Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Replies: 38   Last Post: Apr 13, 2013 11:57 PM

 Messages: [ Previous | Next ]
 Guest
Re: Misner, Thorne and Wheeler, Exercise 8.5 (c)
Posted: Apr 9, 2013 12:40 AM

On 4/8/2013 9:27 AM, Dirk Van de moortel wrote:
> Hetware <hattons@speakyeasy.net> wrote:
>> This is the geodesic equation under discussion:
>>
>> d^2(r)/dt^2 = r(dp/dt)^2
>>
>> d^2(p)/dt^2 = -(2/r)(dp/dt)(dr/dt).
>>
>> r is radius in polar coordinates, p is the angle, and t is a path
>> parameter.
>> The authors ask me to "[S]olve the geodesic equation for r(t) and
>> p(t), and show that the solution is a uniformly parametrized straight
>> line(x===r cos(p) = at+p for some a and b; y===r sin(p) = jt+k for
>> some j and k).

>
> Normally we'd write dotted variables, but with quotes it's easier.
> So write

Like this?

<math xmlns='http://www.w3.org/1998/Math/MathML'
xmlns:mathematica='http://www.wolfram.com/XML/'>
<mrow>
<mrow>
<mover>
<mi>r</mi>
<mo>..</mo>
</mover>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mi>r</mi>
<mo>&#8289;</mo>
<mo>(</mo>
<mover>
<mi>&#981;</mi>
<mo>&#183;</mo>
</mover>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>=</mo>
<mi>r</mi>
</mrow>
</mrow>
<mo>&#8291;</mo>
<msup>
<mi>&#969;</mi>
<mn>2</mn>
</msup>
</mrow>
[/itex]
<math xmlns='http://www.w3.org/1998/Math/MathML'>
<mtext>

</mtext>
[/itex]
<math xmlns='http://www.w3.org/1998/Math/MathML'
xmlns:mathematica='http://www.wolfram.com/XML/'>
<mrow>
<mrow>
<mi>r</mi>
<mo>&#8290;</mo>
<mover>
<mi>&#969;</mi>
<mo fontweight='bold'>&#183;</mo>
</mover>
</mrow>
<mo>=</mo>
<mrow>
<mrow>
<mo>-</mo>
<mn>2</mn>
</mrow>
<mo>&#8290;</mo>
<mi>&#969;</mi>
<mo>&#8290;</mo>
<mover>
<mi>r</mi>
<mo fontweight='bold'>&#183;</mo>
</mover>
</mrow>
</mrow>
[/itex]
<math xmlns='http://www.w3.org/1998/Math/MathML'>
<mtext>
</mtext>
[/itex]
<math xmlns='http://www.w3.org/1998/Math/MathML'
xmlns:mathematica='http://www.wolfram.com/XML/'>
<mrow>
<mrow>
<mfrac>
<mover>
<mi>r</mi>
<mo>..</mo>
</mover>
<mi>r</mi>
</mfrac>
<mo>=</mo>
</mrow>
<mo>&#8291;</mo>
<msup>
<mi>&#969;</mi>
<mn>2</mn>
</msup>
</mrow>
[/itex]
<math xmlns='http://www.w3.org/1998/Math/MathML'>
<mtext>

</mtext>
[/itex]
<math xmlns='http://www.w3.org/1998/Math/MathML'
xmlns:mathematica='http://www.wolfram.com/XML/'>
<mrow>
<mfrac>
<mover>
<mi>&#969;</mi>
<mo fontweight='bold'>&#183;</mo>
</mover>
<mi>&#969;</mi>
</mfrac>
<mo>=</mo>
<mrow>
<mrow>
<mo>-</mo>
<mn>2</mn>
</mrow>
<mo>&#8290;</mo>
<mfrac>
<mover>
<mi>r</mi>
<mo fontweight='bold'>&#183;</mo>
</mover>
<mi>r</mi>
</mfrac>
</mrow>
</mrow>
[/itex]
<math xmlns='http://www.w3.org/1998/Math/MathML'>
<mtext>
</mtext>
[/itex]
<math xmlns='http://www.w3.org/1998/Math/MathML'
xmlns:mathematica='http://www.wolfram.com/XML/'>
<mrow>
<mrow>
<mi>ln</mi>
<mo>&#8289;</mo>
<mo>(</mo>
<mi>&#969;</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mrow>
<mrow>
<mo>-</mo>
<mn>2</mn>
</mrow>
<mo>&#8290;</mo>
<mrow>
<mi>ln</mi>
<mo>&#8289;</mo>
<mo>(</mo>
<mi>r</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>+</mo>
<mi>k</mi>
</mrow>
</mrow>
[/itex]
<math xmlns='http://www.w3.org/1998/Math/MathML'>
<mtext>

</mtext>
[/itex]
<math xmlns='http://www.w3.org/1998/Math/MathML'
xmlns:mathematica='http://www.wolfram.com/XML/'>
<mrow>
<mrow>
<mi>&#969;</mi>
<mo>=</mo>
<mi>a</mi>
</mrow>
<mo>&#8291;</mo>
<msup>
<mi>r</mi>
<mrow>
<mo>-</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
[/itex]
<math xmlns='http://www.w3.org/1998/Math/MathML'>
<mtext>
</mtext>
[/itex]
<math xmlns='http://www.w3.org/1998/Math/MathML'
xmlns:mathematica='http://www.wolfram.com/XML/'>
<mrow>
<mfrac>
<mi>&#969;</mi>
<mi>a</mi>
</mfrac>
<mo>=</mo>
<msup>
<mi>r</mi>
<mrow>
<mo>-</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
[/itex]
<math xmlns='http://www.w3.org/1998/Math/MathML'>
<mtext>
</mtext>
[/itex]
<math xmlns='http://www.w3.org/1998/Math/MathML'
xmlns:mathematica='http://www.wolfram.com/XML/'>
<mrow>
<msqrt>
<mfrac>
<mi>&#969;</mi>
<mi>a</mi>
</mfrac>
</msqrt>
<mo>=</mo>
<msup>
<mi>r</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
[/itex]
<math xmlns='http://www.w3.org/1998/Math/MathML'>
<mtext>
</mtext>
[/itex]
<math xmlns='http://www.w3.org/1998/Math/MathML'
xmlns:mathematica='http://www.wolfram.com/XML/'>
<mrow>
<mrow>
<mover>
<mi>r</mi>
<mo>..</mo>
</mover>
<mo>=</mo>
<msqrt>
<mfrac>
<mi>a</mi>
<mi>&#969;</mi>
</mfrac>
</msqrt>
</mrow>
<mo>&#8291;</mo>
<msup>
<mi>&#969;</mi>
<mn>2</mn>
</msup>
</mrow>
[/itex]