Newtonian mechanics is the minimal dynamical framework in which trajectories $\mathbf{x}(t)$ evolve under forces via $m\mathbf{a}=\mathbf{F}$, with state specified by positions and velocities in time. At graduate level, the central questions are not only how to solve problems, but how to formulate the theory consistently, recognize its symmetries, and understand where it fails.

An inertial frame is one in which a force-free particle moves in a straight line at constant speed. The set of inertial frames is related by Galilean transformations: $\mathbf{x}'=\mathbf{x}-\mathbf{V}t-\mathbf{x}_0$ and $t'=t+t_0$. Newtonian physics is invariant under this group, and the associated conserved quantities emerge from symmetry via Noether’s theorem in its mechanical form.

In practice, a force is a model-dependent construct encoding interaction. The same observed dynamics may admit multiple force decompositions, but the physically meaningful content resides in invariant predictions. At graduate level, one must distinguish fundamental forces from effective forces, constraint forces, and generalized forces in non-Cartesian coordinates.