Given a rotating masses system with lagrangian
$$ L = \frac 12 J_1\omega_1^2+\frac 12 J_2\omega_2^2 -\frac 12 C(\theta_1-\theta_2)^2 $$
where $\omega_i = \dot\theta_i$ assuming the constraint $\omega_1 = A(t)\omega_2$. What is the nature of this constraint? If holonomic it can be integrated into the lagrangian as $L_{\lambda}=L+\lambda(\omega_1-A(t)\omega_2)$ but I am not quite sure about it's nature. Any help will be appreciated.
your constraint equation is:
$$\omega_1=A(t)\,\omega_2~,\text{or}\\ \frac{d}{dt}\theta_1=A(t)\,\frac{d}{dt}\theta_2$$
multiply by dt and integrating
$$\theta_1=\int A(t)\,d\theta_2$$
for holonomic constraint you expected $~\theta_1=A(t)\,\theta_2$ but this is not the case.
hence this constraint is non holonomic.
$$2\,L=J_1\,\dot{\theta}_1^2+J_2\,\dot{\theta}_2^2-C\,(\theta_1-\theta_2)^2 $$
and the non holonomic constraint
$$ \dot\theta_1-A(t)\,\dot\theta_2=0$$
