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I am wondering how to physically interpret the generalized momentum quantity derived from the Euler-Lagrange equations. For some Lagrangians is it equal to the actual momentum for the particle, however i have noticed that for a relativistic particle moving in an electromagnetic field the generalized momentum is not equal to the actual relatvistic momentum.

Could someone explain why this is so, or maybe explain the physical significance of the extra term in the generalized momentum for this EM field case?

Thanks

Ray