Gauss's principle of least constraint
The principle of least constraint is one variational formulation of classical mechanics enunciated by Carl Friedrich Gauss in 1829, equivalent to all other formulations of analytical mechanics. Intuitively, it says that the acceleration of a constrained physical system will be as similar as possible to that of the corresponding unconstrained system. Statement The principle of least constraint is a least squares principle stating that the true accelerations of a mechanical system of masses is the minimum of the quantity where the jth particle has mass , position vector , and applied non-constraint force acting on the mass. The notation indicates time derivative of a vector function , i.e. position. The corresponding accelerations satisfy the imposed constraints, which in general depends on the current state of the system, . It is recalled the fact that due to active and reactive (constraint) forces being applied, with resultant , a system will experience an acceleration . Connections to other formulations Gauss's principle is equivalent to D'Alembert's principle. The principle of least constraint is qualitatively similar to Hamilton's principle, which states that the true path taken by a mechanical system is an extremum of the action. However, Gauss's principle is a true (local) minimal