0
$\begingroup$

For a given Lagrangian $L$, the $i$th generalized momentum is defined as $$p_i = \frac{\partial L}{\partial \dot{q_i}}$$ where $\dot{q_i}$ is the time derivative of the $i$th generalized coordinate (i.e. the $i$th generalized velocity).

I have also seen the above referred to as conjugate momenta, or even generalized conjugate momenta. What exactly is the difference between these terms? Do they all mean the same thing?

$\endgroup$

1 Answer 1

2
$\begingroup$

There is no difference. Generalized emphasizes the fact that the momenta depend on generalized coordinates (therefore, physical dimensions may be different from $M L T^{-1}$), while conjugate refers to the definition, connected to the Legendre transform underlying the introduction of the $ p$s, starting from the Lagrangian and the generalized coordinates $q$s.

$\endgroup$
2
  • $\begingroup$ +1 Worth adding that it is conjugate to corresponding generalized coordinate. It cannot be conjugate by itself. $\endgroup$
    – Roger V.
    Commented Jan 30, 2023 at 10:04
  • $\begingroup$ @RogerVadim Thanks for the comment. I have added a final part of the last sentence. $\endgroup$ Commented Jan 30, 2023 at 10:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.