Problem 6Consider the following Lagrangian describing the two-dimensional motion of a particle of mass m in aninertial system (1, 2),L = m² (1² + ±²) - m² (x² -w² (x² + a x²) - bx₁ x2,2where a and b are constant numbers.(i) Write down the Euler-Lagrange equations for the system.(ii) For generic values of a and b, what are the conserved quantities of the system and what are thecorresponding Noether symmetries? Consider next the case a = 1 and b = 0. Discuss if there are anyadditional symmetries and find the corresponding conserved quantities.(iii) Write down the Hamiltonian of the system and the corresponding Hamilton equations for generic valuesof a and b.(iv) The time evolution of a generic physical observable O(x, p) is described by the equationwhereӦ={O,H},{A, B}=Σ2 ДА ӘВдхп дрпДА ӘВдрп дхпJB )n=1is the Poisson bracket of A with B. x := (x1, x2) and p = (P1, P2) denote the position vector and themomentum of the particle, respectively. Using this result, calculate the time evolution of p₁ and L3,i.e. calculate p₁ and Ĺ3, for generic values of a and b. Here L3 = x1p2x2P1.(v) Consider the case a = 1 and b = -mw². Discuss if there are any additional symmetries, and determinethe corresponding conserved quantities.

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Answered: Consider the following Lagrangian…

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Consider the following Lagrangian describing the two-dimensional motion of a particle of mass m in an inertial system (1, 2), m L = 2 m (à² + i) - w² (x² + ax²) − bx₁ x2,