# Equations of motion with Euler-Lagrange equations

I try to determine the equations of motion of the system defined by the lagrangian:

$$L=\frac{1}{2}m(\overset{.}{x}^2+\overset{.}{y}^2)+\frac{1}{2}I\overset{.}{\theta}^2.$$

I found that the system operates under the following constraint: $$\omega=\sin \theta dx-\cos\theta dy .$$

To do this I think I can use the Euler-Lagrange equations to solve:

$$\frac{d}{dt}\frac{\partial L}{\partial \overset{.}{z}} - \frac{\partial L}{\partial z}=\lambda\omega$$

where $$z=(x,y,\theta)$$.

But I don't understand how to solve these. I also wonder if I can transform this relation into the following way

$$\frac{d}{dt}\frac{\partial \mathscr{L}}{\partial \overset{.}{z}} - \frac{\partial \mathscr{L}}{\partial z}=\lambda\omega$$

with $$\mathscr{L}=L-\lambda\omega$$.

Can someone give me some hints?

Have a nice day!

• Yes, thanks for this remark, I edited my post Commented Apr 21, 2021 at 6:49

I'm guessing the configuration space is $$\Bbb R^2 \times \Bbb S^1$$ and you're modelling the motion of an ice skate with the non-holonomic constraint $$\ker \omega$$. The perfect reaction force will be a multiple of $$\omega$$ itself (by a dimension count), say $$\mathcal{R}=\lambda \omega$$. There is no potential energy, so the external force of the full system vanishes. So the equations of motion become just $$\mu\left(\frac{D\dot{c}}{dt}\right)=\lambda(c(t))\omega_{c(t)}.$$But the left side equals $$\left(\frac{\partial L}{\partial x} -\frac{d}{dt}\frac{\partial L}{\partial \dot{x}}\right)\,dx+\cdots \mbox{similar terms with y and \theta},$$and you know what the right side is. Equate the coefficients of $$dx$$, $$dy$$ and $$d\theta$$ on both sides to get a $$3\times 3$$ differential system. You won't even need to find $$\lambda$$. See section 5.4 of the book by Godinho & Natario for more details.
1. OP's differential constraint is a semi-holonomic constraint, which can equivalently be written as $$f~\equiv~\dot{x}\sin \theta -\dot{y} \cos\theta~=~0.\tag{1}$$
2. The Lagrange equations become $$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}^j}-\frac{\partial L}{\partial q^j}~=~ \lambda \frac{\partial f}{\partial \dot{q}^j} ,\tag{2}$$ cf. e.g. this Phys.SE post. Eq. (2) is equivalent to the last equation in Ivo Terek's answer (v1).
3. Warning: Note that the action $$S[q,\lambda]=\int\! dt ( L-\lambda f)$$ leads to wrong EOMs different from the correct eq. (2)!
• Your third point triggered one more question. Thanks to the answers I found the following equations of motion : $\theta(t) = \theta_0 + Kt$ $x(t)=Rsin(\theta_0 + Kt) + K_1$ $y(t)=-Rcos(\theta_0 +Kt) + K_2$ Now, how can I get the equation of the sub-Riemannian problem $$\int_0^T L(t)dt -> min$$ with the constraint $z(t)\in\mathscr{D}(z(t))$? Commented Apr 22, 2021 at 13:12