# What is the meaning of nonholonomy in a system?

• indianaronald
In summary, the term "nonholonomy" in the context of mechanical systems refers to the type of constraints the system has. If the constraints are functions of coordinates only, the system is considered holonomic and is easier to analyze. However, if the constraints cannot be expressed solely through coordinates and also involve velocity, the system becomes non-holonomic and more complex to study. This can be seen in the example of a ball rolling on a plane surface without slipping. The book "Classical Mechanics" by Goldstein, specifically section 1.3, provides a detailed explanation of this concept.
indianaronald
I keep coming across this term and I cannot understand what this means pertaining to a mechanical system. I'm working on spherical robots and their control and there is mention of nonholonomy in the control of spherical robots. I googled it but I couldn't find a clear starting point to start reading. Someone point me to the fundamentals or something from where this starts?

If I remember correctly, it has to do with the kind of constraints the system has. If the constraints are functions of coordinates only, the system is holonomic. This usually means the system is "simple" to analyze (mass point on a circle).

If the constraints cannot be expressed via coordinates only, but are functions of velocity or even worse, the system is non-holonomic. Then we expect the system to behave in am more complex way. (ball rolling on plane surface without slipping).

https://en.wikipedia.org/wiki/Holonomic_constraints
https://en.wikipedia.org/wiki/Nonholonomic

Jano L. said:
If I remember correctly, it has to do with the kind of constraints the system has. If the constraints are functions of coordinates only, the system is holonomic. This usually means the system is "simple" to analyze (mass point on a circle).

If the constraints cannot be expressed via coordinates only, but are functions of velocity or even worse, the system is non-holonomic. Then we expect the system to behave in am more complex way. (ball rolling on plane surface without slipping).

https://en.wikipedia.org/wiki/Holonomic_constraints
https://en.wikipedia.org/wiki/Nonholonomic

yeah..but I need to understand this fully and properly. I read the wiki page and like all wiki pages it's qualitative at best. I was hoping for some book or topic which covers this extensively. But why is a ball on a plane complex? I don't understand that all.

But why is a ball on a plane complex? I don't understand that all.
Because non-holonomic constraint cannot be removed by coordinate transformation and elimination of the constraint variable. To see the details, try to get Goldstein's textbook, sec. 1.3 - he explains this nicely.

Jano L. said:
Because non-holonomic constraint cannot be removed by coordinate transformation and elimination of the constraint variable. To see the details, try to get Goldstein's textbook, sec. 1.3 - he explains this nicely.

Thank you. That is exactly what I was looking for.

indianaronald said:
Thank you. That is exactly what I was looking for.

what is the book called?

Kidphysics said:
what is the book called?

classical mechanics by Goldstein. Google it. It's available on scribd.

## 1. What is nonholonomy in a system?

Nonholonomy in a system refers to the phenomenon where the motion of a system is restricted or constrained by its structure or geometry, rather than just the forces acting on it. This means that the system's behavior cannot be fully explained by its dynamics or equations of motion alone.

## 2. How is nonholonomy different from holonomy?

Holonomy and nonholonomy are two complementary concepts in mathematics and physics. Holonomy refers to the complete description of a system's behavior based on its dynamics and forces, while nonholonomy refers to the limitations or constraints on that behavior due to the system's geometry or structure.

## 3. What are some examples of nonholonomic systems?

Some examples of nonholonomic systems include rolling objects, such as a ball or wheel, where the direction of motion is constrained by the surface it is rolling on. Other examples include systems with constraints, such as a double pendulum or a car driving on a curved path.

## 4. How does nonholonomy affect the stability of a system?

Nonholonomy can have a significant impact on the stability of a system. In some cases, it can lead to unstable or chaotic behavior, as the system is limited in its range of possible motions. However, in other cases, nonholonomy can actually improve stability by preventing the system from entering unstable states.

## 5. How is nonholonomy relevant in real-world applications?

Nonholonomy has many practical applications in fields such as robotics, control theory, and mechanics. Understanding nonholonomic systems allows us to design more efficient and stable machines and predict their behavior in various environments. It also plays a crucial role in the development of self-driving cars and other autonomous systems.

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