For particles moving independently and freely subject only to specified forces, the motion of each particle obeys Newton’s law F=ma which can be written in terms of the Cartesian coordinates in the form x”=(1/m)dV/dx where V is the potential energy (and the form is the same for all components).
For particles constrained to be parts of a rigid body we might prefer to use more natural variables like the angular orientation of the body, but then the equations become more complicated (eg with centripetal and coriolis “forces” coming in, so the equation for the r component is not just
r”=(1/m)dV/dr ). And it is often not easy to work out the translation of Newton’s eq into the new coordinates.
The Lagrangian approach rewrites the equations of motion in an equivalent form which has the same structure in all coordinate systems – which turns out to be given for every component q by d/dt(dL/dq’)-dL/dq=0 with L=T-V where V is again the potential energy and T the kinetic energy (which in Cartesian coords would be (m(x’)^2)/2 ). This provides a systematic way of getting the appropriate equations rather than having to work them out in terms of the desired coordinates by a complicated conversion process from the Newtonian form.
It turns out that the Lagrangian equations are also equivalent to a variational principle (as mentioned by other responders) – namely that the actual trajectory be a stationary point (eg min or max) of the path integral of the Lagrangian function L.
Source: (1001) Alan Cooper’s answer to In layman’s terms, what is a Lagrangian? – Quora