Generalised Momentum: Defs & Meaning

[Incand]

I have two books that define generalised momentum differently. Either
$p_i = \frac{\partial L}{\partial \dot q_i}$
or
$p_i = \frac{\partial T}{\partial \dot q_i}$.
Is this since defining generalised momentum only make sense when the potential energy is independent of a coordinate $q$ and hence the above definitions are equal? Or is one of these more general than the other?

 

[hilbert2]

The differentiations should be done with respect to the time derivative of $q$, not $q$ itself. Is $T$ the kinetic energy? In most simple mechanical systems the potential energy is not dependent on velocity.

 

[Incand]

The differentiations should be done with respect to the time derivative of $q$, not $q$ itself. Is $T$ the kinetic energy? In most simple mechanical systems the potential energy is not dependent on velocity.

Thanks, missed the dots. Added them now. Yes $T$ is the kinetic energy. Is it possible that the potential energy $V$ depends on $\dot q$? Is one of these definitions correct in that case?

 

[hilbert2]

If there are magnetic fields and electric charges in the system, the potential energy depends on velocities. Then you can't use the definition where you differentiate $T$.

 

[Incand]

Cheers! That was a good example!

 

[vanhees71]

The momentum canonically conjugated to the generalized coordinate $q^i$ is defined by
$$p_i=\frac{\partial L}{\partial \dot{q}^i},$$
and nothing else!