Title: On non-Abelian Lie Bracket of Generalized Covariant Hamilton Systems,
Abstract: This is a theoretical paper, which is a continuation of [arXiv:1710.10597],
it considers the non-abelian Lie algebra $\mathcal{G}$ of Lie groups for
$\left[ {{X}_{i}},{{X}_{j}} \right]=c_{ij}^{k}{{X}_{k}}\in \mathcal{G}$ on the
foundation of the GCHS, where $c_{ij}^{k}\in {{C}^{\infty }}\left( U,R \right)$
are the structure constants. The GPWB [arXiv:1710.10597] is nonlinear bracket
applying to the non-Euclidean space, the second order (2,0) form antisymmetric
curvature tensor ${{F}_{ij}}=c_{ij}^{k}{{D}_{k}}$, and Qsu quantity
${{q}_{i}}=w_{i}^{k}{{D}_{k}}$ are accordingly obtained by using the
non-abelian Lie bracket. The GCHS $\left\{ H,f \right\}\in {{C}^{\infty
}}\left( M,\mathbb{R} \right)$ holds for the non-symplectic vector field
$X_{H}^{M}\in \mathcal{G}$ and $f\in {{C}^{\infty }}\left( M,\mathbb{R}
\right)$ that implies the covariant evolution equation consists of two parts,
NGHS and W dynamics along with the second order invariant operator
$\frac{{\mathcal{D}^{2}}}{d{{t}^{2}}}=\frac{{{d}^{2}}}{d{{t}^{2}}}+2w\frac{d}{dt}+\beta$.