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Concept
Gauss's principle of least constraint
Summary
The principle of least constraint is one variational formulation of classical mechanics enunciated by Carl Friedrich Gauss in 1829, equivalent to all other formulations of analytical mechanics. Intuitively, it says that the acceleration of a constrained physical system will be as similar as possible to that of the corresponding unconstrained system.
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.
Hertz'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.
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