I Is ##p^k = \partial L / \partial \dot{x}^k## true for all ##L##'s?

AI Thread Summary
The discussion centers on the relationship between generalized momentum and the Lagrangian in classical mechanics, specifically questioning whether the equation p^k = ∂L/∂dot{x}^k holds for all Lagrangians. An example using the Lagrangian L = T - U = (1/2)mv^2 - U demonstrates that the equation is valid for this case, yielding p^k = m dot{x}^k. The conversation seeks to establish a general proof for this relationship across all Lagrangians. A reference to generalized coordinates is provided to support the discussion. The inquiry highlights the need for a deeper understanding of momentum definitions in the context of Lagrangian mechanics.
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Is the relation Is ##p^k = \partial L / \partial \dot{x}^k## true for all Lagrangians?
Using the Lagrangian $$L=T-U=\frac{1}{2}mv^2-U$$ we clearly have $$ \frac{\partial L}{\partial \dot{x}^k} = m\dot{x}^k = p^k $$ i.e., the ##k##'th component of momentum. How does one show that the relation $$p^k = \frac{\partial L}{\partial \dot{x}^k} $$ holds for all Lagrangians?
 
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“The generalized momentum "canonically conjugate to" the coordinate qi is defined by

{\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q}}_{i}}}.}
https://en.m.wikipedia.org/wiki/Generalized_coordinates
 
Thanks. I did not phrase the question very well. I have made a more detailed post of the question here:
 
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