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I am trying to describe a system with parametric excitation. Usually this is described as an open system where the time dependence of a parameter is explicitly included in this differential equation: $$\ddot{x}+\omega_0^2 x+\Delta\omega(t) x=0.$$ Here $x$ is the position of the harmonic oscillator, $\omega_0$ is the natural frequency of the harmonic oscillator and $\Delta\omega(t)$ is the time dependent perturbation of the frequency. But I want to describe it as a closed system. As an example I want to consider two harmonic oscillators where the position of one harmonic oscillator drives the parametric variation of the other oscillator: It is obvious to me then that the varied parameter depends on some physical coordinate of the rest of the system such that the Hamiltonian for the total system should look something like this:

$$H_{tot}=H_{sys}+H_{exc}+H_{int}$$ $$H_{sys}=\frac{p_0^2}{2m}+\frac{1}{2}k_0q_0^2$$ $$H_{exc}=\frac{p^2}{2m}+\frac{1}{2}kq^2$$ $$H_{int}=\frac{1}{2}k_1qq_0^2.$$

Is this a realistic and physical Hamiltonian for a system? Or am I missing some terms?

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To my knowledge, the Hamiltonian can be attained from integrating the Hamiltonian Density over space, which is calculated by (for a Scalar Field ϕ): $$ H=Π^{0}∂_{0}ϕ-L$$ Where, H is the Hamiltonian Density, L is the Lagrangian Density, and Π is the momentum density. When ϕ is a vector field, the momentum density is simply: $$Π^{0i}$$ Where 'i' is just an index.

I would say this would be the general 'rule' for writing down Hamiltonians.

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