Gauss’s principle of least constraint is a classical differential variational principle. Due to its generality, it is regarded as the most suitable fundamental principle for dynamics. Meanwhile, its formulation as a minimization problem has, with the rapid advancement of modern computational technology, renewed scholars’ interest in this long-established principle. In textbooks on analytical mechanics, Gauss’s principle is usually introduced in two forms: the minimization of the Gauss constraint and the variational form in the sense of Gauss. The equivalence between these two forms constitutes the most fundamental issue in the theoretical extension of Gauss’s principle. This paper discusses the applicable conditions for the equivalence of the two forms of Gauss’s principle, and clearly states that when the constraints can be fully described by constraint equations, the two forms are both necessary and sufficient for each other. Furthermore, non-ideal constraints are classified according to the representation of constraint forces, and the equivalence as well as the applicable conditions of the extended Gauss’s principle under different constraint force models are examined separately. The results show that the two forms are equivalent only when the non-ideal constraint forces are independent of the ideal constraint forces. Compared with the minimization form, the variational form of Gauss’s principle is more general, with its theoretical foundation resting on Newton’s second law and the assumption of ideal constraints. Using the sliding motion of a simple rigid rod as an example, the existence conditions for the minimization form of Gauss’s principle are demonstrated. The discussion in this paper provides a fundamental analytical basis for extending Gauss’s principle to different constrained systems.