4. Consider a harmonic oscillator in two dimensions described by the Lagrangian mw? L =* +r°*) – m where (r, ø) are the polar coordinates, m > 0 is the mass, and w > 0 the oscillation frequency of the oscillator.
4. Consider a harmonic oscillator in two dimensions described by the Lagrangian mw? L =* +r°*) – m where (r, ø) are the polar coordinates, m > 0 is the mass, and w > 0 the oscillation frequency of the oscillator.
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Answer a b & c ONLY.
Transcribed Image Text:4. Consider a harmonic oscillator in two dimensions described by the Lagrangian
- )ف م + ث(
m
where (r, ø) are the polar coordinates, m > 0 is the mass, and w > 0 the oscillation
frequency of the oscillator.
(a) Find the canonical momenta p, and po, and show that the Hamiltonian is given by
mw?
H(r, 6,Pr, P6):
2m
2mr?
(b) Show that p, is a constant of motion.
(c) In the following we will perform a variable transformation (r, 6, pr, Pa) (Qr, Qo, Pr, P)
defined by
Qo = 20,
lp-
P,
Po
P =2
2r
where l is an arbitrary constant length that just ensures that Q, has also the dimension
of length. Show that this transformation is canonical.
(d) The Hamiltonian becomes in the new variables
H(Q,,Q6, Pr, Pg) =Q,P? +
2P?
+
mw?l
ml
mlQ,
Show that the fact that energy E is conserved and H:
Hamiltonian HK and a new energy EK such that HK
with
E allows us to introduce a new
Ek is equivalent to H = E,
P2
P2
HK(Qr, Q6, Pr, Ps) =
-
2m 2mQ? Q,
Determine the constant a and the new energy EK as a function of m, w, E.
(e) In question 4 (d) just above you will find that one always has Eg < 0 and a > 0. Give
a physical argument why this must indeed be the case.
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