Lagrangian mechanics is one of three major formulations of classical mechanics with the others being Newtonian mechanics and Hamiltonian mechanics. Classical mechanics as a whole is a branch of physics most applicable to the situations that humans most often encounter, as it does not account for quantum randomness (hence cannot be applied to very small scales), nor does it account for relativistic effects (which are most relevant when dealing with strong gravitational fields or velocities close to the speed of light). In Lagrangian mechanics, we define a quantity known as the Lagrangian of our physical system:
L=T−V.(1) Where T is kinetic energy and V is potential energy of our entire physical system. It is important to note that only conservative forces contribute to V. Lagrangian mechanics can account for non-conservative (i.e. dissipative) forces, too, but we will get to how one handles them later.
The distinction between conservative and non-conservative forces is whether they change the net energy of your system. For instance, gravity does not because any increase in kinetic energy it causes is balanced by a corresponding reduction in potential energy. Likewise, any loss of kinetic energy caused by gravity is balanced by an increase in potential energy. Consequently, it is a conservative force. Air resistance, however, is a non-conservative force because the losses it causes in kinetic energy are not balanced by increases in potential energy.