Generalized vs conjugate momenta

Question

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?

Answer 1

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.