Difference between revisions of "Forest UCM NLM Ch1 CoordSys"

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:<math>\frac{ d \hat{r}}{dt} = \frac{d \phi}{dt} \hat{\phi}</math>
 
:<math>\frac{ d \hat{r}}{dt} = \frac{d \phi}{dt} \hat{\phi}</math>
  
Thus for circular motion at a constraint radius we get the familiar expression
+
Thus for circular motion at a constant radius we get the familiar expression
  
 
:<math>\vec{v} = \lim_{\Delta t \rightarrow 0} \frac{\vec{r}(t_2)-\vec{r}(t_1)}{\Delta t}= \lim_{\Delta t \rightarrow 0} \frac{r\left(  \hat{r}(t_2) - \hat{r}(t_1)\right)}{\Delta t} = r \frac{\Delta \phi}{\Delta t} \hat{\phi} = r \omega \hat{\phi}</math>
 
:<math>\vec{v} = \lim_{\Delta t \rightarrow 0} \frac{\vec{r}(t_2)-\vec{r}(t_1)}{\Delta t}= \lim_{\Delta t \rightarrow 0} \frac{r\left(  \hat{r}(t_2) - \hat{r}(t_1)\right)}{\Delta t} = r \frac{\Delta \phi}{\Delta t} \hat{\phi} = r \omega \hat{\phi}</math>
Line 88: Line 88:
  
 
:linear velocity <math>\equiv  v_r </math>  Angular velocity <math>\equiv  v_{\phi} </math>
 
:linear velocity <math>\equiv  v_r </math>  Angular velocity <math>\equiv  v_{\phi} </math>
 +
 +
 +
;Finding the derivative directly
 +
 +
Cast the unit vector in terms of cartesian coordinates and take the derivative.
 +
 +
:<math>\hat r =  \cos \phi \hat{i} + \sin \phi \hat{j}</math>
 +
 +
:<math>\hat \dot{r} = \frac{d \hat{r}}{d \phi} \frac{d \phi}{d t} =\left( -\sin \phi \hat{i} + \cos \phi \hat{j} \right ) \dot{\phi}</math>
 +
:: <math>= \left ( \hat{\phi} \right ) \dot{\phi}  </math>
  
 
====Acceleration in Polar Coordinates ====
 
====Acceleration in Polar Coordinates ====
Line 119: Line 129:
  
 
:<math>\frac{d \hat{\phi}}{dt}= -\dot{\phi} \hat{r}</math>
 
:<math>\frac{d \hat{\phi}}{dt}= -\dot{\phi} \hat{r}</math>
 +
 +
 +
;Finding the derivative of <math>\hat{\phi}</math> directly:
 +
 +
Cast the unit vector in terms of cartesian coordinates and take the derivative.
 +
 +
: <math> \hat{\phi} = -\sin \phi \hat{i} + \cos \phi \hat{j} </math>
 +
 +
:<math>\hat \dot{\phi} = \frac{d \hat{\phi}}{d \phi} \frac{d \phi}{d t} =\left( -\cos \phi \hat{i} - \sin \phi \hat{j} \right ) \dot{\phi}</math>
 +
:: <math>= \left (- \hat{r} \right ) \dot{\phi}  </math>
 +
 +
  
 
Substuting the above into our calculation for acceleration:
 
Substuting the above into our calculation for acceleration:
Line 176: Line 198:
 
another way to determine the unit vector derivative is to cast them in terms of cartesian coordinate.
 
another way to determine the unit vector derivative is to cast them in terms of cartesian coordinate.
  
