A Explicit non-holonomic equations of motion

AI Thread Summary
In the discussion on explicit non-holonomic equations of motion, participants explore the challenges of formulating these equations compared to holonomic cases. It is noted that while Lagrange multipliers are essential for addressing non-holonomic constraints, existing literature, including Landau and Lifshitz, may not provide comprehensive guidance. The focus is on finding a general form for the equations of motion, as non-holonomic constraints are local and cannot be resolved by merely selecting independent coordinates. Participants express a desire for a more explicit formulation beyond the standard Euler-Lagrange equations. The conversation emphasizes the complexity of integrating non-holonomic constraints into the motion equations.
andresB
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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
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Are there such general equations for the non-holonomic case?
 
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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!
 
vanhees71 said:
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.
 
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.
 
vanhees71 said:
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.
 
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