One of the simplest mechanical systems having an SU(3) dynamical symmetry is the three dimensional isotropic harmonic oscillator. Its phase space is $\mathbb{R}^6$ parametrized by 3 position coordinates $x_i$ and 3 momenta $p_i$, $i=1,2,3$.
A fixed energy hypersurface in phase space is given by the constraint (in units where the natural frequency is 1):
$$\sum_i x_i^2 + \sum_i p_i^2 = E $$
This a the $5$-dimensional sphere $S^5$, and the set of classical trajectories (the reduced phase space corresponding to the Hamiltonian action) on this hypersurface is the complex projective space $CP^2$ , obtained from $S^5$ by a Hopf fibration and from the original phase space by Marsden-Weinstein symplectic reduction with respect to the action of the total energy function:
$$CP^2 = S^5/U(1) = \mathbb{C}^3//U(1)$$
Being a $SU(3)$ homogeneous space $CP^2$ has a natural $SU(3)$action:
The functions representing $SU(3)$ in the $CP^2$ Poisson algebra have simple geometric expressions:

- First we have the three angular momentum components:
$$L_{ij} = x_ip_j-x_jp_i$$
- In addition we have 5 quadrupole moments built from the (traceless)symmetric combinations of the coordinates and momenta:
$$Q_{ij} = \frac{3}{2} (x_ip_j+x_jp_i) - \delta_{ij} \sum_k x_kp_k$$
Together they generate the $SU(3)$ Lie algebra. The quadrupole moment space is closed under the $SU(2)$ action generated by the angular momenta. They transform according to the spin-2 representation.

Physically, the quadrupole moments interact with gradients of an external magnetic field. If the harmonic oscillator is charged, then its interaction energy with an external magnetic field has the form:
$$E_{int} = \sum_{ijk} \epsilon_{ijk}L_{ij}B_k+ \sum_{ij} Q_{ij} \partial_{i}B_j$$
Please, see the following Melih Sener lecture note by for a detailed exposition.