The Lagrangian $L(q,\dot{q},t)=T-V$ packages Newtonian dynamics into a variational statement: the physical path extremizes $S[q]=\int_{t_1}^{t_2}L\,dt$. The Euler-Lagrange equations, $\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i}-\frac{\partial L}{\partial q_i}=0$, reproduce Newton’s laws for conservative systems and generalize efficiently to constrained coordinates.

For a system with configuration coordinates $q^i$, the kinetic energy often defines a metric on configuration space: $T=\tfrac12 g_{ij}(q)\dot{q}^i\dot{q}^j$. This geometry clarifies geodesic motion, coordinate invariance, and the treatment of constraints. In modern language, Newtonian mechanics on curved configuration manifolds is naturally a geodesic-plus-potential problem.

Holonomic constraints can be enforced by augmenting the Lagrangian with multipliers: $L' = L + \sum_a \lambda_a f_a(q,t)$. The multipliers represent constraint forces in a coordinate-invariant way. For many-body systems, this method is especially useful when the geometry of constraints is complex.