  In the energy-momentum squared gravity (EMSG), the matter energy-momentum
tensor is not conserved due to nonminimal interaction between the usual and
modified matter fields. For this reason, the $N$-body acceleration may host the
EMSG effects that can be probed at the solar scale by the perihelion shift of
the planets and experimental tests of the Strong Equivalence Principle (SEP).
To clarify this point, in this paper, we introduce the $N$-body equations of
motion in the weak-field limit of the EMSG theory. To do so, the post-Newtonian
(PN) hydrodynamic equations, the viral identities, as well as the corresponding
equilibrium conditions are introduced in this theory. Armed with these
relations, we derive the dynamics of the $N$-body system and its PN inter-body
metric. It is shown that the EMSG theory is not ruled out by the classical
test, the perihelion advance of Mercury, and the test of SEP. In other words,
in the first PN order, it is not possible to constrain the free parameter of
this theory and even distinguish it from GR using these local tests.
