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Action-angle coordinates are also useful in perturbation theory of Hamiltonian mechanics, especially in determining adiabatic invariants. One of the earliest results from chaos theory , for dynamical stability of integrable dynamical systems under small perturbations, is the KAM theorem , which states that the invariant tori are partially stable.
where is the action-angle coordinate, is a positive integer, and and are Maslov indexes. μ i {\displaystyle \mu _{i}} corresponds to the number of classical turning points in the trajectory of q i {\displaystyle q_{i}} ( Dirichlet boundary condition ), and b i {\displaystyle b_{i}} corresponds to the number of reflections with a hard wall ...
A variable J k in the action-angle coordinates, called the "action" of the generalized coordinate q k, is defined by integrating a single generalized momentum around a closed path in phase space, corresponding to rotating or oscillating motion: [15]: 454 =
There then exist, as mentioned above, special sets of canonical coordinates on the phase space known as action-angle variables, such that the invariant tori are the joint level sets of the action variables. These thus provide a complete set of invariants of the Hamiltonian flow (constants of motion), and the angle variables are the natural ...
The inverse square law behind the Kepler problem is the most important central force law. [1]: 92 The Kepler problem is important in celestial mechanics, since Newtonian gravity obeys an inverse square law.
The Hannay angle is defined in the context of action-angle coordinates.In an initially time-invariant system, an action variable is a constant. After introducing a periodic perturbation (), the action variable becomes an adiabatic invariant, and the Hannay angle for its corresponding angle variable can be calculated according to the path integral that represents an evolution in which the ...
In dynamical systems theory, the Liouville–Arnold theorem states that if, in a Hamiltonian dynamical system with n degrees of freedom, there are also n independent, Poisson commuting first integrals of motion, and the level sets of all first integrals are compact, then there exists a canonical transformation to action-angle coordinates in which the transformed Hamiltonian is dependent only ...
Using canonical perturbation theory and action-angle coordinates, it is straightforward to show [1] that A rotates at a rate of, = {()} = {()}, where T is the orbital period, and the identity L dt = m r 2 dθ was used to convert the time integral into an angular integral (Figure 5).