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  "body": "\n## Statement\n\nConsider $N$ particles with masses $m_i \u003e 0$ and applied forces $F_i$, for $i = 1, \\dots, N$, subject to ideal constraints that are holonomic or linear in the velocities.\nFix a time $t$ and the positions and velocities of the particles at time $t$.\nCall 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.\nDefine the *constraint* (Gauss's function)\n$$Z(b_1, \\dots, b_N) = \\sum_{i=1}^{N} m_i \\,\\Big\\lvert b_i - \\frac{F_i}{m_i} \\Big\\rvert^2.$$\nThen the actual accelerations $a_i = \\ddot r_i(t)$ are the unique minimiser of $Z$ among admissible accelerations.\n\n## Remarks\n\nFor such constraints the admissible accelerations form an affine subspace $a + W$, where $W$ is the space of virtual displacements.\nOn it $Z$ is a strictly convex quadratic function, so it has a unique minimiser.\nFor $b = a + w$ with $w \\in W$, one finds $Z(b) - Z(a) = 2\\sum_i (m_i a_i - F_i) \\cdot w_i + \\sum_i m_i \\abs{w_i}^2$.\nThe linear term vanishes by [[d'Alembert's principle|GD776GVZSR]], so $Z(b) \u003e Z(a)$ when $w \\neq 0$.\nExample: a single free particle has all accelerations admissible, and the minimiser is $b = F/m$, which is Newton's law.\n\n## Explanations\n\n### L1 human\n\nWithout constraints, each particle would accelerate as $F_i/m_i$.\nConstraints prevent this.\nGauss's principle says that the real motion deviates from the free motion as little as possible, where the deviation is measured by a mass-weighted sum of squares.\n\n### L2 human\n\nGauss's principle is a minimum principle at each instant, not a stationarity principle over a whole path like [[Hamilton's principle|FAQ9CH2Z76]].\nIt is a quadratic program in the accelerations, and it handles nonholonomic constraints linear in the velocities with no extra work.\nThis makes it useful in the numerical simulation of constrained multibody systems.\nIt is equivalent to d'Alembert's principle for ideal constraints.\n",
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