I am solving dynamics(central orbits) and have a doubt in differential equation of a particle moving in central orbit.
Suppose a particle moves in central orbit under influence of a central force.
Let a particle move in a plane with an acceleration P , always directed towards a fixed point O which is also the centre of force.Let (r,θ) be the polar coordinates of the position P of the moving particle at any instant t.
Radial acceleration of particle is given by :-
d2r/dt2 – r (dθ/dt)2 = -P
Transverse acceleration of particle is given by :-
$\frac{1}{r}$$\frac{d}{dt}$(r2$\frac{dθ}{dt}$)
I know about radial and transverse accelerations in circular motion. Central orbit motion seems similar . But still not able to relate how the author presented these equations.
Any help is appreciated !!
Best Answer
the core of a Central Forces problem is that F(r) = F(r)$ \hat{r}$. Due to this, the choice of spherical coordinates is pretty obvious.
So now, $$\mathbf{\dot{r}} = r\,\mathbf{\hat{r}} + r\dot{\varphi} \,\boldsymbol{\hat{\varphi}} \tag{*}$$
How do we get this? I would suggest checking out a classical mechanics book, like Taylor or Goldstein. Basically, this equation is a vector representation of radial and transverse velocities.
Differentiating (*) you would easily yield both the radial and transverse accelerations. I would highly suggest you try doing it yourself. Once you are done differentiating you would be getting two components, one of them would be the radial acceleration and the other the transverse acceleration.
hint on how to get equation (*).
$\mathbf{r} = (x, y) = (r\sin{\varphi}, r\cos{\varphi}) $
$\mathbf{\dot{r}}=\frac{d\mathbf{r}}{dt}= {\mathbf {v}}={\dot {r}}(\cos \varphi ,\ \sin \varphi )+r{\dot {\varphi }}(-\sin \varphi ,\cos \varphi ) $
From here on hopefully you can clearly see what the unit vectors are.