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.
Kostik
Messages
250
Reaction score
28
TL;DR Summary
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?
 
Physics news on Phys.org
“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:
 
Thread 'Question about pressure of a liquid'
I am looking at pressure in liquids and I am testing my idea. The vertical tube is 100m, the contraption is filled with water. The vertical tube is very thin(maybe 1mm^2 cross section). The area of the base is ~100m^2. Will he top half be launched in the air if suddenly it cracked?- assuming its light enough. I want to test my idea that if I had a thin long ruber tube that I lifted up, then the pressure at "red lines" will be high and that the $force = pressure * area$ would be massive...
I feel it should be solvable we just need to find a perfect pattern, and there will be a general pattern since the forces acting are based on a single function, so..... you can't actually say it is unsolvable right? Cause imaging 3 bodies actually existed somwhere in this universe then nature isn't gonna wait till we predict it! And yea I have checked in many places that tiny changes cause large changes so it becomes chaos........ but still I just can't accept that it is impossible to solve...
Back
Top