What is the Meaning of Lagrangian?

[nakulphy]

what is Lagrangian ?

the Hamiltonian H = T + V represents the total energy of the system, and Lagrangian L = T-V, but what does it actually represents and what is the exact meaning of Lagrangian ? it represents excess energy or energy loss or some thing else ?

 

[psmt]

Well, since we're in the Quantum zone: if you consider the propagator for a single quantum particle between two known times, the Lagrangian integrates to an action, which is related to the amplitude for a particular path in the Feynman path integral formulation, with the action being stationary on paths that correspond to solutions of the classical equations of motion.
If we restrict the discussion to classical mechanics only, i don't think the Lagrangian has the kind of neat interpretation you're hoping for, in terms of energy. However, energy can be defined as the quantity that is conserved as a result of the Lagrangian having no explicit time dependence, so the Lagrangian is really the more fundamental quantity here. Bear in mind also that if V is a function of velocities as well as positions (e.g. for a particle in an EM field), then the Hamiltonian (energy) is still a conserved quantity but is not equal to T+V (the more generally applicable definition of the Hamiltonian from the Lagrangian being the Legendre transform described at http://en.wikipedia.org/wiki/Legendre_transformation#Hamilton-Lagrange_mechanics ).

 

[wotanub]

The Lagrangian is the quantity for which the action is minimized. The principle of least action formulation of mechanics that says the trajectory of a mass is the one taken for which the action is minimized.

The Lagrangian isn't related to an observable like the Hamiltonian is. It minimizes the action, and therefore contains all the information about the equations of motion of a system.

Susskind gives a good explanation in his CM lecture series. Give this video a watch. I skipped to the part you may find interesting.

http://youtu.be/3YARPNZrcIY?t=14m39s