  Scale invariance in the theory of classical mechanics can be induced from the
scale invariance of background fields. In this paper we consider the relation
between the scale invariance and the constants of particle motion in a
self-similar spacetime, only in which the symmetry is well-defined and is
generated by a homothetic vector. Relaxing the usual conservation condition by
the Hamiltonian constraint in a particle system, we obtain a conservation law
holding only on the constraint surface in the phase space. By the conservation
law, we characterize constants of motion associated with the scale invariance
not only for massless particles but for massive particles and classify the
condition for the existence of the constants of motion. Furthermore, we find
the explicit form of the constants of motion by solving the conservation
equations.
