Explicit non-holonomic equations of motion

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• andresB
In summary, in the holonomic case, the Lagrangian can be used in the Lagrange equations to find the explicit equations of motion. However, for the non-holonomic case, the equations must be solved together with the non-holonomic constraint equations. While specific examples can be found in standard books, it is surprising that there is no general explicit formula available.
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?

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!

topsquark
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.

1. What is the difference between explicit and non-holonomic equations of motion?

Explicit equations of motion are those in which the variables and their derivatives are explicitly stated, while non-holonomic equations of motion involve constraints on the motion of a system, such as friction or other external forces.

2. How are explicit non-holonomic equations of motion derived?

Explicit non-holonomic equations of motion are derived using the principles of classical mechanics, such as Newton's laws of motion and the principle of least action.

3. What types of systems can be described using explicit non-holonomic equations of motion?

Explicit non-holonomic equations of motion can be used to describe a wide range of systems, including mechanical systems, electrical circuits, and fluid dynamics.

4. What are some applications of explicit non-holonomic equations of motion?

Explicit non-holonomic equations of motion have many practical applications, such as in robotics, vehicle dynamics, and control systems.

5. Are there any limitations to using explicit non-holonomic equations of motion?

While explicit non-holonomic equations of motion are powerful tools for describing complex systems, they may not be suitable for all situations. They may not accurately capture all aspects of a system's behavior, and may require simplifications or assumptions to be made.

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