# Gauss's principle of least constraint

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 ${\displaystyle N}$ masses is the minimum of the quantity

${\displaystyle Z\ {\stackrel {\mathrm {def} }{=}}\ \sum _{k=1}^{N}m_{k}\left|{\frac {d^{2}\mathbf {r} _{k}}{dt^{2}}}-{\frac {\mathbf {F} _{k}}{m_{k}}}\right|^{2}}$

for all trajectories satisfying any imposed constraints, where ${\displaystyle m_{k}}$, ${\displaystyle \mathbf {r} _{k}}$ and ${\displaystyle \mathbf {F} _{k}}$ represent the mass, position and applied forces of the ${\displaystyle \mathrm {k^{th}} }$ 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

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

${\displaystyle Z=\sum _{k=1}^{N}\left|{\frac {d^{2}\mathbf {r} _{k}}{dt^{2}}}\right|^{2}}$

The kinetic energy ${\displaystyle T}$ is also conserved under these conditions

${\displaystyle T\ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{2}}\sum _{k=1}^{N}\left|{\frac {d\mathbf {r} _{k}}{dt}}\right|^{2}}$

Since the line element ${\displaystyle ds^{2}}$ in the ${\displaystyle 3N}$-dimensional space of the coordinates is defined

${\displaystyle ds^{2}\ {\stackrel {\mathrm {def} }{=}}\ \sum _{k=1}^{N}\left|d\mathbf {r} _{k}\right|^{2}}$

the conservation of energy may also be written

${\displaystyle \left({\frac {ds}{dt}}\right)^{2}=2T}$

Dividing ${\displaystyle Z}$ by ${\displaystyle 2T}$ yields another minimal quantity

${\displaystyle K\ {\stackrel {\mathrm {def} }{=}}\ \sum _{k=1}^{N}\left|{\frac {d^{2}\mathbf {r} _{k}}{ds^{2}}}\right|^{2}}$

Since ${\displaystyle {\sqrt {K}}}$ is the local curvature of the trajectory in the ${\displaystyle 3N}$-dimensional space of the coordinates, minimization of ${\displaystyle K}$ 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.