The principle of least constraint is another formulation of classical mechanics enunciated by Carl Friedrich Gauss in 1829.
The principle of least constraint is a least squares principle stating that the true motion of a mechanical system of
for all trajectories satisfying any imposed constraints, where
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 principle, whereas the other is an extremal principle.
Hertz's principle of least curvature
Hertz's principle of least curvature is a special case of Gauss's principle, restricted by the two conditions that there be no applied forces and that all masses are identical. (Without loss of generality, the masses may be set equal to one.) Under these conditions, Gauss's minimized quantity can be written
The kinetic energy
Since the line element
the conservation of energy may also be written
Dividing
Since