Kostik
- 250
- 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?