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Isotropic stellar mass loss changes the central gravitational parameter $\mu (t)$ while preserving the central direction of the force, provided the escaping material causes no recoil or drag on the orbiting body. The relative equation is $\ddot{\mathbf{r}}=-\mu (t)\mathbf{r}/r^3$. Specific angular momentum remains exactly constant; specific orbital energy obeys $\dot{C}=-\dot{\mu }/r$. Orbital expansion is adiabatic only when the mass-loss law varies slowly on the orbital period.

The kinematic jerk is the time derivative of acceleration. For a Kepler orbit with relative position $\mathbf{r}$, relative velocity $\mathbf{v}$, $r=|\mathbf{r}|$, $\dot{r}=\mathbf{r}\cdot \mathbf{v}/r$ and total mass $M$, differentiation of Newton's law of universal gravitation gives $\mathbf{j}=-GM(\mathbf{v}-3\dot{r}\mathbf{r}/r)/r^3$.

In the flavor convention $K_1=(K^0-{\bar{K}}^0)/\sqrt{2}$ and $K_2=(K^0+{\bar{K}}^0)/\sqrt{2}$, the parameter $ϵ=(a-b)/(a+b)$ writes normalized mass states as $(K_1+ϵK_2)/\sqrt{1+|ϵ{|}^2}$ and $(K_2+ϵK_1)/\sqrt{1+|ϵ{|}^2}$. In the Hermitian mass-only approximation with correlated roots, $ϵ=i\tan (\phi /2)$ and the states are orthogonal. For the full non-Hermitian decay matrix the real part need not vanish. The parameter depends on the flavor-state phase convention; observable CP violation involves convention-invariant combinations with decay amplitudes.

In a Hermitian neutral-kaon mass matrix, independently choosing square-root signs of $z$ and $z^{\ast }$ can interchange the eigenvalue labels. Choose $z=r{e}^{i\phi }$, $a=\sqrt{r}{e}^{i\phi /2}$ and $b=\sqrt{r}{e}^{-i\phi /2}$, so $ab=r>0$, continuously from the desired CP-conserving limit. Then $(a,-b)$ is the lower-mass eigenvector. At a degenerate off-diagonal zero the mass eigenbasis is not fixed by diagonalization alone.

With $a=(1-\lambda )GM$, $h=\lambda E>0$ and $\bar{E}>0$, set $\bar{E}{q}^2=h{r}^2+ar$. If $a\ge 0$, its positive-radius root is $r=\sqrt{\bar{E}/h}(\sqrt{q^2+b^2}-b)$, with $b=a/(2\sqrt{h\bar{E}})$. The transformed Newtonian gravitational potential is an additive constant plus $-G\bar{M}/(b+\sqrt{q^2+b^2})$, the spherical isochrone model. The mass sign and branch conditions must be checked.

For unit mass in $\Phi =-GM/r$, conservation of angular momentum splits the velocity norm into ${\dot{r}}^2+L^2/r^2$. Conservation of energy then gives $\frac{1}{2}{\dot{r}}^2+L^2/(2r^2)=E+GM/r$. It separates radial dynamics from the reconstruction of the orbital angle.

For fixed particle masses and positive total mass, write ${\mathbf{x}}_i=\mathbf{R}+{\mathbf{y}}_i$ with $\mathbf{R}$ the centre of mass. The weighted relative velocities sum to zero, so expanding the kinetic energy eliminates the mixed term and yields the displayed decomposition. For a rigid body, the relative contribution is the rotational kinetic energy $I{\omega }^2/2$ about the chosen centre-of-mass axis.

For the signed inverse-square force $\mathbf{F}=-\kappa \mathbf{r}/r^3$ on a particle of mass $m$, with Hamiltonian $H=p^2/(2m)-\kappa /r$, the Laplace-Runge-Lenz vector $\mathbf{A}=\mathbf{p}\times \mathbf{L}-m\kappa \hat{\mathbf{r}}$ is conserved. Attraction has $\kappa >0$ and repulsion has $\kappa <0$. It obeys $\mathbf{A}\cdot \mathbf{L}=0$ and $A^2=m^2{\kappa }^2+2mH{L}^2$. For unit mass in a Kepler orbit, $\kappa =GM$.

A mass eigenstate diagonalizes the Hermitian mass operator or mass matrix and has a definite mass. In a flavor basis with off-diagonal mass terms, it is a linear combination of flavor states. For unstable particles, decay propagation instead involves an effective complex matrix; its eigenvectors need not inherit the orthogonality of a Hermitian mass problem.

Mass flux is the mass crossing a unit area per unit time. In a one-dimensional flow it is $\rho u$.

The mass shell is the locus of four-momenta satisfying the relativistic dispersion relation for a fixed mass $m$. With mostly-minus Minkowski metric, $p^2=m^2$; with mostly-plus Minkowski metric, $p^2=-m^2$. The positive-energy sheet represents particle four-momenta. For $m=0$ this is the light cone. The condition of being on shell refers more generally to satisfying the field equations.

Let $R$ be the actual largest pairwise separation of positive masses in their centre of mass frame. The pairwise moment-of-inertia identity gives $m_{(1)}{m}_{(2)}{R}^2/(2M)\le I\le R^2{\sum }_{j<k}{m}_j{m}_k/(2M)$. Consequently the scalar inertia is comparable to $R^2$, independently of configuration.

The mass-normalized first velocity moment of a distribution on phase space. For a velocity-only distribution, omit the position integral. Nonzero finite mass and a finite first moment are required. For a probability distribution this is its expected value; its unnormalized counterpart is the momentum.