Acceleration of Point in Plane in Intrinsic Coordinates

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Theorem

Let $\mathbf a$ be the acceleration of a particle $P$ in space.

Let $P$ be moving in a plane $\PP$.


Then the motion of $P$ can be expressed in intrinsic coordinates as:

$\mathbf a = \dfrac {\d \map v t} {\d t} \mathbf s + \dfrac {\paren {\map v t}^2} \rho \bspsi$

where:

$\mathbf s$ denotes the unit vector along the tangential direction of $P$
$\bspsi$ denotes the unit vector toward the center of curvature of the motion of $P$
$\map v t$ is the speed of $P$ at the time $t$
$\rho$ is the radius of curvature of the motion of $P$ at time $t$.


Proof


This theorem requires a proof.
In particular: Needs some more background work on intrinsic coordinates for this to make sense
See https://dspace.mit.edu/bitstream/handle/1721.1/60691/16-07-fall-2004/contents/lecture-notes/d4.pdf
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Sources

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