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Lecture Note Sketches Hermann Riecke - ESAM Home Page

Lecture Note Sketches Hermann Riecke - ESAM Home Page

• total energy H H

• total energy H H note that (35) can be written as Note: to compute d dt � ∂tψ ψ ∗ + ψ∂tψ ∗ dx = � ∂tψ δN {ψ, ψ∗ ) δψ(x) + δN {ψ, ψ∗ ) δψ∗ ∂tψ (x) ∗ dx i.e. think of ψ(x, t) as a vector with components labeled by x, each of which depends on t, i.e. ψ(x, t) ∼ ψx(t), and then use chain rule on N{ψ, ψ∗ }. Analogously, d H = dt = � δH{ψ, ψ ∗ ) � δψ ∂tψ + δH{ψ, ψ∗ ) δψ ∗ −i∂tψ ∗ ∂tψ + i∂tψ ∂tψ ∗ dx = 0 ∂tψ ∗ dx = • because of the factor i in (34) the energy of the system does not decrease with time as it does in systems with a Lyapunov functional. Instead it is conserved. • for a system with a Lyapunov functional (variational system) one has ∂tψ = − δF{ψ} δψ resulting in a non-increasing dependence of F on t: Significance of conserved quantities: Example: with F ∈ R d δF F{ψ} = dt δψ ∂tψ � �2 δF = − ≤ 0 δψ • Newton’s equation of motion conserves total energy multiply by d dt x i.e. 1 d m 2 dt m d2 d dt2x = F(x) = − V (x) dx m d d2 d d x dt dt2x = − x V (x) � dt dx � d dt x � � 2 = − d V (x) dt � d 1 dt 2 m ˙x2 � + V (x) = 0 78

Notes: Thus: ˙x = 1 2 m ˙x2 + V (x) = E = const. � � 2 dx (E − V (x)) ⇒ t = � m 2 (E − V (x)) m – using energy conservation reduces the order of the differential equation: expresses ˙x as a function of x – solution can be obtained by simple integration (quadrature): the system is called integrable – for two interacting particles x1(t) and x2(t) energy conservation alone leads to a single relation between the two velocities 1 2 m ˙x2 1 + 1 2 m ˙x2 2 + V (x1, x2) = E = const. to express each velocity ˙xi in terms of the positions xi we would need a second equation: a second conserved quantity – in general: for Newton’s equations of motion with N degrees of freedom to be integrable one needs N independent, conserved quantities. • Hamiltonian systems with N degress of freedom are integrable if they have N independent conserved quantities. • the NLS has infinitely many degrees of freedom and infinitely many conserved quantities. It can be shown to be integrable. Exact solutions can be obtained by the inverse scattering transform (well beyond this class). • the existence of completely integrable nonlinear systems like the NLS was found after a lot of effort in the wake of the numerical simulations by Fermi, Pasta, and Ulam of a nonlinear one-dimensional lattice model in which they were trying to identify the approach of such a system to thermal equilibrium. They found, however, that the nonlinear system they investigated did not approach equilibrium, but instead the system repeatedly returned to a state very close to the initial condition. This was quite surprising since it often had been assumed that even a small nonlinearity would generically make systems non-integrable and have them approach states corresponding to thermal equilibrium. For some more details and a historical overview have a look at the overviews given in [8, 2]. 79

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