# Gauss's principle of least constraint

The principle of least constraint is one formulation of classical mechanics enunciated by Carl Friedrich Gauss in 1829.

## Statement

The principle of least constraint is a least squares principle stating that the true accelerations of a mechanical system of $n$ masses is the minimum of the quantity

$Z\,{\stackrel {\mathrm {def} }{=}}\sum _{j=1}^{n}m_{j}\cdot \left|\,{\ddot {\mathbf {r} }}_{j}-{\frac {\mathbf {F} _{j}}{m_{j}}}\right|^{2}$ where the jth particle has mass $m_{j}$ , position vector $\mathbf {r} _{j}$ , and applied non-constraint force $\mathbf {F} _{j}$ acting on the mass.

The notation ${\dot {\mathbf {r} }}$ indicates time derivative of a vector function $\mathbf {r} (t)$ , i.e. position. The corresponding accelerations ${\ddot {\mathbf {r} }}_{j}$ satisfy the imposed constraints, which in general depends on the current state of the system, $\{\mathbf {r} _{j}(t),{\dot {\mathbf {r} }}_{j}(t)\}$ .

It is recalled the fact that due to active $\mathbf {F} _{j}$ and reactive (constraint) $\mathbf {F_{c}} _{j}$ forces being applied, with resultant $\mathbf {R} =\sum _{j=1}^{n}\mathbf {F} _{j}+\mathbf {F_{c}} _{j}$ , a system will experience an acceleration ${\ddot {\mathbf {r} }}=\sum _{j=1}^{n}{\frac {\mathbf {F} _{j}}{m_{j}}}+{\frac {\mathbf {F_{c}} _{j}}{m_{j}}}=\sum _{j=1}^{n}\mathbf {a} _{j}+\mathbf {a_{c}} _{j}$ .

### Connections to other formulations

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 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

$Z=\sum _{j=1}^{n}\left|{\ddot {\mathbf {r} }}_{j}\right|^{2}$ The kinetic energy $T$ is also conserved under these conditions

$T\ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{2}}\sum _{j=1}^{n}\left|{\dot {\mathbf {r} }}_{j}\right|^{2}$ Since the line element $ds^{2}$ in the $3N$ -dimensional space of the coordinates is defined

$ds^{2}\ {\stackrel {\mathrm {def} }{=}}\ \sum _{j=1}^{n}\left|d\mathbf {r} _{j}\right|^{2}$ the conservation of energy may also be written

$\left({\frac {ds}{dt}}\right)^{2}=2T$ Dividing $Z$ by $2T$ yields another minimal quantity

$K\ {\stackrel {\mathrm {def} }{=}}\ \sum _{j=1}^{n}\left|{\frac {d^{2}\mathbf {r} _{j}}{ds^{2}}}\right|^{2}$ Since ${\sqrt {K}}$ is the local curvature of the trajectory in the $3n$ -dimensional space of the coordinates, minimization of $K$ 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.