# Regarding the constraint $\omega_1 = A(t)\omega_2$ what it's nature? Holonomic or non-holonomic?

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.

• I think that you have non holonomic constraint
– Eli
Oct 12, 2021 at 14:49

$$\omega_1=A(t)\,\omega_2~,\text{or}\\ \frac{d}{dt}\theta_1=A(t)\,\frac{d}{dt}\theta_2$$

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

$$\dot\theta_1-A(t)\,\dot\theta_2=0$$

• Thank you for your considerations! Oct 12, 2021 at 15:22