The generalized covariant Hamilton system in complex coordinates

Abstract

Imitating methods of working on the GCHS and GSPB defined on ${{\mathbb{R}}^{r}}$ in real coordinates, an attempt to follow this way in complex coordinates is considered, then we try to generalize the PB on ${{\mathbb{C}}^{n}}$ in complex coordinates to the GSPB with zero restriction of the non-degeneracy expressed in complex coordinates that is compatible with the real case in formulas. Thusly, then the GCHS in complex coordinates defined by the GSPB is self-consistent to the real situation. Meanwhile, we find some difference of the GCHS between the real and complex case in formula. Much of what distinguishes a GCHS in real coordinates from a GCHS in complex coordinates is that the TGHS in complex form has an extra expression.