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}.
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