Generalized coordinates in Lagrangian mechanics

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Generalized coordinates in Lagrangian mechanics can include quantities with various dimensions, such as energy or length squared, rather than being limited to traditional lengths and angles. Lagrange's equations remain valid regardless of the units used for these coordinates. An arbitrary change of variables allows for the transformation of standard coordinates into those with different dimensions, such as converting lengths into time by dividing by the speed of light. Angles are inherently dimensionless, so they do not pose any issues in this context. The flexibility in choosing generalized coordinates enhances the versatility of Lagrangian mechanics.
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In some texts about Lagrangian mechanics,its written that the generalized coordinates need not be length and angles(as is usual in coordinate systems)but they also can be quantities with other dimensions,say,energy,length^2 or even dimensionless.
I want to know how will be the Lagrange's equations in such coordinates?
Could you give an example whose proper coordinates are as such?
Thanks
 
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Lagranges equations are unchanged regardless of the units of the coordinates. You can take any problem with lengths and angles and simply do an arbitrary change of variables to get a coordinate in any units you like. E.g. Take any length and divide by the speed of light and you have a new coordinate with units of time.
 
tbf, angles are dimensionless so there's no issue there.
 

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