Problem 4 Consider the Lagrangian of the one-dimensional harmonic oscillator, m L = -2 m 2 (i) Write down the Euler-Lagrange equation for the system. (ii) Write down the Hamiltonian of the system, and the corresponding Hamilton equations. (iii) Consider the following complex quantities a == mw 2 гр mw гр x + a* := x- mw. 2 mw Calculate aa* and re-express the Hamiltonian in terms of a and a*. (iv) For two given functions A(x, p) and B(x,p), the Poisson bracket {A, B} is defined as ДАӘВ {A, B} : === дх др ӘАӘВ др дх Calculate the Poisson brackets {a, a*}, {a, H}, and {a*, H}. (v) Write down the time evolution equation for a(t), and its solution.

Problem 4 Consider the Lagrangian of the one-dimensional harmonic oscillator, m L = -2 m 2 (i) Write down the Euler-Lagrange equation for the system. (ii) Write down the Hamiltonian of the system, and the corresponding Hamilton equations. (iii) Consider the following complex quantities a == mw 2 гр mw гр x + a* := x- mw. 2 mw Calculate aa* and re-express the Hamiltonian in terms of a and a. (iv) For two given functions A(x, p) and B(x,p), the Poisson bracket {A, B} is defined as ДАӘВ {A, B} : === дх др ӘАӘВ др дх Calculate the Poisson brackets {a, a}, {a, H}, and {a*, H}. (v) Write down the time evolution equation for a(t), and its solution.