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Change of Coordinates Lagrangian

Consider a Lagrangian system, with configuration space R^n, given by (x^1, ... x^n); and Lagrangian L(x', ..., x^n; v^1, ... v^n). Now consider a new system of coordinates, (y^1,... ^n), for this same system, so the y's are functions of the x's; and, inverting, the x's are also functions of the y's. Find the Lagrangian in the y-coordinate system, and show that it produces the same physical equations of motion for the system as does the Lagrangian in the x-coordinate system.

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We use the general formulas of multivariate calculus and the ...

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