Central-force trajectories: the central-force equation, angular momentum, energy and the effective potential

Let $\mu >0$ be a real number, let $J_0\subseteq (0,\mathrm{\infty })$ be an open interval, let $U:J_0\to \mathbb{R}$ be of class $C^2$ ($C^k$ Euclidean maps and diffeomorphisms), let $I\subseteq \mathbb{R}$ be an interval with at least two points (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), and let $r:I\to {\mathbb{R}}^3$ be twice differentiable with $\mathrm{\mid }r(t)\mathrm{\mid }\in J_0$ for every $t\in I$ (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral). The curve $r$ is a central-force trajectory for $(\mu ,U)$ when

$\mu r^{\mathrm{\prime }\mathrm{\prime }}(t)=-\frac{U^{\mathrm{\prime }}(\mathrm{\mid }r(t)\mathrm{\mid })}{\mathrm{\mid }r(t)\mathrm{\mid }}r(t)(t\in I),$

where $\mathrm{\mid }x\mathrm{\mid }:=\parallel x{\parallel }_2$ is the Euclidean norm (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$) and $U^{\mathrm{\prime }}$ is the derivative of $U$ on the open interval $J_0$.

Attached objects. The angular momentum vector of such a trajectory is the curve $j:I\to {\mathbb{R}}^3$,

$j(t):=\mu r(t)\times r^{\mathrm{\prime }}(t),$

with $\times$ the cross product of ${\mathbb{R}}^3$ (The cross product in ${\mathbb{R}}^3$); its energy is the function $E:I\to \mathbb{R}$,

$E(t):=\frac{1}{2}\mu \parallel r^{\mathrm{\prime }}(t){\parallel }_2^2+U(\mathrm{\mid }r(t)\mathrm{\mid });$

and for every $J\ge 0$ the effective potential of momentum magnitude $J$ is the function $U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}:J_0\to \mathbb{R}$,

$U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}(\rho ):=U(\rho )+\frac{J^2}{2\mu {\rho }^2}\mathrm{.}$

Why the objects are defined. Since $\rho >0$ throughout $J_0$, the right-hand side of the central-force equation is defined, the map $\rho \mapsto J^2\mathrm{/}(2\mu {\rho }^2)$ is $C^2$ on $(0,\mathrm{\infty })$ (a quotient of $C^2$ functions with nowhere vanishing denominator), and $U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}$ is again $C^2$; in particular second derivatives of $U_J^{\mathrm{e}\mathrm{f}\mathrm{f}}$ are available. The vector $j(t)$ is defined for every $t\in I$ because $r$ is differentiable there.

Mathematical status. The data $\mu$, $U$ and $J$ are real numbers and functions; the equation and the attached objects are mathematical statements about curves. Nothing here asserts that the equation describes a physical interaction, and the names are conventional labels for the mathematical objects only.