Gauss's principle of least constraint

Carl Friederich Gauss

Heinrich Hertz

Gauss and Hertz devised variational principles of mechanics

Theprinciple of least constraintis onevariational formulationofclassical mechanicsenunciated byCarl Friedrich Gaussin 1829, equivalent to all other formulations ofanalytical mechanics.Intuitively, it says that the acceleration of aconstrainedphysical systemwill be as similar as possible to that of the corresponding unconstrained system.

Statement

The principle of least constraint is aleast squaresprinciple 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 thejth particle hasmass$m_{j}$,position vector$\mathbf {r} _{j}$,and applied non-constraint force$\mathbf {F} _{j}$acting on the mass.

The notation${\dot {\mathbf {r} }}$indicatestime derivativeof a vector function$\mathbf {r} (t)$,i.e. position. The correspondingaccelerations${\ddot {\mathbf {r} }}_{j}$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${\textstyle \mathbf {R} =\sum _{j=1}^{n}\mathbf {F} _{j}+\mathbf {F_{c}} _{j}}$,a system will experience an acceleration${\textstyle {\ddot {\mathbf {r} }}=\sum _{j=1}^{n}{\frac {\mathbf {F} _{j}}{m_{j}}}+{\frac {\mathbf {F_{c}} _{j}}{m_{j}}}=\sum _{j=1}^{n}\mathbf {a} _{j}+\mathbf {a_{c}} _{j}}$.

Connections to other formulations

Gauss's principle is equivalent toD'Alembert's principle.

The principle of least constraint is qualitatively similar toHamilton's principle,which states that the true path taken by a mechanical system is an extremum of theaction.However, Gauss's principle is a true (local)minimalprinciple, whereas the other is anextremalprinciple.

Hertz's principle of least curvature

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 apotential 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

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

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

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

theconservation of energymay 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 localcurvatureof the trajectory in the$3n$-dimensional space of the coordinates, minimization of$K$is equivalent to finding the trajectory of least curvature (ageodesic) that is consistent with the constraints.

Hertz's principle is also a special case ofJacobi's formulation ofthe least-action principle.

Philosophy

Hertz designed the principle to eliminate the concept of force and dynamics, so that physics would consist exclusively of kinematics, of material points in constrained motion. He was critical of the "logical obscurity" surrounding the idea of force.

I would mention the experience that it is exceedingly difficult to expound to thoughtful hearers that very introduction to mechanics without being occasionally embarrassed, without feeling tempted now and again to apologize, without wishing to get as quickly as possible over the rudiments, and on to examples which speak for themselves. I fancy that Newton himself must have felt this embarrassment...

To replace the concept of force, he proposed that the acceleration of visible masses are to be accounted for, not by force, but by geometric constraints on the visible masses, and their geometric linkages to invisible masses. In this, he understood himself as continuing the tradition ofCartesian mechanical philosophy,such asBoltzmann's explaining of heat by atomic motion, andMaxwell's explaining of electromagnetismbyethermotion. Even though both atoms and the ether were not observable except via their effects, they were successful in explaining apparently non-mechanical phenomena mechanically. In trying to explain away "mechanical force", Hertz was "mechanizing classical mechanics".

See also

• Appell's equation of motion

Literature

• 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.S2CID199545985.
• 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.ISBN978-0-486-65067-8.
• 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.ISBN978-981-4338-71-4.

External links

• [1]A modern discussion and proof of Gauss's principle
• Gauss principlein theEncyclopedia of Mathematics
• Hertz principlein the Encyclopedia of Mathematics