By Sylvio Ferraz-Mello
The ebook is written in most cases to complex graduate and post-graduate scholars following classes in Perturbation thought and Celestial Mechanics. it's also meant to function a consultant in study paintings and is written in a truly specific approach: all perturbation theories are given with information permitting its instant program to genuine difficulties. furthermore, they're via examples displaying all steps in their program. The ebook isn't really meant to discover the math of Hamiltonian structures, yet could be helpful to mathematicians in loads of conditions as a reference at the useful software of the theories. within the related manner, it can be a resource publication at the difficulties of degeneracy and small divisors, which impact using perturbation theories to boot in Celestial Mechanics as in Physics.
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Extra info for Canonical Perturbation Theories: Degenerate Systems and Resonance
72) ( = 2, · · · , N ). Φ Fig. 2. cls date/time: 20-Oct-2006/9:21 20 1 The Hamilton–Jacobi Theory Hamilton–Jacobi mappings, when they exist, are ﬁnite transformations generally including singularities due to the topological diﬀerences between the surfaces H(q, p) = E and the planes p∗1 = E. As a consequence, they are more general than the Lie mappings that will be introduced in Chap. 5. All Lie mappings may be written as Hamilton–Jacobi mappings, but the converse is not true. Lie mappings are inﬁnitesimal (near-identity) homeomorphisms of the phase space into itself.
84) k=1 amongst the frequencies of the system. Degeneracy may be essential or accidental. A degeneracy is said to be essential when it does not depend on the initial conditions. We shall stress that this does not mean that the frequencies themselves are independent of the initial conditions. The Keplerian motion is a good example: the frequencies νr , νθ , νφ (deﬁned by the derivatives of E with respect to Jr , Jθ , Jφ ) depend on the initial conditions but they are always equal, regardless of the initial conditions.
28) j=1 (by construction, δi qj∗ = γji ). Therefore, in a partial cyclic variation of qi , wi increases of 2π while the others wk (k = i) remain unchanged. 2 The Actions The next step is to ﬁnd the action variables Jk canonically conjugate to the angle variables wk . To do this, we introduce the Jacobian generating function ˜ J). 4 Separable Multiperiodic Systems 39 and N N dwk = j=1 ∂ 2 S˜ ∂ 2 S˜ dqj + dJj . 30) In a partial cyclic variation of qk , wk increases 2π. Besides, along the given path, dqj = 0 for j = k and dJi = 0.