  We explore the effects of dark matter and dark energy on the dynamical
scaling properties of galaxy clusters. We investigate the cluster Faber-Jackson
(FJ), Kormendy and Fundamental Plane (FP) relations between the mass, radius
and velocity dispersion of cluster size halos in cosmological $N$-body
simulations. The simulations span a wide range of cosmological parameters,
representing open, flat and closed Universes. Independently of the cosmology,
we find that the simulated clusters are close to a perfect virial state and do
indeed define a Fundamental Plane. The fitted parameters of the FJ, Kormendy
and FP relationships do not show any significant dependence on $\Omega_m$
and/or $\Omega_{\Lambda}$. The one outstanding effect is the influence of
$\Omega_{m}$ on the thickness of the Fundamental Plane. Following the time
evolution of our models, we find slight changes of FJ and Kormendy parameters
in high $\Omega_m$ universe, along with a slight decrease of FP fitting
parameters. We also see an initial increase of the FP thickness followed by a
convergence to a nearly constant value. The epoch of convergence is later for
higher values of $\Omega_m$ while the thickness remains constant in the low
$\Omega_m$ $\Lambda$-models. We also find a continuous increase of the FP
thickness in the Standard CDM (SCDM) cosmology. There is no evidence that these
differences are due to the different power spectrum slope at cluster scales.
From the point of view of the FP, there is little difference between clusters
that quietly accreted their mass and those that underwent massive mergers. The
principal effect of strong mergers is to change significantly the ratio of the
half-mass radius $r_{half}$ to the harmonic mean radius $r_h$.
