Hamiltonian systems with several degrees of freedom and which contain a periodic orbit. More generally, we can work with tori of any dimension from zero (i.e. fixed points) through to full dimension. The following section gives a full description of the conditions we require the Hamiltonian system to satisfy.
4.3
Framework & the P1 & P2 conditions
Here, and in sections 4.4 and 4.5, we use a slightly different notation from the rest of this thesis in order that the notation here is consistent with the article [73] which we refer to at several points. We useHto denote the original, unperturbed Hamiltonian function while
H appears as a term in the perturbation, and subsequent modification, ofH.
The periodic orbits (or lower dimensional invariant tori) we work with are assumed to be non-degenerate. That is, they are not contained within a resonant invariant torus of higher dimension. For periodic orbits, this requirement is ensured by the assumption that the Floquet multipliers — the eigenvalues of the monodromy matrix — of the original system are distinct, and, therefore, by the canonical structure, are non-zero (cf. §2.3).
The exact conditions necessary to ensure non-degeneracy of lower dimensional tori, in general, are given explicitly in theorem 4.6.
We will assume that the Hamiltonian systemHwe are working with has the following properties.
1. The initial HamiltonianHhasddegrees of freedom and is autonomous, and analytic with respect to all its variables. It contains an invariant torus with linear, quasi- periodic flow with rationally independent frequencies ˆω(0) ∈ Rr, 0 ≤ r ≤ d(r = 1
corresponds to a periodic orbit).
2. The invariant torus is reducible; that is, the time-dependent linear equations which describe the linearization of the flow on the torus (e.g. x˙ = A(φ+ωt)x) can be transformed into linear, constant coefficient equations ˙y = ˆAy. It is known that reducibility holds automatically for periodic orbits due to Floquet theory (cf. §4.1). For invariant tori with more degrees of freedom there are various positive results concerning when a system is reducible (see, for example, [69, 26, 36, 71, 72, 70, 101, 123]); however, the question of reducibility remains open in general.
3. The initial periodic orbit or invariant torus is isotropic; that is the symplectic form evaluates to zero everywhere on it. Any one-dimensional manifold of a symplectic vector space is isotropic so the property holds automatically for periodic orbits. There is a canonical change of coordinates such that the initial Hamiltonian can be written as a function of the coordinates ˆθ,I, x, yˆ with ˆθ,Iˆ ∈ Cr, x, y ∈
Cm, d = r +m, and
z>= (x>, y>). Here ˆθandxare the position variables, with ˆθbeing the angle coordinates on the periodic orbit or torus. Iˆ and y are their respective conjugate momenta. (We
80 Chapter 4. Preservation of Periodic Orbits
use hats to denote variables pertaining to the initial torus. The x and y coordinates are the “normal” directions to the torus.) Since the original Hamiltonian was assumed to be analytic, it has a Taylor expansion (about z = 0, ˆI = 0). For periodic orbits, Floquet theory ensures that the expansion has constant coefficients for the ˆI and z>z terms and that the only linear term is ˆω(0)Iˆwhere ˆω(0) ∈ R is the frequency of the periodic orbit. (For invariant tori of dimension two, or greater, the frequency is replaced by a frequency vector ˆω(0) ∈Rr and the linear term is ˆω(0)>Iˆ.) More generally, the assumption of linear,
reducible flow on the invariant torus ensures that the initial Hamiltonian can be put into the semi-normal form
H(ˆθ, x,I, yˆ ) = ˆω(0)>Iˆ+1 2z
>B
z+H∗(ˆθ, x,I, yˆ ). (4.2)
This is sometimes referred to as Floquet form. The terms in the Taylor expansion of H∗
begin at second order in ˆI and z. The assumption that the flow on the torus can be reduced to the case of constant coefficients means that H∗ has no quadratic terms in the
z variables — all such terms are included in 12z>Bz. In these variables B is a symmetric 2m×2mcomplex matrix. H∗ is analytic with respect to all its arguments and is periodic
in ˆθ.
We also assume that:
4. The analyticity of H∗ holds in a neighbourhood of z = 0, Iˆ = 0 (the periodic
orbit/torus is assumed to be centered about this point — if it is not, then a change of variables can be used to reduce to this case) and in a complex strip about the variable ˆθ, that is for |Im(ˆθj)| ≤ ρ, j= 1,2, . . . , r, ρ∈R.Also, the matrix JmB
is diagonal with distinct eigenvalues
λ>= (λ1, . . . , λm,−λ1, . . . ,−λm),
where Jm is the canonical symplectic form on C2m (cf. equation (1.13)).