:<math>\hat{r} = \cos \phi \sin \theta \hat{i} + \sin \phi \sin \theta  \hat{j} + \cos \theta \hat{k}</math>
+
:<math>\hat{r} = \frac{\vec{r}}{|\vec{r}|} =\cos \phi \sin \theta \hat{i} + \sin \phi \sin \theta  \hat{j} + \cos \theta \hat{k}= \frac{\frac{\partial \vec{r}}{\partial r}}{\left | \frac{\partial \vec{r}}{\partial r}\right | } </math>
 
 
:<math>\hat{\phi} = - \sin \phi \hat{i} + \cos \phi \hat{k}</math>
 
  
:<math>\hat{\theta} = -\cos \phi \cos \theta \hat{i} + \sin \phi \sin \theta  \hat{j} + \cos \theta \hat{k}</math>
+
:<math>\hat{\phi} = \frac{\frac{\partial \vec{r}}{\partial \phi}}{\left | \frac{\partial \vec{r}}{\partial \phi}\right | } =- \sin \phi \hat{i} + \cos \phi \hat{j}= \frac{\hat {z} \times \hat {r} }{\sin \theta}</math>
  
::<math> =\dot{r}\hat{r} + r\dot{\theta}\sin \theta\hat{\theta} + r \dot{\phi} \hat{\phi}</math>  
+
:<math>\hat{\theta} = -\hat{\phi} \times \hat {r} =\cos \phi \cos \theta \hat{i} + \sin \phi \cos \theta \hat{j} - \sin \theta \hat{k}</math>
  
 +
The derivative of the above unit vectors are
  
cartesian unit vectors do not change with time (unit vectors for other coordinate system types do)
+
::<math>\dot{\hat{r}} = \dot{\theta} \hat{\theta} + \dot{\phi} \sin \theta \hat {\phi}</math>
 +
::<math>\dot{\hat{\phi}} = -\dot{\phi} \left ( \sin \theta \hat{r} + \cos \theta \hat {\theta} \right )</math>
 +
::<math>\dot{\hat{\theta}} = -\dot{\theta} \hat{r} + \dot{\phi} \cos \theta \hat {\phi}</math>
  
  
:<math>\frac{d \hat{i}}{dt} =0 =\frac{d \hat{j}}{dt} =\frac{d \hat{k}}{dt}</math> 
+
substituting the above into the definition of velocity
  
:<math>\vec{v}</math> = <math>\frac{d \vec{r}}{dt}</math> = <math>\frac{d x}{dt}\hat{i} + \frac{d y}{dt}\hat{j} + \frac{d z}{dt}\hat{k} </math>
+
:<math>\vec{v}</math> = <math>\frac{d r}{dt}\hat{r} + r\frac{d \hat{r}}{dt} </math>  
 +
::= <math>\dot{r} \hat{r} + r \left ( \dot{\theta} \hat{\theta} + \dot{\phi} \sin \theta \hat {\phi} \right )</math>
  
 
====Acceleration vector in Spherical  coordinates====
 
====Acceleration vector in Spherical  coordinates====
  
  
:<math>\vec{a}</math> = <math>\frac{d \vec{v}}{dt}</math> = <math>\frac{d^2 x}{dt^2}\hat{i} + \frac{d^2 y}{dt^2}\hat{j} + \frac{d^2 z}{dt^2}\hat{k} </math>
+
:<math>\vec{a}</math>= <math>\frac{d}{dt} \left ( \dot{r} \hat{r} + r \left ( \dot{\theta} \hat{\theta} + \dot{\phi} \sin \theta \hat {\phi} \right ) \right )</math>
::<math>\ddot{r} -r \dot{\theta}^2 \cos^2{\phi} -r \dot{\phi}^2) \frac{d^2 x}{dt^2}\hat{i} + \frac{d^2 y}{dt^2}\hat{j} + \frac{d^2 z}{dt^2}\hat{k} </math>
 
  
 +
::=<math> \ddot{r} \hat{r}  +  \dot{r} \dot{\hat{r}}  </math>
 +
:: <math>+ \dot{r} \dot{\theta} \hat{\theta}  + r \ddot{\theta} \hat{\theta}  + r \dot{\theta} \dot{\hat{\theta}}</math>
 +
:: <math>+ \dot{r}  \dot{\phi} \sin \theta \hat {\phi} + r  \ddot{\phi} \sin \theta \hat {\phi} + r  \dot{\phi} \dot{\sin \theta} \hat {\phi} + r  \dot{\phi} \sin \theta \dot{\hat {\phi}}</math>
 +
:: = <math>\left ( \ddot{r} - r \dot{\theta}^2 - r \dot{\phi}^2 \sin \theta \right) \hat{r}
 +
+ \left( r \ddot{\theta} + 2 r \dot{\theta} - r \dot{\phi}^2 \sin \theta \cos \theta \right) \hat{\theta}
 +
+ \left ( r \ddot{\phi} \sin \theta + 2 r \dot{\theta} \dot{\phi} \cos \theta + 2 \dot{r} \dot{\phi} \sin \theta \right ) \hat{\phi}
 +
</math>
  
