What is the oscillator model in a generalized Snyder scheme？

[Zhiping Lai]

Homework Statement

Graduation thesis topic

Relevant Equations

$$H= \frac{1}{4} \sum_{\mu v}\left(\frac{\hat{\rho}_{\mu v}^{2}}{M}+M \omega^{2} \hat{x}_{\mu v}^{2}\right)+\lambda \hat{x}^{4},$$

What is the oscillator model in a generalized Snyder scheme？How to derive the formula？

 

[Nate810]

The oscillator model in a generalized Snyder scheme is a mathematical model that describes the dynamics of an oscillator, i.e. a system which can oscillate between two states over time. It is usually derived from a system of coupled differential equations, in which the oscillator is driven by a forcing term. The model is most commonly used to describe the behavior of nonlinear systems, such as those found in physical, chemical, and biological processes.The formula for the oscillator model in a generalized Snyder scheme can be derived by starting with the following system of coupled differential equations: \begin{alignat}{3}\frac{dX}{dt} &= A X + F(t) \\\ \frac{dY}{dt} &= B Y + G(t)\end{alignat}Where X and Y are the two states of the oscillator, A and B are the coefficients of the coupling between the two states, and F(t) and G(t) are the forcing terms. Solving the above system of equations yields the following equation for the oscillator model: \begin{equation}X(t) = X_0 e^{At} + \int_0^t e^{A(t-\tau)}F(\tau) d\tau\end{equation}Where X_0 is the initial condition of the oscillator at time t=0.