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lecnotes4

# lecnotes4 - Calculate Pictorially x1 x2 f = = 0 0 x1 x2 x2...

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Calculate ψ ψ ωx ˙ 1 ωx ˙ 2 � · f = ωx 1 + ωx 2 = 0 + 0 Pictorially 2 1 x x 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 4.1 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 . 22

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Consider again the undamped pendulum: d 2 β + γ 2 sin β = 0 . d t 2 Let t ∗ − t and thus ω/ωt ∗ − ω/ωt . There is no change—the equation is invariant under the transformation.
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