When we use arc length as a parameter

[Castilla]

If we have a fly in a room, its position respect to some frame of reference will change with time, so if we want to describe the fly's movement with a parametrized curve, it is easy to see the convenience of taking time as the parameter.

I read that we can also take the length of the curve as a parameter and it is not dificult to follow formally the equations by which we put the original parameter -time- in terms of this new parameter. My question is: when or why is better to work with arc length as parameter, instead of time?

Thanks in advance.

 

[arildno]

From a geometrical point of view, it is simpler to use the arc length parameter in order to understand the concepts of curvature and torsion, for example.

Curvature of a curve is simply the rate of (locally planar) turning of the unit tangent when working with the arc length parameter.

 

[HallsofIvy]

If we use arclength, s, as parameter, then the derivative, dr(s)/dx is the unit tangent vector. From that it follows that the second derivative, d²r(s)ds², is the normal vector. With any other parameter, the derivative is tangent but not of unit length and the second derivative is not normal to the curve. Also, with arclength as parameter the length of the second derivative, |d²r(s)ds²| is the curvature of the graph.

 

[Castilla]

Thanks Arildno and Hallsoftivy. I am beginning to read about curvature, so your answers help me.

 

[wofsy]

Also using arc length is often easier. In situations where the speed of a particle doesn't matter it is easier to calculate when the motion is uniform. For instance the work done by an electric field on an electron that moves through a closed circuit does not depend on the speed of the electron. It can traverse the loop in any way. The same is true for integrals of differential forms over manifolds. Parameterization doesn't matter. Perhaps this is the essence of your question: when does speed matter and when is it irrelevant?

 

[Tac-Tics]

My question is: when or why is better to work with arc length as parameter, instead of time?

It's better when the time isn't important to the question you are asking. When you're concerned about the shape of the path, use arc length. When you're concerned about motion, use time. The time parametrization contains more "information" than the arc parametrization. You can construct the path from the motion, but not the motion from the path.