Newton’s second law $m\ddot{\mathbf{x}}=\mathbf{F}$ defines an initial-value problem: given $\mathbf{x}(t_0)$ and $\dot{\mathbf{x}}(t_0)$, the trajectory is determined under suitable regularity conditions. For many-body systems, the state is a point in a $6N$-dimensional phase space, although the Newtonian formulation lives naturally on configuration space and its tangent bundle.

Holonomic constraints can be written as equations among coordinates and time, $f_a(q_1,\dots,q_n,t)=0$, while nonholonomic constraints involve velocities and are not reducible to purely coordinate conditions. The distinction matters because holonomic constraints can often be eliminated, whereas nonholonomic systems require careful treatment of admissible virtual displacements and constraint reactions.

Integrating $m\dot{\mathbf{v}}=\mathbf{F}$ over a short time interval yields impulse, $\Delta\mathbf{p}=\int \mathbf{F}\,dt$. This is the key tool for collisions, where forces may be large but act over a short duration. In research settings, impulsive approximations are used carefully when separation of timescales is justified.