Abstract
The Gauss principle of least constraint is derived from a new point of view. Then, an extended principle of least constraint is derived to cover the case of nonideal constraints. Finally, a version of the principle for general underdetermined systems is adumbrated. Throughout, the notion of generalized inverses of matices plays a prominent role.
Highlights
In his epochal paper of 1829, Gauss (Ref. 1) began by remarking that the D’Alembert principle reduced all of dynamics to statics and that the principle of virtual works reduced all of statics to a mathematical problem
He went on to state his own new principle, the principle of least constraint reducing all of mechanics, dynamics, and statics, to a single principle
The entire analysis presented in this paper was started directly with the constraints on the system, which emphasizes again the importance of the constraints in a mechanical system
Summary
In his epochal paper of 1829, Gauss (Ref. 1) began by remarking that the D’Alembert principle reduced all of dynamics to statics and that the principle of virtual works reduced all of statics to a mathematical problem. He observed that every new principle is not without merit, especially if it can shed new light on mechanical processes and perhaps render the solution of certain problems simpler to obtain He went on to state his own new principle, the principle of least constraint reducing all of mechanics, dynamics, and statics, to a single principle. This paper, besides being a tribute to Gauss in the 175th anniversary of the principle bearing his name, will present a rederivation of the Gauss principle and an extension of his principle to cases in which the standard principle of virtual works is not applicable These cases include those in which sliding friction (for example) is significant and cannot be neglected
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