Question

1. (Hamilton System) Show that the system a a12yAr2 - 2BxyCy? (1) a1yDx- 2 Ary By2 + is a Hamiltonian system, i.e. find the Hamiltonian function H(r, y) for this system

Answer 1

1. 
   - A system of ordinary differential equations of the form à (ocx) = f(x,y) 4 ( ) - 81, 8). niin valled Hamiltonian system if there exists a function Henry) such that fla,y) = 2 Hex,y) D. gem,y) = -2 Hwy) To check whether a system is Hamiltonian or not, check the following : de f(x,y) + 2 ga,y) = 0. In our case; d' and taky + Ax?- 2 Bay + Cy? y = Ana - ang tDx2 - 2Any +By? fin, y) = ana tapy a AR-2Bay + Cy? geny) = Anja &- any + Dart-2 Any + By Thus, ® means (ali +242 - 2By ) + (-01. - 2 Ax +2By) = 0 Hence, the system is Hamiltonian For finding Hex,y); we make use of O and @ ; Thus, Hary) satisfies both O and @
   
2. 
   44) 44 - 0,14 4 аль х" - na - x + ca and s-gegylda = - [ Part 2 - Anya + D23 - Ang tagja - ? tanya - DnB + Arty - Byr. Amm2 Thus Heary) has tems from both of the above equations so as to satisfy both O and @ Hex,y) = Andy + any? - Asin? + Arty - Bay=+Cy? - DX3 2 This is the required Hamiltonian function.
   
3. Keep in mind that the terms which repeat in both are to be taken once only in the Hamiltonian function .