Gauss's principle of least constraint
The principle of least constraint is a least squares principle stating that the true motion of a mechanical system of masses is the minimum of the quantity
for all trajectories satisfying any imposed constraints, where , and represent the mass, position and applied forces of the mass.
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 is also conserved under these conditions
Since the line element in the -dimensional space of the coordinates is defined
the conservation of energy may also be written
Dividing by yields another minimal quantity
Since is the local curvature of the trajectory in the -dimensional space of the coordinates, minimization of is equivalent to finding the trajectory of least curvature (a geodesic) that is consistent with the constraints. Hertz's principle is also a special case of Jacobi's formulation of the least-action principle.
- Gauss, C. F. (1829). "Über ein neues allgemeines Grundgesetz der Mechanik". Crelle's Journal. 4: 232. doi:10.1515/crll.1829.4.232.
- Gauss, C. F. Werke. 5. p. 23.
- Hertz, H. (1896). Principles of Mechanics. Miscellaneous Papers. III. Macmillan.