In this short note, I'll prove an identity regarding differentiable curves. Let be a differentiable curve. For every , the following identity holds
The left hand side equals
On the other hand, we have
thus
Therefore,
This result has several implications. First note that by the Cauchy-Schwarz inequality,
Thus the above identity implies
If we apply the above inequality to and integrate both sides over we get
In words, a straight line has the shortest distance among all curves connecting two points.
Second, if represents the position of a particle at time , then its velocity and acceleration vectors at time are given by and , respectively. If we apply the identity to the velocity vector we get
From this identity we deduce the following:
1) the velocity of a particle is perpendicular to its acceleration vector if and only if speed is constant (e.g., a particle in uniform circular motion), and
2) at any given moment, speed is increasing (decreasing) if and only if is positive (negative) at that moment.
InfoTheorist (talk) 06:30, 28 September 2014 (UTC)