Transformations of Coordinates,Conic Sections,Lagrange's Equations,

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Position of an object in space can be described with reference to a fixed point and some coordinate axes in different systems. The same position of the object is described in different ways in different systems of coordinates, Cartesian, spherical polar, cylindrical polar etc. , the parameters used are different in different systems. The transformation of coordinates from one set of Cartesian System to another is done through a transformation matrix. Transformation of orthogonal curvilinear coordinates from one system into another is done through partial differentiation. The transformation matrix is also a simple case of partial differentiation.
Another aspect of coordinate transformation is Compression of coordinates ( Change of scale in graph) resulting in deformation of graphs. A grand example of this is given by deriving the equations of conic sections from the equation of circle.
Finally a grand example of transformation of coordinates would be Lagrange’s equations. The Newton’s Laws of motion have been transformed from Cartesian coordinates into generalized coordinates, the coordinates of position, angles, and velocities of a system of particles treated on equal footing even if they are of different dimensions. The method of derivation through partial differentiation can be applied to transformation of any set of coordinate system into another. The beauty of Lagrange’s equations lies in the fact that they are invariant in form in any coordinate system. The wonder about it is, that the for divergent problems of quantum mechanics and electromagnetic theory and optics are easily derived from the same Lagrange’s equations using ; the total energy functions in each area.
Moreover, the Lagrange’s equations also are derived from d’ Alembert’s “principle of least action”, “light traverses the minimum path between two points” etc. the principles which dictates the direction in which events move naturally; a standard technique of “Calcu


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narayana dash
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