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User:InfoTheorist/Differentiable curves

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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)