# Frenet-serret formulas, local reference frame

1. Oct 8, 2009

### fisico30

Hello Forum,

Using a fixed, Cartesian reference frame (i.e., we, observers, are stationary and located at the origin of the system), the trajectory of a particle would be easily described by the parametric (parameter time or arch length) equations [x(t), y(t), z(t)] or [x(s),y(s),z(s)].

If, instead, we sit on the moving particle and move with it in space, the reference frame that is local to the particle is the moving triad described by T,B, N unit vectors.
When the particle curves, to us, sitting on the particle, the local frame of reference appears to always point in the same direction (since the triad T,N,B turns with the particle too). There is no change between the new direction of the particle and the direction of T,B, N, since T is always tangent to the trajectory, while B,N perpendicular....

How can this local reference system be useful to detect changes? Unless we look at how T,B,N change with respect to the fixed Cartesian frame.... Then what is the point in using the local frame?
What am I so obviously missing?
Can you give me a simple example?

thanks
fisico30

2. Oct 8, 2009

Hello

I think you misunderstand the defenition of "Local frame". Local frame is an inertial frame that at any time the veclicity of particle respect to it is zero vector. But this frame is inertial so you can not rotate it's axis regard to an inertial frame's axis. In other word you can not take T,N,B az three axis of frame because they rotate in space.
Notice that the velocity of particle regard to local frame is zero but it doesn't mean that acceleration vector is too.
So it may be easier and more precise to use local frame in some mechanics problems. Especially relativistic mechanics.
Good luck

3. Oct 8, 2009