Question

C. Noether Theorem. In class we discussed how conservation of total momentum, angular momentum and energy are consequences of certain symmetries of the Lagrangian. More generally assume now that under a transformation of the form qi qi + eK (9), the lagrangian L = L(9.4) is invariant (meaning dL/de = 0). The functions Ki are functions of all the qi (denoted here collectively by 9). 1) Show that, if the Euler-Lagrange equations are satisfied then the quantity p(g, g) = King) does not change with time.

Answer 1

Ars) Guien the lograngian is invariant under bransformation 0 of form & &; – qit & Ki (q) 2 qit & Ki Cq) de DE = 0 = & (al dai + a2 da. ) I aq; TEé aq; del (2= 2 (9,2)) - El al Ki tal ki) - © i lari 29; (using 1) ng Now di Ca o Euler Jagrange equation using this can becomes o- & (a ( 2 ) K + 2 ) K = dki and a = f ( 3 ) >= 2530* + $(#)] - Ki (q) = P1914)