Gauss's principle of least constraint

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

${\displaystyle Z\,{\stackrel {\mathrm {def} }{=}}\sum _{j=1}^{n}m_{j}\cdot \left|\,{\ddot {\mathbf {r} }}_{j}-{\frac {\mathbf {F} _{j}}{m_{j}}}\right|^{2}}$

where the jth particle has mass ${\displaystyle m_{j}}$, position vector ${\displaystyle \mathbf {r} _{j}}$, and applied non-constraint force ${\displaystyle \mathbf {F} _{j}}$ acting on the mass.

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

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

Hertz's principle of least curvature is a special case of Gauss's principle, restricted by the three 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

${\displaystyle Z=\sum _{j=1}^{n}\left|{\ddot {\mathbf {r} }}_{j}\right|^{2}}$

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

${\displaystyle T\ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{2}}\sum _{j=1}^{n}\left|{\dot {\mathbf {r} }}_{j}\right|^{2}}$

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

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

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

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

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.

• Appell's equation of motion

• Gauss, C. F. (1829). "Über ein neues allgemeines Grundgesetz der Mechanik". Crelle's Journal. 1829 (4): 232–235. doi:10.1515/crll.1829.4.232. S2CID 199545985.
• Gauss, Carl Friedrich. Werke [Collected Works]. Vol. 5. p. 23–28.
• Hertz, Heinrich (1896). Principles of Mechanics. Miscellaneous Papers. Vol. III. Macmillan.
• Lanczos, Cornelius (1986). "IV §8 Gauss's principle of least constraint". The variational principles of mechanics (Reprint of University of Toronto 1970 4th ed.). Courier Dover. pp. 106–110. ISBN 978-0-486-65067-8.
• Papastavridis, John G. (2014). "6.6 The Principle of Gauss (extensive treatment)". Analytical mechanics: A comprehensive treatise on the dynamics of constrained systems (Reprint ed.). Singapore, Hackensack NJ, London: World Scientific Publishing Co. Pte. Ltd. pp. 911–930. ISBN 978-981-4338-71-4.

• A modern discussion and proof of Gauss's principle
• Gauss principle in the Encyclopedia of Mathematics
• Hertz principle in the Encyclopedia of Mathematics