Proving Euler-Lagrange for constrained system

[railblue]

Homework Statement

Given two Euler-Lagrange systems with generalized coordinates $r_1$ and $r_2,$ and input $u_1$ and $u_2$. Suppose now that a constraint is placed on them such that $r_1 = f_1(q)$ and $r_2 = f_2(q)$.

Propose a Lagrangian for the constrained system and show that is is also Euler-Lagrange

Homework Equations

Where should I even be starting on this type of proof?

The Attempt at a Solution

I do know that the generalized coordinate themselves are constrained now, but do they contribute at all to reducing the degree of freedom?

 

[mark726]

Hello, great question! Yes, the constrained generalized coordinates do contribute to reducing the degree of freedom in the system. This is because the constraint equations $r_1 = f_1(q)$ and $r_2 = f_2(q)$ effectively eliminate two degrees of freedom from the original system. This means that the Lagrangian for the constrained system must also take into account these constraints in order to accurately describe the dynamics of the system.

To propose a Lagrangian for the constrained system, we can start with the original Lagrangian for the unconstrained system and add in terms that take into account the constraints. For example, we can add a term $\lambda_1 (r_1 - f_1(q))$ to the Lagrangian, where $\lambda_1$ is a Lagrange multiplier. This term ensures that the constraint $r_1 = f_1(q)$ is satisfied in the dynamics of the system. Similarly, we can add a term $\lambda_2 (r_2 - f_2(q))$ to take into account the second constraint.

To show that this proposed Lagrangian is also Euler-Lagrange, we can use the same method as for an unconstrained system. We can derive the equations of motion using the constrained Lagrangian and show that they are equivalent to the equations of motion derived from the original unconstrained Lagrangian. This will demonstrate that the proposed Lagrangian accurately describes the dynamics of the constrained system.

I hope this helps! Let me know if you have any further questions.