Mechanics of particles and systems Constraints and generalized coordinates Fact

Gauss's principle of least constraint

Statement

Consider $N$ particles with masses $m_i > 0$ and applied forces $F_i$ , for $i = 1, \dots, N$ , subject to ideal constraints that are holonomic or linear in the velocities. Fix a time $t$ and the positions and velocities of the particles at time $t$ . Call a family $(b_1, \dots, b_N)$ of vectors of $\R^3$ an admissible acceleration if it is compatible with the constraints at these positions and velocities. Define the constraint (Gauss's function) $Z(b_1, \dots, b_N) = \sum_{i=1}^{N} m_i \,\Big\lvert b_i - \frac{F_i}{m_i} \Big\rvert^2.$ Then the actual accelerations $a_i = \ddot r_i(t)$ are the unique minimiser of $Z$ among admissible accelerations.

Remarks

For such constraints the admissible accelerations form an affine subspace $a + W$ , where $W$ is the space of virtual displacements. On it $Z$ is a strictly convex quadratic function, so it has a unique minimiser. For $b = a + w$ with $w \in W$ , one finds $Z(b) - Z(a) = 2\sum_i (m_i a_i - F_i) \cdot w_i + \sum_i m_i \abs{w_i}^2$ . The linear term vanishes by d'Alembert's principle, so $Z(b) > Z(a)$ when $w \neq 0$ . Example: a single free particle has all accelerations admissible, and the minimiser is $b = F/m$ , which is Newton's law.

Going deeper

For a student

Gauss's principle is a minimum principle at each instant, not a stationarity principle over a whole path like Hamilton's principle. It is a quadratic program in the accelerations, and it handles nonholonomic constraints linear in the velocities with no extra work. This makes it useful in the numerical simulation of constrained multibody systems. It is equivalent to d'Alembert's principle for ideal constraints.

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Euler–Lagrange equation concept
d'Alembert's principle statement
Gauss's principle of least constraint statement

Tag	`FZWAHMFZM3`
URI	`m2://FZWAHMFZM3`
Kind	statement, fact
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MSC	70F25, 70H30, 49S05
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About	d'Alembert's principle
Uses	d'Alembert's principle
Same as	Wikidata Q1477313 Wikipedia Gauss's principle of least constraint