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Transcribed Image Text:Consider the Lagrangian of the one-dimensional harmonic oscillator,
m
L =
12
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
ip
mw
+
a* :=
mw
2
(x — —³) ·
ip
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
ДА ӘВ
ДА ДВ
{A, B} ==
Эх др
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.
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