Holonomic constraints and non-holonomic system

In summary, holonomic constraints are restrictions that can be expressed as equations involving the coordinates and time, allowing the system to be reduced to a smaller number of independent variables. These constraints are integrable and can be derived from a potential function. In contrast, non-holonomic systems have constraints that depend on the velocities and cannot be expressed solely in terms of the coordinates, often leading to more complex dynamics. Non-holonomic constraints are typically non-integrable and represent conditions that cannot be simplified into position-based equations, influencing the system's motion and behavior in unique ways.
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Homework Statement
A disk of uniform mass density, mass M, and radius R sits at rest on a frictionless floor. The disk is attached to the floor by a frictionless pivot at its center, which keeps the center of the disk in place, but allows the disk to rotate freely. An ant of mass m ##\ll## M is initially standing on the edge of the disk; ou may give your answers to leading order in m/M.

The ant walks an angular displacement ##\theta## along the edge of the disk. Then it walks radially inward by a distance h ##\ll## R, tangentially through an angular displacement −##\theta##, then back to its starting point on the disk. Assume the ant walks with constant speed v.

Through what net angle does the disk rotate throughout this process, to leading order in h/R?
Relevant Equations
The disk will rotate $$\frac{4 m h \theta}{M R}$$
The solution is given. What makes this solution unique is that there is a net turn for the disk. The note of the solution mentions this system is non-holonomic. My question is that are there other non-holonomic examples. What makes this particular set up non-holonomic? Thanks!
 
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Not sure about your holonomics question, but the process is essentially the same as for cat dropped upside down. It spreads out its back legs, pulls in its front legs, and twists. The back legs having greater MoI twist through the smaller angle. It can then swap over the leg postures and twist the other way.
Net result, cat turns in mid air.
Astronauts use the same trick.
For the ant on the disc, it could walk in small circles near one edge of the disc and the disc would gradually rotate the other way.
 
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That's right, the cat example is also non-holonomic. I think I am starting to get the idea. Looks like I would need to work through the Pfaffian form of the system to rigorously show it's non-holonomic. Any system where the constraint ##f(x_i, t) = 0## is not integrable is non-holonomic because such system's evolution through states depends on the path of evolution.
 

FAQ: Holonomic constraints and non-holonomic system

What are holonomic constraints?

Holonomic constraints are restrictions on a system that can be expressed as equations involving the coordinates and time. These constraints reduce the number of degrees of freedom of the system. They are integrable and can be written in the form \( f(q_1, q_2, ..., q_n, t) = 0 \), where \( q_i \) are the generalized coordinates.

What are non-holonomic constraints?

Non-holonomic constraints are restrictions on a system that cannot be expressed solely as functions of the coordinates and time. They often involve inequalities or differential equations that cannot be integrated to yield a constraint equation in terms of the coordinates alone. An example is the rolling without slipping condition of a wheel, which involves the velocities of the contact point.

How do holonomic and non-holonomic constraints affect the degrees of freedom of a system?

Holonomic constraints reduce the number of degrees of freedom by providing explicit equations that the coordinates must satisfy. Non-holonomic constraints, on the other hand, limit the motion of the system without necessarily reducing the number of degrees of freedom in a straightforward manner. They impose conditions on the velocities or higher derivatives, which can complicate the analysis of the system's motion.

Can you provide an example of a system with holonomic constraints?

An example of a system with holonomic constraints is a double pendulum. The lengths of the rods provide constraints that can be written as \( l_1^2 = x_1^2 + y_1^2 \) and \( l_2^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 \), where \( l_1 \) and \( l_2 \) are the lengths of the rods, and \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the pendulum bobs.

What is an example of a non-holonomic system?

An example of a non-holonomic system is a wheeled robot that must roll without slipping. The constraint involves the velocities of the wheels and the body of the robot, and it can be expressed as \( v_x - r\omega = 0 \), where \( v_x \) is the linear velocity of the robot, \( r \) is the radius of the wheel, and \( \omega \) is the angular velocity of the wheel. This constraint cannot be integrated into a simple equation involving only the coordinates.

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