Pauca Sed Matura (Few, but Ripe)”
Gauss provided the principle of least constraint, which is an ingenious reinterpretation of d’Alembert’s principle [1], [2]. Consider a system of \(N\) point particles. Gauss defined the quantity \(Z\) called constraint defined by: \[ Z = \sum_{i=1}^{N}\frac{1}{2m_i} ||\mathbf{F}_i-m_i\ddot{\mathbf{r}}_i||^2 \] In this equation, \(\mathbf{F}_i\in \mathbb{R}^3\) is the sum of forces experienced by the \(i\)-th particle; \(\ddot{\mathbf{r}}_i\in\mathbb{R}^3\) is the acceleration of the \(i\)-th particle; \(m_i\in\mathbb{R}\) is the mass of the \(i\)-th particle.
Gauss principle of least constraint states that under the given kinematical conditions, the actual motion (or actual acceleration \(\ddot{\mathbf{r}}_i\)) occurring in nature is such that the constraint \(Z\) becomes as small as possible [2]. The proof of this principle is provided by Lanczos’ fantastic book [2].
It is immediate to check that under no kinematic restriction, the minimum of \(Z\) happens when: \[ \mathbf{F}_i = m_i\ddot{\mathbf{r}}_i, ~~~~ i = 1,2,\cdots, N \] which is simply the Newton’s Second Law of motion.