  
 
[[Forest_UCM_NLM#Space]]
 
[[Forest_UCM_NLM#Space]]

Latest revision as of 20:23, 27 August 2021

Cartesian, Spherical, and Cylindrical coordinate systems are commonly used to describe three-dimensional space.

Cartesian

TF UCM CartCoordSys.png


Vector Notation convention:

Position:

r=xˆi+yˆj+zˆk=(x,y,z)=31riˆei

Velocity and Acceleration vector in cartesian coordinates

v = drdt = dxdtˆi+xdˆidt+


cartesian unit vectors do not change with time (unit vectors for other coordinate system types do)


dˆidt=0=dˆjdt=dˆkdt
v = drdt = dxdtˆi+dydtˆj+dzdtˆk


Similarly Acceleration is given by


a = dvdt = d2xdt2ˆi+d2ydt2ˆj+d2zdt2ˆk

Polar

TF UCM PolarCoordSys.png Vector Notation convention:

Position:

Because ˆr points in a unique direction given by ˆr=r|r| we can write the position vector as

r=rˆr
rrˆr+ϕˆϕ: ϕ does not have the units of length


The unit vectors (ˆr and ˆϕ ) are changing in time. You could express the position vector in terms of the cartesian unit vectors in order to avoid this

r=rcos(ϕ)ˆi+rsin(ϕ)ˆj

The dependence of position with ϕ can be seen if you look at how the position changes with time.

Velocity in Polar Coordinates

Consider the motion of a particle in a circle. At time t1 the particle is at r(t1) and at time t2 the particle is at r(t2)


TF UCM PolarVectDiff.png


If we take the limit t2t1 ( or Δt0) then we can write the velocity of this particle traveling in a circle as

ˆr(t2)ˆr(t1)Δˆr=Δϕˆϕ
or
dˆrdt=dϕdtˆϕ

Thus for circular motion at a constant radius we get the familiar expression

v=limΔt0r(t2)r(t1)Δt=limΔt0r(ˆr(t2)ˆr(t1))Δt=rΔϕΔtˆϕ=rωˆϕ
v=rdϕdtˆϕ


If the particle is not constrained to circular motion ( i.e.: r can change with time) then the velocity vector in polar coordinates is



v = drdtˆr+rdϕdtˆϕ
or in more compact form
v=˙r=˙rˆr+r˙ϕˆϕ=vrˆr+vϕˆϕ


linear velocity vr Angular velocity vϕ


Finding the derivative directly

Cast the unit vector in terms of cartesian coordinates and take the derivative.

ˆr=cosϕˆi+sinϕˆj
\hat \dot{r} = \frac{d \hat{r}}{d \phi} \frac{d \phi}{d t} =\left( -\sin \phi \hat{i} + \cos \phi \hat{j} \right ) \dot{\phi}
=(ˆϕ)˙ϕ

Acceleration in Polar Coordinates

Taking the derivative of velocity with time gives the acceleration


a=dvdt=¨r
=d(˙rˆr+r˙ϕˆϕ=vrˆr+vϕˆϕ)dt
=(d˙rdtˆr+˙rdˆrdt)+(drdt˙ϕˆϕ+rd˙ϕdtˆϕ+r˙ϕdˆϕdt)
=(¨rˆr+˙r˙ϕˆϕ)+(˙r˙ϕˆϕ+r¨ϕˆϕ+r˙ϕdˆϕdt)


We need to find the derivative of the unit vector ˆϕ with time.

