Gauss's principle of least constraint

The principle of least constraint is another 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

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

where the kth particle has mass $m_{k}$, position vector $\mathbf {r} _{k}$, and applied non-constraint force $\mathbf {F} _{k}$ acting on the mass. The notation d/dt indicates a time derivative. The corresponding accelerations ${\frac {d^{2}\mathbf {r} _{k}}{dt^{2}}}$ satisfy the imposed constraints, which in general depends on the current state of the system, $(\mathbf {r} _{k}(t),{\frac {d\mathbf {r} _{k}(t)}{dt}})$.

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 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

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

The kinetic energy $T$ is also conserved under these conditions

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

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

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

the conservation of energy may also be written

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

Dividing $Z$ by $2T$ yields another minimal quantity

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

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.

See also

• Appell's equation of motion

References

• Gauss, C. F. (1829). "Über ein neues allgemeines Grundgesetz der Mechanik". Crelle's Journal. 4: 232. doi:10.1515/crll.1829.4.232.
• Gauss, C. F. Werke. Vol. 5. p. 23.
• Hertz, H. (1896). Principles of Mechanics. Miscellaneous Papers. Vol. III. Macmillan.
• Lanczos, Cornelius (1952). The Variational Principles of Mechanics. Dover. p. 106.

External links

• [1] A modern discussion and proof of Gauss's principle
• [2] Gauss's principle of least constraint
• [3] Hertz's principle of least curvature