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Lecture

Periodic Orbits of Hamiltonian Systems

Description

This lecture covers the concept of periodic orbits in Hamiltonian systems, focusing on fixed points and their stability. It discusses the behavior of periodic orbits under different conditions and their relation to symplectic manifolds.

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Related concepts (32)

Periodic table

The periodic table, also known as the periodic table of the elements, arranges the chemical elements into rows ("periods") and columns ("groups"). It is an organizing icon of chemistry and is widely used in physics and other sciences. It is a depiction of the periodic law, which says that when the elements are arranged in order of their atomic numbers an approximate recurrence of their properties is evident. The table is divided into four roughly rectangular areas called blocks.

Periodic point

In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Given a mapping f from a set X into itself, a point x in X is called periodic point if there exists an n so that where f_n is the nth iterate of f. The smallest positive integer n satisfying the above is called the prime period or least period of the point x.

Periodic trends

In chemistry, periodic trends are specific patterns that are present in the Periodic table that illustrate different aspects of certain elements when grouped by period and/or group. They were discovered by the Russian chemist Dmitri Mendeleev in the year 1863. Major periodic trends include atomic radius, ionization energy, electron affinity, electronegativity, valency and metallic character. These trends exist because of the similar electron configurations of the elements within their respective groups or periods; they reflect the periodic nature of the elements.

Block (periodic table)

A block of the periodic table is a set of elements unified by the atomic orbitals their valence electrons or vacancies lie in. The term appears to have been first used by Charles Janet. Each block is named after its characteristic orbital: s-block, p-block, d-block, f-block and g-block. The block names (s, p, d, and f) are derived from the spectroscopic notation for the value of an electron's azimuthal quantum number: sharp (0), principal (1), diffuse (2), or fundamental (3).

Hamiltonian mechanics

Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics.

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Related concepts (32)

Periodic table

Periodic point

Periodic trends

Block (periodic table)

Hamiltonian mechanics