By Povh B., Rith K., Scholz C., Zetsche F.
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Additional info for Particles and nuclei: an introduction
If the system consists of two non-interacting parts A and B, each of which has its Lagrangian, then the L of the whole system is equal to the sum of the two Lagrangians: L = LA + LB . 14), it is obvious, that multiplying of the Lagrangian by a constant C does not change them. Hence, for the description of motion, with the same success one can use the function CL . We note that when we consider a single isolated system we can multiply L by different constants. But when we consider different isolated parts of one mechanical system (or different isolated mechanical systems), their Lagrangian functions can not be multiplied by different constants but only with one and the same constant.
Df (r ,t) , without changing 6. Why can we add to the Lagrangian a function dt the equations of motion? 7. Why can we add any constant to the potential energy of a system and how does this affect the Lagrangian function and Lagrange's equations? Chapter 30 8. Why does the Lagrangian function of a free particle depend only on v 2 but not on v and r ? 9. What does it mean that the Lagrangian function is an invariant with respect to the Galilean transformation? Give an example. 10. What is a potential?
14) in both systems would not change, as the Galilean principle demands, if the second term in the right is the total time derivative of a function of the coordinates and the time. , ∂L (v′2 ) ∂v′2 = C , L ′ = L (v′2 ) = Cv′2 . 24) satisfies Galileo's principle. The Lagrangian function is an invariant also at a finite velocity u. In fact L = Cv 2 = C ( v′ + u ) = Cv′2 + 2Cv′u + Cu 2 , 2 L = L′+ d 2Cr ′u + Cu 2t ) . 25) Since the second term in the right of the last equation is a total time derivative, it can be neglected and consequently L = L ′.