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When the parameter for a curve $\mathbf{R}\left(p\right)$ is actually the time, $t$, then the decomposition for $\mathbf{R}\u2033$ becomes
$\stackrel{..}{\mathbf{R}}\=\mathbf{a}\=\stackrel{\.}{v}\mathbf{T}plus;\mathrm{kappa;}{v}^{2}\mathbf{N}$
where $\mathbf{a}$ is the acceleration vector for a particle moving along the curve described by $\mathbf{R}\left(t\right)$. This is a remarkable result. It says that motion (as described on the left side by the the acceleration vector $\mathbf{a}$) is determined by the geometry on the right side (as described by the geometric quantities T, the unit tangent vector; N, the principal normal; $\mathrm{\κ}$, the curvature; and $v$, the length of $\mathbf{R}\prime$).
This decomposition for the acceleration vector can be used to solve problems in dynamics where, primarily, the geometry of the path of motion is known. When combined with Newton's second law, $\mathbf{F}\=m\mathbf{a}$, several interesting applications become accessible.
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