Lagrangians with Redundancy

 In (1), the following Lagrangian is presented:  L= mm/12 vvvv + U(x)mvv - U(x)U(x) where v is velocity. The equations of motion of this Lagrangian are identical to those of the simpler L1= m/2 vv - U(x). We argue in this note, that given a simple L1, one may be able to construct a more complicated one such that O[L2] = H1 O[L1]. Here H1 is the Hamiltonian associated with L1 and is a constant, namely energy. OL = d/dt dL/dv - dL/dx. 

    As a result, we argue one must be careful in interpreting Lagrangians. The form L=T-V is often quoted, but one wishes V(x) to be a physical potential which delivers momentum changing hits to a particle. This is particularly important when considering quantum mechanics for which we argue V(x)= Sum over k Vk exp(ikx), the exp(ikx) being virtual photon hits. One may take the Fourier transform of any function, but not any function represents virtual photon hits, so one must take care. For L2, U(x)U(x) is the purely spatial potential piece, but it is U(x), as seen from the equations of motion, which delivers momentum changes. The equations of motion may give an indication of a simple constant (say energy) and show explicitly what “potential function” changes v, the velocity.