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Floquet theory shows stability in Hill differential equation (introduced by George William Hill) approximating the motion of the moon as a harmonic oscillator in a periodic gravitational field. Bond softening and bond hardening in intense laser fields can be described in terms of solutions obtained from the Floquet theorem.
The periodic solution is one of the (,) and (,), called a Mathieu function of the first kind of integral order. The nonperiodic one is denoted either fe n ( x , q ) {\displaystyle {\text{fe}}_{n}(x,q)} and ge n ( x , q ) {\displaystyle {\text{ge}}_{n}(x,q)} , respectively, and is called a Mathieu function of the second kind (of integral order).
A periodic motion is a closed curve in phase space. That is, for some period, ′ = (,), = (). The textbook example of a periodic motion is the undamped pendulum.. If the phase space is periodic in one or more coordinates, say () = (+), with a vector [clarification needed], then there is a second kind of periodic motions defined by
Hill's equation is an important example in the understanding of periodic differential equations. Depending on the exact shape of f ( t ) {\displaystyle f(t)} , solutions may stay bounded for all time, or the amplitude of the oscillations in solutions may grow exponentially. [ 3 ]
The theorem partly resolves the small-divisor problem that arises in the perturbation theory of classical mechanics. The problem is whether or not a small perturbation of a conservative dynamical system results in a lasting quasiperiodic orbit. The original breakthrough to this problem was given by Andrey Kolmogorov in 1954. [1]
In particular, the motion of the body in free space (obtained by integrating ()) is exactly the same, just completed faster by a ratio of . Consequently, we can analyze the geometry of motion with a fixed value of L 2 {\displaystyle L^{2}} , and vary ω ( 0 ) {\displaystyle \omega (0)} on the fixed ellipsoid of constant squared angular momentum.
Apparent retrograde motion is the periodic, apparently backwards motion of planetary bodies when viewed from the Earth (an accelerated reference frame). Satellite is an object that orbits another object (known as its primary). The term is often used to describe an artificial satellite (as opposed to natural satellites, or moons).
Periodic motion. Add languages. Add links. ... Print/export Download as PDF ... In other projects Appearance. move to sidebar hide. From Wikipedia, the free ...
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