Toda Oscillator - Energy

Energy

Rigorously, the oscillation is periodic only at . Indeed, in the realization of the Toda oscillator as a self-pulsing laser, these parameters may have values of order of ; during several pulses, the amplitude of pulsation does not change much. In this case, we can speak about period of pulsation, function is almost periodic.

In the case, the energy of oscillator does not depend on, and can be treated as constant of motion. Then, during one period of pulsation, the relation between and can be expressed analytically:

z=\pm\int_x^{x_\max}\!\!\frac{{\rm d}a} {\sqrt{2}\sqrt{E-\Phi(a)}}

where and are minimal and maximal values of ; this solution is written for the case when .

however, other solutions may be obtained using the translational invariance.

The ratio is a convenient parameter to characterize the amplitude of pulsation, then, the median value \delta=\frac{x_\max -x_\min}{1} can be expressed as \delta= \ln\frac{\sin(\gamma)}{\gamma} ; and the energy E=E(\gamma)=\frac{\gamma}{\tanh(\gamma)}+\ln\frac{\sinh \gamma}{\gamma}-1 also is an elementary function of . For the case, an example of pulsation of the Toda oscillator is shown in Fig. 1.

In application, the quantity has no need to be physical energy of the system; in these cases, this dimensionless quantity may be called quasienergy.

Read more about this topic:  Toda Oscillator

Famous quotes containing the word energy:

    “But often the presence of mind and energy of a person remote from the spotlight decide the course of history for centuries to come.”
    —Stefan Zweig (18811942)

    “The welfare, the happiness, the energy and spirit of the men and women who do the daily work ... is the underlying necessity of all prosperity.... There can be nothing wholesome unless their life is wholesome; there can be no contentment unless they are contented.”
    —Woodrow Wilson (1856–1924)

    “In the west, Apollo and Dionysus strive for victory. Apollo makes the boundary lines that are civilization but that lead to convention, constraint, oppression. Dionysus is energy unbound, mad, callous, destructive, wasteful. Apollo is law, history, tradition, the dignity and safety of custom and form. Dionysus is the new, exhilarating but rude, sweeping all away to begin again. Apollo is a tyrant, Dionysus is a vandal.”
    —Camille Paglia (b. 1947)