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 .
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.