Hamiltonian and reversible systems with smooth families of invariant tori
Section snippets
Kronecker tori
Finite-dimensional invariant tori carrying conditionally periodic motions are among the key elements of the structure of smooth dynamical systems with continuous time. The importance and ubiquity of such tori stems, in the long run, from the fact that any finite-dimensional connected compact Abelian Lie group is a torus [1], [17], [42]. By definition, given a certain flow on a certain manifold, an invariant -torus carrying conditionally periodic motions ( being a non-negative integer) is an
Preliminaries
Given non-negative integers and , we designate the identity matrix as and the zero matrix as . In fact, the symbols , , and correspond to no actual objects and will only be used for unifying the notation.
Let , , and be non-negative integers and consider a skew-symmetric matrix of the form where is an matrix of rank and is a skew-symmetric matrix (the superscript “t” denotes transposing). If then
The system
Now let , , be arbitrary non-negative real constants and let be an arbitrary smooth function. Consider the Hamilton function on the symplectic manifold (5). The term is automatically absent for . According to (6), the equations of motion afforded by take the form where , , , and is the Kronecker
Compact phase spaces
Like in the setting of our note [50], the general construction of Sections 2 Preliminaries, 3 The main construction admits an analogue with a compact phase space. Consider the symplectic manifold with the same structure matrix (4). Of course, now the corresponding symplectic form is always non-exact. The -tori (7) with , , are again isotropic for , are coisotropic for , and are atropic for . For any angular variable introduce the notation
Reversible analogues
Both the Hamiltonian systems (9), (11) are reversible with respect to the phase space involution of type , so that , , and , where . However, , so that not all the -tori (7) are invariant under . In the case of the system (9), a torus is invariant under if and only if . Consequently, the statement “each torus is symmetric” is valid if
Compactified reversible analogues
The system (14) can be compactified in the same way as the system (9), cf. [50]. Consider the manifold (3) equipped with the involution given by the same formula (2) and having the same type . For any and , consider the -torus given by the same expression (13). Since , a torus is invariant under if and only if , i.e., if each component of is equal to either or .
Now let again , be arbitrary non-negative real
Acknowledgment
I am grateful to B. Fayad for fruitful correspondence and sending me the breakthrough preprint [21] prior to submission to arXiv.
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