The dynamics of many Newtonian systems can be obtained from stationary action with $$S=\int_{t_1}^{t_2}L(q,\dot q,t)\,dt,$$ where $L=T-V$. The Euler–Lagrange equations are equivalent to Newton's second law for appropriately chosen generalized coordinates.

The canonical momentum is $p_i=\partial L/\partial \dot q_i$, and the Hamiltonian is $$H(q,p,t)=\sum_i p_i\dot q_i-L.$$ For standard conservative systems, $H$ coincides with total energy and generates time evolution through Hamilton's equations.

Advanced Newtonian mechanics underlies orbital dynamics, spacecraft attitude control, molecular dynamics, granular materials, active matter models, and high-accuracy numerical integration schemes such as symplectic methods. Structure-preserving computation is crucial when long-time energy behavior matters.