Question

what is the lagrangian of a free particle?

Answer 1

The  Lagrangian of a system of particles is defined as
 
  L = T - V
         T = total kinetic energy of all particles
         V = total potential energy

A free particle means there is no external force acting on the system ie., the particle.

    Then the Lagrangian is a constant. (wrt time)

For a freely falling particle under gravity, the Lagrangian is
   L = 1/2 m v² -  m g h 
     constant  because  KE increases by an amount by which PE decreases.

Answer 2

Lagrangian is generally defined by the difference between the kinetic energy Ec = T and the potential energy Ep = V:

L = Ec - Ep = T - V

Lagrangian is constant.