The evolution of energy, momentum, and spin parameter in
  self-gravitating collisionless dark matter flow and integral constants on
  large and small scales
  N-body equations of motion in comoving system and expanding background is
recast in a transformed system with static background. The energy and momentum
evolution in dark matter flow is formulated for both systems. In transformed
system, 1) energy equation is identical to that of a damped harmonic oscillator
and consistent with cosmic energy equation; 2) two-body collapse model (TBCM)
predicts an exponential evolution of energy. Combining both leads to a
power-law energy evolution in comoving system and suggests an effective
potential exponent $n_e$=-10/7 for virial theorem (-1.38 from simulation) that
deviates from -1. With $n_e$=-10/7, kinetic/potential energy (K$_p$ and P$_y$)
increase linearly with time t such that K$_p$=$\epsilon_u$t and
P$_y$=-7$\epsilon_u$t/5, where $\epsilon_u$ is a constant rate of energy
production. On halo scale, halo size, energy and mass are related by two
constants $\alpha_s^*$ and $\beta_s^*$, whose mean values are independent of
time and mass. Their dispersion decreases with halo mass due to short lifespan
of large halos, whose effective potential exponent $n_s^*$=-1.3. Simulation
suggests scaling $\propto a^{3/2}$ and $\propto a^{5/2}$ for radial and angular
momentum of entire system, and $\propto a^{3/2}$ for momentum of halos. Halo
momentum can be modeled by two mass-dependent coefficients $\tau_s^*$ and
$\eta_s^*$. Halo spin $\lambda_p$ is determined by $\alpha_s^*$, $\eta_s^*$,
and $n_s^*$ and decreases with halo mass with $\lambda_p$=0.09 and 0.031 for
small and large halos. The radial/angular momentum (G and H) are closely
related to constant $I_m$ that is defined as integral of velocity correlation
or $m$th derivative of energy spectrum at small k. On large scale, H is
negligible, $I_2$=0 reflects conservation of linear momentum, while $I_4$
reflects fluctuation of G. On halo scale, $I_4$ is determined by both G and H
that are comparable.
