  Recent observations of young embedded clumpy clusters and statistical
identifications of binary star clusters have provided new insights into the
formation process and subsequent dynamical evolution of star clusters. The
early dynamical evolution of clumpy stellar structures provides the conditions
for the origin of binary star clusters. Here, we carry out $N$-body simulations
in order to investigate the formation of binary star clusters in the Milky Way
and in the Large Magellanic Cloud (LMC). We find that binary star clusters can
form from stellar aggregates with a variety of initial conditions. For a given
initial virial ratio, a higher degree of initial substructure results in a
higher fraction of binary star clusters. The number of binary star clusters
decreases over time due to merging or dissolution of the binary system.
Typically, $\sim 45\%$ of the aggregates evolve into binary/multiple clusters
within $t=20$~Myr in the Milky Way environment, while merely $\sim30\%$
survives beyond $t=50$~Myr, with separations $\lesssim 50$~pc. On the other
hand, in the LMC, $\sim 90\%$ of the binary/multiple clusters survive beyond
$t=20$~Myr and the fraction decreases to $\sim 80\%$ at $t=50$~Myr, with
separations $\lesssim 35~$pc. Multiple clusters are also rapidly formed for
highly-substructured and expanding clusters. The additional components tend to
detach and the remaining binary star cluster merges. The merging process can
produce fast rotating star clusters with mostly flat rotation curves that speed
up in the outskirts.
