Independent variables of the Lagrangian

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SUMMARY

The discussion centers on the treatment of y and y' as independent variables in the context of the Lagrangian formulation of mechanics. Participants explore the implications of differentiating functions and the independence of these variables, particularly at the boundaries of a path. The consensus is that y and y' are considered independent algebraically, which is crucial for the calculus of variations. This understanding is foundational for analyzing physical systems using Lagrangian mechanics.

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  • Understanding of Lagrangian mechanics
  • Familiarity with calculus of variations
  • Knowledge of independent and dependent variables in physics
  • Basic differentiation techniques in calculus
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  • Study the principles of Lagrangian mechanics in detail
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  • Explore the implications of independent variables in physical systems
  • Investigate the role of boundary conditions in variational problems
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This discussion is beneficial for physicists, mathematicians, and students studying mechanics, particularly those interested in the foundations of Lagrangian dynamics and the calculus of variations.

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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