Understanding Conjugate Momentum: Exploring the Nomenclature

[snatchingthepi]

So I'm studying for my CM final and in looking over the section on ignorable coordinates and generalized momenta I come across the definition for conjugate momentum $$p_{i} = \frac{\partial L}{\partial \dot{q_{i}}}$$

My question is simple - why is it called 'conjugate'? I just never really thought to question to the name.

 

[jedishrfu]

The best I can come up,with that since the momentum is paired with the generalized coordinates of the system then conjugate is used to describe that fact.

http://www.merriam-webster.com/dictionary/conjugate

 

[Matterwave]

You need a word to differentiate it from a physical momentum, I guess "generalized momentum" might also work, but the "conjugate" part comes into play more when you move to Hamilonian mechanics where your phase space coordinates are these (p,q) conjugate pairs.

 

[snatchingthepi]

Thank you both.

 

[PJC01]

The term "conjugate" in conjugate momentum refers to the relationship between the generalized coordinates q and the corresponding generalized momenta p. This relationship is described by Hamilton's equations, which state that the time derivative of q is equal to the partial derivative of the Hamiltonian with respect to p, and the time derivative of p is equal to the negative partial derivative of the Hamiltonian with respect to q.

This relationship is known as a "conjugate" relationship because the two quantities, q and p, are dependent on each other and cannot be considered separately. This is similar to the concept of conjugate variables in mathematics, where two variables are related in a way that makes it impossible to change one without affecting the other.

In the context of classical mechanics, the term "conjugate" also refers to the fact that the product of the generalized coordinate and the corresponding generalized momentum is equal to the canonical momentum, p_{i}q_{i}, which is a conserved quantity in Hamiltonian systems.

In summary, the term "conjugate" in conjugate momentum highlights the interdependent and complementary nature of the generalized coordinates and momenta in classical mechanics.