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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.
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 =
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 point P has coordinates (x, y) with respect to the original system and coordinates (x′, y′) with respect to the new system. [1] In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle . A rotation of axes in more than two dimensions is defined similarly.
the azimuthal angle φ, which is the angle of rotation of the radial line around the polar axis. [b] (See graphic regarding the "physics convention".) Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates.
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 ...
Geodetic latitude and geocentric latitude have different definitions. Geodetic latitude is defined as the angle between the equatorial plane and the surface normal at a point on the ellipsoid, whereas geocentric latitude is defined as the angle between the equatorial plane and a radial line connecting the centre of the ellipsoid to a point on the surface (see figure).
In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force that varies in strength as the inverse square of the distance between them.