Gauss's principle of least constraint

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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 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[edit]

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.

See also[edit]

References[edit]

  • 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. 

External links[edit]

  • [1] Gauss's principle of least constraint
  • [2] Hertz's principle of least curvature