词条 | Gauss's principle of least constraint |
释义 |
The principle of least constraint is one formulation of classical mechanics enunciated by Carl Friedrich Gauss in 1829. StatementThe 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 kth particle has mass , position vector , and applied non-constraint force acting on the mass. The notation d/dt indicates a time derivative. The corresponding accelerations satisfy the imposed constraints, which in general depends on the current state of the system, . Connections to other formulationsGauss'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 curvatureHertz's principle of least curvature is a special case of Gauss's principle, restricted by the two conditions that there are no externally applied forces, no interactions (which can usually be expressed as a potential energy), and all masses are equal. 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
References
External links
1 : Classical mechanics |
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