coordinates of a pointCoordinatesOfAPoint22026-08-19 13:47:372026-08-19 13:47:37Definition% this is the default PhysicsLibrary preamble. as your knowledge
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% define commands here\section*{Coordinates of a Point}
The position of a moving particle may be given at any time by giving its
rectangular coordinates $x,y,z$ referred to a set of rectangular axes fixed
in space. It may be given equally well by giving the values of any three
specified functions of $x,y,$ and $z$, if from the values in question the
corresponding values of $x,y,$ and $z$ may be obtained uniquely. These
functions may be used as coordinates of the point, and the values of $x,y,$
and $z$ expressed explicitly in terms of them serve as formulas for
transformation from the rectangular system to the new system.
Familiar examples are polar coordinates in a plane, and cylindrical and
spherical coordinates in space, the formulas for transformation of
coordinates being respectively
\[
\left.
\begin{aligned}
x&=r\cos\phi,\\
y&=r\sin\phi,
\end{aligned}
\right\}
\tag{1}
\]
\[
\left.
\begin{aligned}
x&=r\cos\phi,\\
y&=r\sin\phi,\\
z&=z,
\end{aligned}
\right\}
\tag{2}
\]
\\
and
\\
\[
\left.
\begin{aligned}
x&=r\cos\theta,\\
y&=r\sin\theta\cos\phi,\\
z&=r\sin\theta\sin\phi.
\end{aligned}
\right\}
\tag{3}
\]
It is clear that the number of possible systems of coordinates is unlimited.
It is also clear that if the point is unrestricted in its motion, three
coordinates are required to determine it. If it is restricted to moving in
a plane, since that plane may be taken as one of the rectangular coordinate
planes, two coordinates are required.
The number of independent coordinates required to fix the position of a
particle moving under any given conditions is called the number of
\emph{degrees of freedom} of the particle, and is equal to the number of
independent conditions required to fix the point.
Obviously these coordinates must be numerous enough to fix the position
without ambiguity and not so numerous as to render it impossible to change
any one at pleasure without changing any of the others and without violating
the restrictions of the problem.
\section*{Source}
William Elwood Byerly, \emph{An Introduction to the Use of Generalized Co\"ordinates
in Mechanics and Physics}, Ginn and Company, 1916. Chapter I, ``Introduction.''
The 1916 source work is in the public domain in the United States. This
PhysicsLibrary transcription converts the typography and equations to LaTeX
while preserving the historical exposition and notation except where a
transcription correction is explicitly documented in the package README.