Gauss's principle of least constraint#Hertz's principle of least curvature

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 $\backslash mathbf\{r\}\_j$, and applied non-constraint force $\backslash mathbf\{F\}\_j$ acting on the mass.

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

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

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 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

:$$

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 $\backslash sqrt\{K\}$ 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.

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...

• Appell's equation of motion

• {{cite journal | last =Gauss | first = C. F. | year = 1829 | title = Über ein neues allgemeines Grundgesetz der Mechanik | journal = Crelle's Journal | volume = 1829 | issue = 4 | pages = 232–235 | doi=10.1515/crll.1829.4.232 | s2cid=199545985 | url = https://zenodo.org/record/1448816 }}
• {{cite book | last = Gauss | first = Carl Friedrich | title = Werke | trans-title = Collected Works | volume = 5 | pages = 23–28 }}
• {{cite book | last = Hertz | first = Heinrich | year = 1896 | title = Principles of Mechanics | series = Miscellaneous Papers | volume = III | publisher = Macmillan }}
• {{cite book | title = The variational principles of mechanics | last = Lanczos | first = Cornelius | author-link = Cornelius Lanczos | pages = 106–110 | chapter = IV §8 Gauss's principle of least constraint | isbn = 978-0-486-65067-8 | publisher = Courier Dover | year = 1986 | edition = Reprint of University of Toronto 1970 4th}}
• {{cite book |title=Analytical mechanics: A comprehensive treatise on the dynamics of constrained systems |last=Papastavridis |first=John G. |pages=911–930 |chapter=6.6 The Principle of Gauss (extensive treatment) |edition=Reprint |isbn=978-981-4338-71-4 |publisher=World Scientific Publishing Co. Pte. Ltd. |year=2014 |location=Singapore, Hackensack NJ, London |url=https://books.google.com/books?id=UgW3CgAAQBAJ}}

{{Reflist}}

• [https://preetum.nakkiran.org/misc/gauss/] A modern discussion and proof of Gauss's principle
• [https://encyclopediaofmath.org/wiki/Gauss_principle Gauss principle] in the Encyclopedia of Mathematics
• [https://encyclopediaofmath.org/wiki/Hertz_principle Hertz principle] in the Encyclopedia of Mathematics

Category:Classical mechanics