Action-angle coordinates

In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequencies of oscillatory or rotational motion without solving the equations of motion. Action-angle coordinates are chiefly used when the Hamilton–Jacobi equations are completely separable. (Hence, the Hamiltonian does not depend explicitly on time, i.e., the energy is conserved.) Action-angle variables define an invariant torus, so called because holding the action constant defines the surface of a torus, while the angle variables parameterize the coordinates on the torus. The Bohr–Sommerfeld quantization conditions, used to develop quantum mechanics before the advent of wave mechanics, state that the action must be an integral multiple of Planck's constant; similarly, Einstein's insight into EBK quantization and the difficulty of quantizing non-integrable systems was expressed in terms of the invariant tori of action-angle coordinates. 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 the non-linear perturbations of dynamical systems with a small number of degrees of freedom is the KAM theorem, which states that the invariant tori are stable under small perturbations. The use of action-angle variables was central to the solution of the Toda lattice, and to the definition of Lax pairs, or more generally, the idea of the isospectral evolution of a system. (Wikipedia).

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👉 Learn how to define angle relationships. Knowledge of the relationships between angles can help in determining the value of a given angle. The various angle relationships include: vertical angles, adjacent angles, complementary angles, supplementary angles, linear pairs, etc. Vertical a

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(November 1, 2010) Leonard Susskind discusses the specifics of strings including Feynman diagrams and mapping particles. String theory (with its close relative, M-theory) is the basis for the most ambitious theories of the physical world. It has profoundly influenced our understanding of

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Lecture 8 | String Theory and M-Theory

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Alex Wright - Minicourse - Lecture 3

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