Newton's equation

Theorem 3.2 (Newton’s equation). Let a mechanical system of \(n\) points have configuration \(\mathbf x(t) \in \R ^N\), where \(N = 3n\). By the determinacy principle (Definition 3.1) there is a function \(\mathbf F : \R ^N \times \R ^N \times \R \to \R ^N\) such that every motion of the system satisfies

Conversely, if \(\mathbf F\) is smooth, then a solution is uniquely determined by \(\mathbf x(t_0)\) and \(\dot {\mathbf x}(t_0)\).

Source: [1, ch. 1]