Editing Curvilinear coordinates (section)

==Fictitious forces in general curvilinear coordinates==
By definition, if a particle with no forces acting on it has its position expressed in an inertial coordinate system, (''x''<sub>1</sub>,&nbsp;''x''<sub>2</sub>,&nbsp;''x''<sub>3</sub>,&nbsp;''t''), then there it will have no acceleration (d<sup>2</sup>''x''<sub>''j''</sub>/d''t''<sup>2</sup>&nbsp;=&nbsp;0).<ref>{{cite book | first1=Michael | last1=Friedman | title=The Foundations of Space–Time Theories | publisher=Princeton University Press | year=1989 | isbn=0-691-07239-6 }}</ref> In this context, a coordinate system can fail to be "inertial" either due to non-straight time axis or non-straight space axes (or both). In other words, the basis vectors of the coordinates may vary in time at fixed positions, or they may vary with position at fixed times, or both. When equations of motion are expressed in terms of any non-inertial coordinate system (in this sense), extra terms appear, called Christoffel symbols. Strictly speaking, these terms represent components of the absolute acceleration (in classical mechanics), but we may also choose to continue to regard d<sup>2</sup>''x''<sub>''j''</sub>/d''t''<sup>2</sup> as the acceleration (as if the coordinates were inertial) and treat the extra terms as if they were forces, in which case they are called fictitious forces.<ref>{{cite book | title=An Introduction to the Coriolis Force | url=https://archive.org/details/introductiontoco0000stom | url-access=registration | first1=Henry M. | last1=Stommel | first2=Dennis W. | last2=Moore | year=1989 | publisher=Columbia University Press | isbn=0-231-06636-8}}</ref> The component of any such fictitious force normal to the path of the particle and in the plane of the path's curvature is then called [[centrifugal force]].<ref>{{cite book | title=Statics and Dynamics | last1=Beer | last2=Johnston | publisher=McGraw–Hill | edition=2nd | page=485 | year=1972 | isbn=0-07-736650-6 }}</ref>

This more general context makes clear the correspondence between the concepts of centrifugal force in [[rotating reference frame|rotating coordinate system]]s and in stationary curvilinear coordinate systems. (Both of these concepts appear frequently in the literature.<ref>{{cite book | title=Methods of Applied Mathematics | author-link=Francis B. Hildebrand | first=Francis B. | last=Hildebrand | year=1992 | publisher=Dover | page=[https://archive.org/details/methodsofapplied00hild/page/156 156] | isbn=0-13-579201-0 | url=https://archive.org/details/methodsofapplied00hild/page/156 }}</ref><ref>{{cite book | title=Statistical Mechanics | url=https://archive.org/details/statisticalmecha00mcqu_0 | url-access=registration | first=Donald Allan | last=McQuarrie | year=2000 | publisher=University Science Books | isbn=0-06-044366-9}}</ref><ref>{{cite book | title=Essential Mathematical Methods for Physicists | first1=Hans-Jurgen | last1=Weber | first2=George Brown | last2=Arfken | author-link2 = George B. Arfken | publisher=Academic Press | year=2004 | page=843 | isbn=0-12-059877-9}}</ref>) For a simple example, consider a particle of mass ''m'' moving in a circle of radius ''r'' with angular speed ''w'' relative to a system of polar coordinates rotating with angular speed ''W''. The radial equation of motion is ''mr''”&nbsp;=&nbsp;''F''<sub>''r''</sub>&nbsp;+&nbsp;''mr''(''w''&nbsp;+&nbsp;''W'')<sup>2</sup>. Thus the centrifugal force is ''mr'' times the square of the absolute rotational speed ''A''&nbsp;=&nbsp;''w''&nbsp;+&nbsp;''W'' of the particle. If we choose a coordinate system rotating at the speed of the particle, then ''W''&nbsp;=&nbsp;''A'' and ''w''&nbsp;=&nbsp;0, in which case the centrifugal force is ''mrA''<sup>2</sup>, whereas if we choose a stationary coordinate system we have ''W''&nbsp;=&nbsp;0 and ''w''&nbsp;=&nbsp;''A'', in which case the centrifugal force is again ''mrA''<sup>2</sup>. The reason for this equality of results is that in both cases the basis vectors at the particle's location are changing in time in exactly the same way. Hence these are really just two different ways of describing exactly the same thing, one description being in terms of rotating coordinates and the other being in terms of stationary curvilinear coordinates, both of which are non-inertial according to the more abstract meaning of that term.

When describing general motion, the actual forces acting on a particle are often referred to the instantaneous [[osculating circle]] tangent to the path of motion, and this circle in the general case is not centered at a fixed location, and so the decomposition into centrifugal and Coriolis components is constantly changing. This is true regardless of whether the motion is described in terms of stationary or rotating coordinates.