We will also require that the periodic orbit/torus satisfies a strong non-resonance condition and that the normal “frequencies” λj, j = 1, . . . ,2m satisfy a non-degeneracy
condition. We delay giving the details of these conditions until section 4.5 where they arise naturally.
The method is as follows: we begin with the Hamiltonian in the form of equation (4.2). Before considering any perturbation it is helpful to put the initial Hamiltonian into a (semi-)normal form. One does this by expanding H∗, the higher order part of (4.2), as
a power series in ˆI and zabout ˆI = 0, z= 0.We get
H∗ =
X
p≥2
H(0)p ,
4.3. Framework & the P1 & P2 conditions 81
homogeneous polynomials of degreep;
H(0)p = X
l∈N2m, j∈Nr,
|l|1+2|j|1=p
h(0)l,j(ˆθ)zlIˆj.
The periodic coefficientsh(0)l,j(ˆθ) are defined by their Fourier series,
h(0)l,j(ˆθ) = X
k∈Zr
h(0)l,j,kexp(ik>θˆ). (4.3)
It is then possible to use three steps of an iterative KAM-like procedure to rewrite the initial Hamiltonian (4.2). Individual monomials in the expansion ofH∗ can be eliminated
with a convergent change of variables using a generating function. Each step involves a generating function of the form
S(n)(ˆθ, x,I, yˆ ) = X
l∈N2m, j∈
Nr,
|l|1+2|j|1=n
s(l,jn)(ˆθ)zlIˆj, n= 3,4,5, (4.4)
where the periodic coefficientss(l,jn)(ˆθ) are defined by their Fourier coefficients allowing us to give an expansion forS(n) based onH(nn−3);
s(l,j,kn) = h (n−3)
l,j,k
ik>ωˆ(0)+l>λ. (4.5)
The procedure is similar to that used at each step of the usual KAM method (see section 3.3) with a point of difference being that the small divisors which appear in the construc- tion of the generating function take the form ik>ωˆ(0)+l>λ, with k∈Zr\ {0}, l∈ N2m.
To ensure convergence, the Diophantine condition
ik >ωˆ(0)+l>λ ≥ µ0 |k|γ1, |l|1 ≤2 (4.6)
is assumed to hold for k ∈ Zr+s\ {0}, l ∈
N2m, |l|1 ≤ 2. This differs from the usual non-resonance condition of KAM theory in that it includes the effect of the normal fre- quenciesλ. At each stepS(n) is constructed so that the termH∗satisfies the two following
conditions for monomials of degree= 3,4,5:
P1 The coefficients of the monomials (z,Iˆ) (degree 3) and (z,I,ˆ Iˆ) (degree 5) are zero.
P2 The coefficients of the monomials (z, z,Iˆ) and ( ˆI,Iˆ) (both of degree 4) do not depend on ˆθ and the coefficients of (z, z,Iˆ) vanish, except for the trivial resonant terms. The Diophantine condition (4.6) must hold in order that the procedure converges and that
82 Chapter 4. Preservation of Periodic Orbits
the above two conditions can be satisfied. The resulting Hamiltonian has the form
H= ˆω(0)>Iˆ+ 1 2z >Bz+1 2 ˆ I>CIˆ+H∗(ˆθ, x,I, yˆ ) (4.7)
where C is a constant matrix with det(C)6= 0 and where H∗(ˆθ, x,I, yˆ ) satisfies the condi-
tions P1and P2.
These manipulations of the Hamiltonian to reach the form (4.7) are purely formal in the sense that they are necessary to the proof of theorem 4.6 and to precisely state the assumptions of that theorem, but the symplectic integrator, whose behaviour we are ultimately interested in, is applied directly to the original Hamiltonian without first ma- nipulating it into the form (4.2) or (4.7).
In the following section we consider the effect of applying a symplectic integrator to the Hamiltonian system: we try to find a modified Hamiltonian which is an O(hp) perturbation of the original, and whose time-h flow is given by the trajectory of the symplectic integrator.