Definition of Kinetic energy In class we had that $ T= \frac{1}{2}T_{ij}v_iv_j$ where we used the Einstein summation convention. Hitherto we only discussed examples where the kinetic energy was dependent of the square of one coordinate and our teacher assured us that this is the case in most examples. Now I was wondering whether anybody if you knows an example where the kinetic energy depends in ONE term on different $ v_i , v_j$?
 A: The rotational kinetic energy of a rigid object is given by $ T = \frac{1}{2} \sum_{i \space j} T_{i \space j} \omega_i \omega_j $, where in this case $ T_{i \space j} = I_{i \space j} $ is the inertia tensor and the $ \omega_i $ are the components of the angular velocity vector of the rigid object.  And as pointed out by  Qmechanic in their comment, since this $ I_{i \space j} $ is symmetric it can always be diagonalized by an orthogonal change of coordinates to principal axis.  But if one does not use such coordinates it will in general have cross terms ("ONE term on different $ v_i , v_j $").

As another possibility, suppose for some reason you want to work with coordinates $ q_1 $ and $ q_2 $ defined by
$$ \begin{cases}
q_1 = x + y \\
q_2 = y
\end{cases} $$
so that
$$ \begin{cases}
x = q_1 - q_2 \\
y = q_2
\end{cases} $$
and
$$ \begin{cases}
v_x = v_1 - v_2 \\
v_y = v_2
\end{cases} $$
which gives
$$ T = \frac{1}{2}(v_x^2 + v_y^2) = \frac{1}{2}((v_1 - v_2)^2 + v_2^2) = \frac{1}{2}(v_1^2 - 2v_1 v_2 + 2v_2^2) = \frac{1}{2} \sum_{i \space j} T_{i \space j} v_i v_j $$
with $ T_{1 \space 1} = 1 $, $ T_{1 \space 2} = T_{2 \space 1} = -1 $, and $ T_{2 \space 2}=2 $.
