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Orthogonal curvilinear coordinate system is made of intersecting sets of confocal ellipses & hyperbolas.

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But, I have found no book that would describe this how to use this coordinate system and where to use. Only did I get the above quoted definition from A.P.French's Newtonian Mechanics.

Has it any uses in Newtonian Mechanics? How should I use this? Plz help.

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The co-ordinates you describe are only a special case of orthogonal curvilinear co-ordinates and are known, not surprisingly, as Elliptic Co-ordinates.

They were more useful in the days before widespread computing when analytical solutions of physical problems was more needed to help, e.g. visualise slightly eccentric systems, i.e. those nominally circular but slightly off-circular.

They are useful, for example, in solving Helmholtz's equation for an elliptical cross-section wave-guide.

Another physical interpretation is that the ellipses are the equipotential surfaces and the hyperbolas the field lines for an electrostatic field from a thin, charged plate stretching between $(-1,\,0)$ and $(1,\,0)$.

The Laplace equation for these co-ordinates is unchanged in form, which is a result of the next interesting property. These co-ordinates are special amongst orthogonal co-ordinates insofar that they can be visualised as the level surfaces for a holomorphic complex function - in this case $\Omega:\mathbb{C}\to\mathbb{C};\;\Omega(z)=\cosh z$. The level curves $\mathrm{Re}(\Omega(z))=\textrm{const}$ in the $z$-plane are the hyperbolas, whilst the curves $\mathrm{Im}(\Omega(z))=\textrm{const}$ are the ellipses. Note that such level curves for holomorphic functions are always orthogonal, but not every system of orthogonal families of curves are level curves of holomorphic functions. An interesting paper on this topic is:

Irl C. Bivens, "When Do Orthogonal Families of Curves Possess a Complex Potential?", Mathematics Magazine, Vol. 65, No. 4. (Oct., 1992), pp. 226-235

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Curvilinear coordinates are a coordinate system where the coordinate lines may be curved. A Cartesian coordinate system offers the unique advantage that all three unit vectors, x, y, and z, are constant in direction as well as in magnitude. Unfortunately, not all physical problems are well adapted to solution in Cartesian coordinates.

For instance, if we have a central force problem, such as gravitational or electrostatic force, Cartesian coordinates may be unusually inappropriate. Depending on the application, a curvilinear coordinate system may be simpler to use than the Cartesian coordinate system. For instance, a physical problem with spherical symmetry defined in R3 (for example, motion of particles under the influence of central forces) is usually easier to solve in spherical polar coordinates than in Cartesian coordinates.

The point is that the coordinate system should be chosen to fit the problem, to exploit any constraint or symmetry present in it. Then, hopefully, it will be more readily soluble than if we had forced it into a Cartesian framework.

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