  N-body equations of motion in comoving system and expanding background are
reformulated in a transformed system with static background and fixed damping.
The energy and momentum evolution in dark matter flow are rigorously formulated
for both systems. The energy evolution in transformed system has a simple form
that is identical to the damped harmonic oscillator. The cosmic energy equation
can be easily derived in both systems. For entire N-body system, 1) combined
with the two-body collapse model (TBCM), kinetic and potential energy increase
linearly with time $t$ such that $K_p=\varepsilon_ut$ and
$P_y=-7\varepsilon_ut/5$, where $\varepsilon_u$ is a constant rate of energy
cascade; 2) an effective gravitational potential exponent $n_e=-10/7\ne-1$
($n_e=-1.38$ from simulation) can be identified due to surface energy of fast
growing halos; 3) the radial momentum $G\propto a^{3/2}$ and angular momentum
$H\propto a^{5/2}$, where $a$ is the scale factor. On halo scale, 1) halo
kinetic and potential energy can be modelled by two dimensionless constants
$\alpha_s^*$ and $\beta_s^*$. Both constants are independent of time and halo
mass; 2) both halo radial and angular momentum $\propto a^{3/2}$ and can be
modeled by two mass-dependent coefficients $\tau_s^*$ and $\eta_s^*$; 3) halo
spin parameter is determined by $\alpha_s^*$ and $\eta_s^*$ and decreases with
halo mass with derived values of 0.09 and 0.031 for small and large halos.
Finally, the radial and angular momentum are closely related to the integral
constants of motion $I_m$, i.e. the integral of velocity correlation or the
$m$th derivative of energy spectrum at long wavelength limit. On large scale,
angular momentum is negligible, $I_2$=0 reflects the conservation of linear
momentum, while $I_4$ reflects the fluctuation of radial momentum $G$. On halo
scale, $I_4$ is determined by both momentum that are comparable with each
other.
