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Let an incentive impulse be equal to zero: P = 0. In this case the system of equations, describing the evolution of variables M , γ, becomes separated, and the Hamiltonian of such a reduced system will explicitly depend on time H ∗ = 1 (M , AM ) + 1 µ2 t2 (γ, Cγ), 2 2 7. Rigid Body Motion in a Perfect Incompressible Fluid (Kirchhoff’s Equations) H = 1 (M , AM ) + (M , Bp) + 1 (p, Cp) + U (α, β, γ, x). 2 2 8. 22) where, as it’s clear from the explanation above, A is a tensor of associated moments of inertia, and C is a tensor of associated masses (see also [95]).

Cn ) these values form bifurcation surfaces, whose explicit form has been studied for the most of known integrable systems [25] (see ch. 2). In this case the Hamiltonian system is called integrable according to Liouville (or completely integrable). One can show that for such a system in the vicinity of each torus there exist variables, called “action-angle” (I, ϕ mod 2π) = (I1 , . . , In , ϕ1 mod 2π, . . , ϕn mod 2π), where Hamiltonian H(I) doesn’t depend on angular variables ϕ mod 2π, and the equations of motion have the form From the theoretic perspective the integrability of the Hamiltonian system in quadratures may not necessarily be connected with the presence of necessary quantity of the first integrals.

Complex methods, based on the study of the Laurant full-parametric expansions, also seem to be effective [243]. Like the Lax spectrum representation, they are capable of representing a spectrum curve in a hyperelliptic case; here it’s possible to restore separating transformations uniquely and obtain the Abel – Jacobi equations (M. Adler, P. van Moerbeke [186, 188], P. Vanaecke [279]). However, such an approach didn’t help to integrate any new system. 83 § 1. 4) where A = I−1 . Remark 1. 1) were known even to Euler (1758), who also found the simplest case of integrability when a rigid body coasts ( = = 0).

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Beam Dynamics Issues for 1 TeV Linear Collider


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