Is the LaGrangian tautological?

[oldphysicist]

This issue has always bothered me, and I would like to hear a logical resolution. The classical prescription for finding it is L=T-V. From the LaGrangian, the equations are motion are then deduced using the Euler-LaGrange eqs. But - the equations are motion are required in order to determine T and V, since they (eqs. of motion) are necessary to determine the forces, which will be used to calculate T and V. So it looks like a big circle to me ?

Also, when applied to field theory (classical or quantized) how do you identify T and V, if you want to use the basic definition L=T-V?

 

[Dale]

I would say it is empirical rather than tautological. You use a given Lagrangian because it gives equations of motion that match what is observed empirically.

 

[voko]

No, forces are not used in the "pure" Lagrangian formulation. Kinetic energies are obtained directly via $T= \frac {m v^2} {2}$ in whatever coordinate systems employed, potential energies are considered given empirically.