Consider the position change below in terms of only the unit vector ˆϕ


TF UCM PolarPhiUnitVectDiff.png


Using the same arguments used to calculate the rate of change in ˆr:

If we take the limit t2t1 ( or Δt0) then we can write the velocity of this particle traveling in a circle as

ˆϕ(t2)ˆϕ(t1)Δˆϕ=Δϕ(ˆr)
or
dˆϕdt=dϕdtˆr
dˆϕdt=˙ϕˆr


Finding the derivative of ˆϕ directly

Cast the unit vector in terms of cartesian coordinates and take the derivative.

ˆϕ=sinϕˆi+cosϕˆj
\hat \dot{\phi} = \frac{d \hat{\phi}}{d \phi} \frac{d \phi}{d t} =\left( -\cos \phi \hat{i} - \sin \phi \hat{j} \right ) \dot{\phi}
=(ˆr)˙ϕ


Substuting the above into our calculation for acceleration:


a=(¨rˆr+˙r˙ϕˆϕ)+(˙r˙ϕˆϕ+r¨ϕˆϕ+r˙ϕdˆϕdt)
=(¨rˆr+˙r˙ϕˆϕ)+(˙r˙ϕˆϕ+r¨ϕˆϕ+r˙ϕ(˙ϕˆr))
=(¨rr˙ϕ2)ˆr+(2˙r˙ϕ+r¨ϕ)ˆϕ

For the case of circular motion at constant r(˙r=0)

a=r˙ϕ2ˆr+r¨ϕˆϕ

radial (centripetal, center seeking) acceleration r˙ϕ2ˆr=rω2ˆr


angular (tangential) acceleration r¨ϕˆϕ=rαˆϕ

If ˙r0

Then there are two additional terms

¨rˆr = radial acceleration
2˙r˙ϕˆϕ = Coriolis acceleration (to be described later)

Cylindrical

Cylindrical coordinates are polar coordinates with a third dimension usually labeled (z)

change picture so angle is [math]\phi[/math] not [math]\theta[/math]

TF UCM CylCoordSys.png


We just need to add zˆk to all the vectors (remember ˆ˙k=0)

r=rˆr+zˆk
v=˙r=˙rˆr+r˙ϕˆϕ+˙zˆk
a=(¨rr˙ϕ2)ˆr+(2˙r˙ϕ+r¨ϕ)ˆϕ+¨zˆk

Spherical

TF UCM SphericalCoordSys.png


Position:

r=rˆr

Velocity vector in Spherical coordinates

v = drdt = drdtˆr+rdˆrdt


another way to determine the unit vector derivative is to cast them in terms of cartesian coordinate.

ˆr=r|r|=cosϕsinθˆi+sinϕsinθˆj+cosθˆk=rr|rr|
ˆϕ=rϕ|rϕ|=sinϕˆi+cosϕˆj=ˆz׈rsinθ
ˆθ=ˆϕ׈r=cosϕcosθˆi+sinϕcosθˆjsinθˆk

The derivative of the above unit vectors are

˙ˆr=˙θˆθ+˙ϕsinθˆϕ
˙ˆϕ=˙ϕ(sinθˆr+cosθˆθ)
˙ˆθ=˙θˆr+˙ϕcosθˆϕ


substituting the above into the definition of velocity

v = drdtˆr+rdˆrdt
= ˙rˆr+r(˙θˆθ+˙ϕsinθˆϕ)

Acceleration vector in Spherical coordinates

a= ddt(˙rˆr+r(˙θˆθ+˙ϕsinθˆϕ))
=¨rˆr+˙r˙ˆr
+˙r˙θˆθ+r¨θˆθ+r˙θ˙ˆθ
+˙r˙ϕsinθˆϕ+r¨ϕsinθˆϕ+r˙ϕ˙sinθˆϕ+r˙ϕsinθ˙ˆϕ
= (¨rr˙θ2r˙ϕ2sinθ)ˆr+(r¨θ+2r˙θr˙ϕ2sinθcosθ)ˆθ+(r¨ϕsinθ+2r˙θ˙ϕcosθ+2˙r˙ϕsinθ)ˆϕ


Forest_UCM_NLM#Space