ISSN:
1089-7658
Source:
AIP Digital Archive
Topics:
Mathematics
,
Physics
Notes:
We show that the dynamical system characterized by the (complex) equations of motion q¨j+iΩq(overdot)j=∑k=1,k≠jnq(overdot)jq(overdot)kf(qj−qk), j=1,...,n, with f(x)=−λ℘′(λx)/[℘(λx)−℘(λμ)], is Hamiltonian and integrable, and we conjecture that all its solutions qj(t), j=1,...,n are completely periodic, with a period that is a finite integral multiple of T=2π/Ω. Here n is an arbitrary positive integer, Ω is an arbitrary (nonvanishing) real constant, ℘(y)≡℘(y|ω,ω′) is the Weierstrass function (with arbitrary semiperiods ω,ω′), and λ,μ are two arbitrary constants; special cases are f(x)=2λ coth(λx)/[1+r2 sinh2(λx)], f(x)=2λ coth(λx), f(x)=2λ/sinh(λx), f(x)=2/[x(1+λ2x2)], and of course f(x)=2/x. These findings, as well as the conjecture (which is shown to be true in some of these special cases), are based on the possibility to recast these equations of motion in the modified Lax form L[underaccent underbar-underbar [below](overdot)+iΩL[underaccent underbar-underbar [below]=[L[underaccent underbar-underbar [below],M[underaccent underbar-underbar [below]] with L[underaccent underbar-underbar [below] and M[underaccent underbar-underbar [below] appropriate (n×n)-matrix functions of the n dynamical variables qj and of their time-derivatives q(overdot)j. © 1997 American Institute of Physics.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1063/1.532182
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