Adiabatic invariant ($I$)

A quantity conserved to leading order when a parameter varies slowly compared with the natural motion period. The action variable of a regular periodic Hamiltonian orbit is a standard example, as explained by adiabatic invariance of the action.

For slow variation of the field compared with the gyration period, the magnetic moment $m{v}_{\perp }^2/(2B)$ is approximately conserved. Thus transverse random energy scales with $B$, and remains unchanged only when the relevant field strength is unchanged.

The magnetic-moment invariant of rapid cyclotron motion in slowly varying fields. Here the per-unit-mass convention is used: the physical magnetic moment is $m\mu$. In drift kinetics, using $(\mu ,v_{\parallel })$ as coordinates removes the leading perpendicular acceleration and leaves a velocity-space measure proportional to the exact magnetic field strength.

In a slowly changing spherical gravitational potential, angular momentum is exactly conserved and the radial action is an adiabatic invariant. An initially circular orbit remains circular in the adiabatic limit, so $J_r=0$ and $L^2=GrM(r)$ give the displayed shell relation. If dark-matter shells do not cross, their enclosed dark mass is also conserved. A density profile alone does not specify this orbital model: eccentric-orbit populations require their full action distribution.

An inner NFW profile has $M_i(r_i)=A{r}_i^2$, where $A=2\pi {\rho }_s{r}_s$. If the final total circular speed is $V$, circular-shell adiabatic contraction gives $r_i=(V^2/(GA){)}^{1/3}{r}^{2/3}$ and $M_D^{\prime }=A^{1/3}(V^2/G{)}^{2/3}{r}^{4/3}$. Thus ${\rho }_D^{\prime }=A^{1/3}(V^2/G{)}^{2/3}{r}^{-5/3}/(3\pi )$. The total density producing the flat curve is $V^2/(4\pi G{r}^2)$, so the dark fraction tends to zero as $r^{1/3}$ despite its increase relative to the original cusp.

An elastic collision with a wall moving toward the particle at speed $V$ changes the particle speed by approximately $2V$ when recoil is negligible. The collision rate is approximately $v/\ell$ for $v\gg V$, and $\dot{\ell }=-2V$, yielding $\dot{v}/v=-\dot{\ell }/\ell$. For a freely moving finite wall of mass $M$, the neglected recoil changes the increment by order $(m/M)v$, so relative accuracy also requires $(m/M)(v/V)\ll 1$.

// Target: classical-mechanics.bigb