# Gauss principle

*principle of least forcing*

One of the fundamental and most general differential variational principles of classical mechanics, established by C.F. Gauss [1] and expressing an extremum property of a real motion of a system in the class of admissible motions, corresponding to the ideal constraints imposed on the system and to the conditions of constancy of positions and velocities of the points in the system at a given moment of time.

According to the Gauss principle, "the motion of a system of material points, constrained in an arbitrary manner, and subjected to arbitrary forces at any moment of time, takes place in a manner which is as similar as possible to the motion that would be performed by these points if they were free, i.e. with least-possible forcing — the measure of forcing during the time dt being defined as the sum of the products of the mass of each point and the square of the distance of the point from the position which it would occupy if it were free" [1].

The Gauss principle is equivalent with the d'Alembert–Lagrange principle and is applicable both to holonomic and to non-holonomic systems. It has been generalized in various ways [2], [3], such as to systems subject to non-ideal constraints, as well as to the case of continuous media [4].

#### References

[1] | C.F. Gauss, "Ueber ein allgemeines Grundgesetz der Mechanik" J. Reine Angew. Math. , 4 (1829) pp. 232–235 |

[2] | E.A. Bolotov, "On Gauss' principle" Izv. Fiz.-Mat. Obshch. Kazan Univ. (2) , 21 : 3 (1916) pp. 99–152 (In Russian) |

[3] | N.G. Chetaev, "On Gauss' principle" Izv. Fiz.-Mat. Obshch. Kazan Univ. (3) , 6 (1932–1933) pp. 68–71 (In Russian) |

[4] | V.V. Rumyantsev, "On some variational principles in mechanics of continuous media" J. Appl. Math. Mech. , 37 (1973) pp. 917–926 Priklad. Mat. Mekh. , 37 : 6 (1973) pp. 963–973 |

#### Comments

#### References

[a1] | V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian) |

[a2] | E.T. Whittaker, "Analytical dynamics of particles and rigid bodies" , Dover, reprint (1944) |

[a3] | R.B. Lindsay, H. Margenau, "Foundations of physics" , Dover, reprint (1957) |

**How to Cite This Entry:**

Gauss principle. V.V. Rumyantsev (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Gauss_principle&oldid=16810