文档介绍：Calculate
ωx˙ωx˙
ψ fψ= 1 + 2 = 0 + 0
· ωx1 ωx2
Pictorially
x 2
x 1
Note that the area is conserved.
Conservation of areas holds for all conserved systems. This is conventionally
derived from Hamiltonian mechanics and the canonical form of equations of
motion.
In conservative systems, the conservation of volumes in phase space is known
as Liouville’s theorem.
4 Damped oscillators and dissipative systems
General remarks
We have seen how conservative systems behave in phase space.
What about dissipative systems?
What is a fundamental difference between dissipative systems and conserva­
tive systems, aside from volume contraction and energy dissipation?
Conservative systems are invariant under time reversal.
•
Dissipative systems are not; they are irreversible.
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Consider again the undamped pendulum:
d2β
+ γ2 sin β= 0.
dt2
Let t t and thus ω/ωt ω/ωt.
∗−∗−
There is no change—the equation is invariant under the transformation.
The fact that most systems are dissipative is obvious if we run a movie
backwards (ink drop, car crash, cigarette smoke...)
Formally, how may dissipation be represented? Include terms propor­
tional to odd time derivatives., ., break time-reversal invariance.
In the linear approximation, the damped pendulum equation is
d2β dβ
+ ρ+ γ2β= 0
dt2 dt
where
γ2 = g