Question

14. The Lagrangian for a system can be written as <= ax + b3 + e*j + fy+si+g3 – W/22+y2, where a, b, c, f. 8, and k are constants. What is the Hamiltonian? What quantities are conserved?

Answer 1

Ans: The Lagran The Lagrangian of the system is given as. 1 = aä 2+b ý + cà ý + fy2gi Ż + g 4 - K[X2+42 OL a bicifig and k are constants Here, no, of generalized co-ordinates are three, so there are three degree's of freedom for a given syskom Lagrangian 1=T-v=aå 2+ by tc ý + fyzi2 tgý -kxZ+42 Now Pa = 21. Py = 27 7 ,P2 = 8 h 1. Pa = 29ä + cÝ + fyż – 0 Py = b + c5i +9 - C Pe= fyzi - Now lagrange's equation of motton can be given as. de cou) - = 0 for general sed co-ozdinate 09 (208 ++4422)+( by + ka 296i + cÝ + fy2 ₂ + 2 fyży + by + ka de 102442) = 0 - 22 Jx2x42 for generalized co-ordinale 'y ) -

a (blat cätg) + KY -0. JxZ+42 -bod+ cx+ ky = 0 Jx2+42 cë - bitky = 0 X2 Jx2+ y2 - noo for generalized co-ordinale 2 (2) - 23= 0 = 0 a 1f42a)-0=0 se apz=0 d LaL 이비32) 이 dt & Pz = constant say c ie Pz is constant in time, so it is conserved. which says. or. 2ty ý åt fyzść = 0 F y = x = 0 Now Hamiltonian of the system, H = { Pių, -L - Polát py ý + P2 : -L = (29år cý+f42₂) x + ( b +ä+g) Ý + (6 426) Ź -aa²+bý + cx ý + fy2xż + gj - kJ x2 +421 - 29x² + cÝă + fy²żä + by + (x Ý tgÝ + $42% ² – ad2 by - (x - f 42% z – gý + kJ x2 + y2 20 TH= this qa² + cx ya + fy2żät + KJxZ+42/ is the Hamiltenian of green system,