Explicit non-holonomic equations of motion

[andresB]

In the holonomic case, we can put the Lagrangian in the Lagrange equations to obtain the explicit form of the equations of motion. From Greenwood's classical dynamics book, the equations are

Are there such general equations for the non-holonomic case?

 

[vanhees71]

Have a look in Landau Lifshitz vol. 1, who gets the non-holonomic constraints right. You have to introduce Lagrange multipliers in the right way!

 

[andresB]

Have a look in Landau Lifshitz vol. 1, who gets the non-holonomic constraints right. You have to introduce Lagrange multipliers in the right way!

Long time without reading Landau, and I have to say that its treatment of the non-holonomic constraint seems disappointingly scarce.

In any case, I'm not looking for the Euler-Lagrange+ lagrange multipliers equations, they are in every book. Instead I'm lookinf for the final general form of the equation of motion.

 

[vanhees71]

But these are the final general form of the equation of motion. Non-holonomic constraints are local constraints, and you cannot satisfy them by simply choosing a set of independent coordinates as for holonomic constraints.

 

[andresB]

But these are the final general form of the equation of motion. Non-holonomic constraints are local constraints, and you cannot satisfy them by simply choosing a set of independent coordinates as for holonomic constraints.

But the Lagrange equations are just a step in the final solution of the problem. They have to be solved togheter with the non-holonomic constraint equations. I know how to do it in specific examples that can be found in the standard books, but I would be surprised if no general explicit formulat exist.