# Eikonal Blog

## 2010.07.20

### Conservative motion of a 3D particle in potential-like force fields

Filed under: Uncategorized — Tags: , — sandokan65 @ 09:20

Let’s consider the right-hand oriented orthonormal triad ${\cal B} :\equiv \{\underline{n},\underline{m},\underline{k}\}$ ($\underline{n}\cdot\underline{m}= \underline{n}\cdot\underline{k} = \underline{m}\cdot\underline{k}= 0$ and $\underline{n}^2 = \underline{m}^2 = \underline{k}^2 = 1$) assigned to a curve $c$ in 3-dimensional Euclidean space ${\Bbb R}^3$. Let this curve be parametrized by the “time” parameter $t$ associated to the “current” position $\underline{r}(t)$ of a particle moving along that curve. Instead of defining the triad ${\cal T}$ as a tangential vectors (i.e. the tangential, normal and bi-normal vectors) to the $c$ at $\underline{r}$, define it based on the radius vector $\underline{r}$ by $\underline{r} \equiv: \underline{n} r$.

The triad vectors satisfy following evolution equations:

• $\underline{n}' \equiv: \varphi' \underline{m}$,
• $\underline{m}' \equiv: - \varphi' \underline{n} + \theta' \underline{k}$,
• $\underline{k}' = -\theta' \underline{m}$,

where $\phi(t)$ and $\theta(t)$ are two angles defining the position of the particle.

In following we will look at the several choices for the force field $\underline{F}$ that guides the motion of the particle via Newton’s equation $\underline{r}'' = \underline{F}(\underline{r}, \underline{r}')$.

## The simplest case

For $\underline{r}'' = f(r) \underline{r}$ one finds:

• $r'' = f(r) r + \frac{l^2}{r^3}$ (radial equation of motion),
• $r^2\phi' = l (= const|_{t})$ (preservation of angular momentum),
• $\theta' \equiv 0$ (i.e. the trajectory is planar).

## A simple generalization

In force field $\underline{F} = f(r) \underline{r} + g(r) \underline{r}'$ one finds:

• $r'' = f(r) r + g(r) r' + r \varphi'^2$ (radial equation of motion),
• $l(t) :\equiv r^2\phi'$ (definition of non-constant angular momentum),
• $l' = g(r) l$ (equation guiding the evolution of the angular momentum),
• $\theta' \equiv 0$, i.e. the trajectory is (still) planar.

This system can be somehow simplified by use of $\varphi$ instead of $t$, and by introduction of the Binet variable $u(\varphi) :\equiv \frac1{r(t)}$:

• $- \frac{d^2 u}{d\varphi^2} = u + \frac{f[u]}{l^2 u^3}$
• $\frac{d l}{d\varphi} = \frac{g[u]}{u^2}$.

with $f[u] :\equiv f(r)$, $g[u] :\equiv g(r)$, $\frac{d}{dt} = l u^2 \frac{d}{d\varphi}$, $r'(t) = - l \frac{d u}{d\varphi}$ and $r''(t) = - l u^2 \frac{d}{d\varphi}\left(l \frac{d u}{d\varphi}\right)$.

## Further generalization

For $\underline{r}'' = f(r) \underline{r} + g(r) \underline{r}' + h(r) \underline{r}\times\underline{r}'$ we get:

• $r'' = f r + g r' + r \varphi'^2$,
• $l :\equiv r^2 \varphi'$,
• $l' = g(r) l$,
• $\theta' = h(r) r$ – the trajectory is not planar any more.

which, in Binet coordinates has following form

• $- \frac{d^2 u}{d\varphi^2} = u + \frac{f[u]}{l^2 u^3}$
• $\frac{d l}{d\varphi} = \frac{g[u]}{u^2}$.
• $\frac{d \theta}{d\varphi} = \frac{h[u]}{l u^3}$.