Independent variables of the Lagrangian

AI Thread Summary
In the discussion, y and y' are treated as independent variables in the context of the Lagrangian because they are considered independent algebraically, despite being related through differentiation. The question arises about the implications of differentiating f' to g' and whether df/dg is independent of g', leading to a deeper examination of the relationship between variables in calculus of variations. Some participants suggest that y and y' represent boundary values, reinforcing their perceived independence. The conversation also explores the idea of binding velocities instead of positions, prompting further inquiry into the nature of these variables. Overall, the treatment of y and y' as independent is a foundational aspect of the mathematical framework used in the Lagrangian formulation.
JanEnClaesen
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Why are y and y' treated as independent variables, while they are not?

Another slightly related question:
if ' = d/dt then df'/dg' = df/dg because f' = df/dg g', but if we differentiate f' to g' we implicitly assume that df/dg is independent of g', is it?
 
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Perhaps because y and y' are the values of y and y' at the boundaries of a path, in which case they do seem independent.
Hence a natural follow-up question, instead of binding the positions, why don't we bind the velocities?
 

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