The article is aimed at introducing the concept of virtual displacement for non-linear connections with the purpose of keeping both the d'Alambert principle and the Gauss principle and showing one common assumption of dynamics. The proven variation of Gauss principle allows extending the nature of usually considered mechanical systems through the application of Carnot principle from thermodynamics. On the other hand, the proposed variation of Gauss principle is intriguing by direct modification of the Herman and Euler idea which was developed by Lagrange in his description of the dynamics principle. The subject of mechanical systems with non-linear connections is an acute problem in mechanics. On the one hand, the realness of such connections is not clear, and on the other hand, two basic principles of analytical dynamics - d'Alambert principle and Gauss principle (according to Appell and DeLassu) are proved to be incompatible in this case. Appell and DeLassu considered such system motion from different points of view following basically one idea, i.e. to deduce the basic principle of analytical mechanics of systems with linear connections.
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