How to Treat ω in the Legendre Transform of a Lagrangian?

[Uku]

Homework Statement

I am given a Lagrangian, which, per assignment text, describes a single degree of freedom:

L= \frac{I}{2}(\dot{q}+\omega)^2-kq^2

I need to find the Hamiltonian.

Now, what I am wondering, when performing the Legrende transform:

H=\sum_{j}p_{j}\dot{q}_{j}-L(q_{j},\dot{q}_{j},t)

Do I consider \omega as velocity, eg. there are two members of the sum: \sum_{j}p_{j}\dot{q}_{j}? The assignment states one degree of freedom.. so I'm a bit insecure on that.

U.

 

[TSny]

If there is only one degree of freedom ($q$), then $\omega$ would just be a constant or some parameter. So, only one term $p\dot q$ in the transformation.