Eikonal Blog


Vector bases for 3D kinematics

Filed under: mathematics — Tags: — sandokan65 @ 16:02

Relevant sets of basis vectors

There are four relevant sets of orthonormal basis vectors (=triads) here:

  • {\cal B}_1 :\equiv \{\underline{e}_x,\underline{e}_y,\underline{e}_x\} – orts corresponding to the Cartesian coordinates \{x,y,z\},
  • {\cal B}_3 :\equiv \{\underline{n},\underline{p},\underline{q}\}, defined by:
    • \underline{n} :\equiv \frac1{r}\underline{r} = (\cos(\theta)\cos(\varphi),\cos(\theta)\sin(\varphi),\sin(\theta)) ,
    • \underline{p} :\equiv \frac{\partial}{\partial \theta} \underline{n} = (-\sin(\theta)\cos(\varphi),-\sin(\theta)\sin(\phi),\cos(\theta)),
    • \underline{q} :\equiv \frac1{\cos(\theta)}\frac{\partial}{\partial \varphi} \underline{n} = (-\sin(\varphi),\cos(\phi),0),

    Note that here: \underline{n}\times\underline{p} = - \underline{q}, ie. that this basis is left-handed.

  • {\cal B}_3 :\equiv \{\underline{n},\underline{m},\underline{k}\} – radius-based basis, defined by:
    • \underline{n} :\equiv \frac1{r}\underline{r},
    • \underline{m} :\equiv \hbox{Normalized}(\underline{n}'),
    • \underline{k} :\equiv \underline{n}\times\underline{m}.
  • {\cal B}_4 :\equiv \{\underline{T},\underline{O},\underline{B}\} – tangent-based basis (“frame”), defined by:
    • \underline{T} :\equiv \hbox{Normalized}(\underline{r}') = \frac1{v}\underline{v} = \frac{r'}{v} \underline{n} + \frac{r \Phi'}{v} \underline{m},
    • \underline{N} :\equiv \hbox{Normalized}(\underline{T}')
    • \underline{B} :\equiv \underline{T}\times\underline{N}.

    where \underline{v} :\equiv \underline{r}' = r' \underline{n} + r \Phi' \underline{m} and v(t) = \sqrt{r'^2+r^2\Phi'^2}.
    Note that \underline{N} \ne \underline{n}.

Triad kinematic formulas

{\cal B}_2:

  • \underline{n}' = \theta' \underline{p} + \cos(\theta)\varphi' \underline{q},
  • \underline{p}' = -\theta' \underline{n} - \sin(\theta)\varphi' \underline{q},
  • \underline{q}' = -\cos(\theta) \theta' \underline{n} + \sin(\theta)\varphi' \underline{p}.

{\cal B}_3:

  • \underline{n}' = \Phi' \underline{m},
  • \underline{m}' = -\Phi' \underline{n} + \Theta' \underline{k},
  • \underline{k}' = -\Theta' \underline{m}.

Note that \Phi and \Theta are not the same as \varphi and \theta. They are defined by these equations.

A vector \underline{A} = A_n\underline{n}+A_m\underline{m}+A_k\underline{k} evolves according to:
\underline{A}' = [A'_n - \Phi' A_m]\underline{n} + [A'_m + \Phi' A_n - \Theta' A_k]\underline{m} + [A'_k + \Theta' A_m]\underline{k}.

{\cal B}_4 is guided by the Frenet–Serret formulas:

  • \frac{d}{ds}\underline{T} = \kappa \underline{N},
  • \frac{d}{ds}\underline{N} = -\kappa  \underline{T} + \tau \underline{B},
  • \frac{d}{ds}\underline{B} = -\tau \underline{N}.

where s(t) :\equiv \int_0^t dt' v(t').

Relations between triads

{\cal B}_3 in terms of {\cal B}_2:

  • \underline{n} =  \underline{n},
  • \underline{p} = \frac{\theta'}{\Phi'} \underline{m} + \cos(\theta)\frac{\varphi'}{\Phi'} \underline{k},
  • \underline{q} = \cos(\theta) \frac{\varphi'}{\Phi'} \underline{m} - \frac{\theta'}{\Phi'} \underline{k}.

{\cal B}_2 in terms of {\cal B}_3:

  • \underline{n} =  \underline{n},
  • \underline{m} = \frac{\theta'}{\Phi'} \underline{p} + \cos(\theta)\frac{\varphi'}{\Phi'} \underline{q},
  • \underline{k} = \cos(\theta) \frac{\varphi'}{\Phi'} \underline{p} - \frac{\theta'}{\Phi'} \underline{q}.

Note: Here

  • \Phi'^2 :\equiv \theta'^2 + \cos(\theta)^2 \varphi'^2,
  • \Theta' :\equiv \sin(\theta) \varphi'  \left(1+\frac{\theta'^2}{\Phi'^2}\right) + \cos(\theta) \frac{\theta''\varphi'-\theta'\varphi''}{\Phi'^2}.

Note: if we replace t with \Phi, then

  • \left(\frac{d\theta}{d\Phi}\right)^2 + \cos(\theta)^2 \left(\frac{d\varphi}{d\Phi}\right)^2=1,
  • \frac{d\Theta}{d\Phi} = \sin(\theta) \left(\frac{d\varphi}{d\Phi}\right)  \left[1+\left(\frac{d\theta}{d\Phi}\right)^2\right] + \cos(\theta) \left[\frac{d^2\theta}{d\Phi^2}\frac{d\varphi}{d\Phi}-\frac{d\theta}{d\Phi}\frac{d^2\varphi}{d\Phi'^2}\right].


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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