angle variable 예문

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In Hamiltonian dynamics, Liouville also introduced the notion of action-angle variables as a description of completely integrable systems.
The Hamiltonian flow is the integral of the Hamiltonian vector field, and so one writes, using traditional notation for action-angle variables:
In fact, the Robertson uncertainty relation is false if \ hat { A } is an angle variable and \ hat { B } is the derivative with respect to this variable.
Integrability means that there is a change of variables ( action-angle variables ) such that the evolution equation in the new variables is equivalent to a linear flow at constant speed.
Electrical length can also be expressed as the angle units . \ theta, the mathematical symbol for an angle variable, is used as the symbol for electrical length when expressed as an angle.
An elegant action-angle variables solution for the Kepler problem can be obtained by eliminating the redundant four-dimensional coordinates \ boldsymbol \ eta in favor of elliptic cylindrical coordinates ( ?, ?, ?)
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.
In some cases, this may even be seen as a transformation to action-angle variables, although typically only a finite number of the " position " variables are actually angle coordinates, and the rest are noncompact.
The angle variable comes back to itself after 1 unit of increase, so the geometry of phase space in coordinates is that of a half-cylinder, capped off at = 0, which is the motionless orbit at the lowest value of the energy.