitrary-order Gauss principles in generalized coordinates are derived. Thirdly, by constructing the generalized compulsion function of nonholonomic systems, the arbitrary-order Gauss principles are extended to nonholonomic systems, and Appell equations, Lagrange equations and Nielsen equations are derived.</p>