Newtonian mechanics describes motion through forces. Lagrangian mechanics packages the same dynamics into a scalar function of generalized coordinates and velocities.

For many mechanical systems,

$$L(q,\dot q,t)=T-V,$$

where $T$ is kinetic energy and $V$ is potential energy.

Generalized coordinates

The coordinates $q_1,\ldots,q_n$ can be chosen to describe the system's independent degrees of freedom. They need not be Cartesian positions.

For a particle in one dimension,

$$L(x,\dot x)=\frac12m\dot x^2-V(x).$$

Why use a scalar function?

The Lagrangian can encode constraints and complicated coordinate choices without resolving every force component separately. The equations of motion follow from how the action built from $L$ changes when a trajectory is varied.

Lagrangian mechanics is therefore not a different physical theory from ordinary classical mechanics in its shared domain; it is a formulation that makes the structure behind the equations of motion especially clear.