I'm currently playing with a toy model given by the Lagrangian $$L=\frac{m\dot{x}^2}{2}+\frac{m\dot{y}^2}{2}+\frac{1}{2}m\omega^2x^2+x y,$$ which is basically a free particle (described by $y(t)$) and a harmonic oscillator (described by $x(t)$) coupled in a simple way.
My question is if you guys know any system that could correspond to such a Lagrangian? Basically this is a free particle that is forced to oscillate by a harmonic oscillator that pulls it...
First of all I guess that what you wrote is the Hamiltonian and not the Lagrangian of the system and $\dot{x}$ stays for $p_x$ and $\dot{y}$ stays for $p_y$.
You can decouple the problem redefining $$(X,Y)^t = R(x,y)^t$$ for a suitable $R\in O(2)$ diagonalizing the symmetric matrix in the potential part of your Hamiltonian. This way you see the final Hamiltonian is $$\left(\frac{p^2_X}{2m} + \lambda_+ X^2\right) + \left(\frac{p^2_Y}{2m} + \lambda_- Y^2\right)$$ where $\lambda_\pm$ are the eigenvalues of the above symmetric matrix.
In the considered case (up to the dimensional problem already stressed) you find that $ \lambda_+ \lambda_-<0$ (because the determinant of the symmetric matrix is negative nomatter the sign in front of $xy$).
So you have a pair of 1D non-mutually interacting particles, one subjected to a standard harmonic potential and the other subjected to a repulsive harmonic potential (like the one of centrifugal force for a constant angular speed).
For example, if the first particle is moving on a spring and its position sets the electric potential that controls the second, electrically charged, particle. This way you'd have the potential energy of the coupling in the form of the product of coordinates.
A similar Lagrangian pops up in the Caldeira-Legget model to describe quantum dissipation. This model is given by a particle with mass M connected to a bath of harmonic oscillators with mass m_k and frequency w_k. The only difference is that the CL-model takes multiple oscillators into account.
