Advanced Constraints

Differential constraints and advanced restrictions. Covers: Nonholonomic Constraints.

General idea of the topic

A constraint is holonomic if it can be written as a relation between coordinates, f(q, t) = 0, and it then reduces the number of independent coordinates. A nonholonomic constraint restricts the velocities and cannot be integrated to eliminate coordinates: the wheel rolling without slipping on a plane is the classic example.

The topic shows why these constraints require their own treatment: the system reaches positions that seem forbidden, but does so along different paths, and Lagrange's equations are completed with multipliers.

What you will truly understand

By the end you will be able to distinguish a holonomic from a nonholonomic constraint, decide whether it is integrable and set up the equations of motion with Lagrange multipliers.

You will also know that the number of degrees of freedom can be smaller than the number of coordinates needed to describe the configuration.

Recommended learning route

1. Nonholonomic Constraints: constraints on velocities that cannot be integrated, examples of rolling and skating, and equations with multipliers.

Physical criterion you must acquire

The essential criterion is to check whether a relation between velocities can be integrated into a relation between coordinates. If it can, the constraint is actually holonomic; if not, multipliers appear and the constraint forces enter the equations.

Connection with real problems

• wheels, skates and skateboards.
• mobile robots and vehicles.
• balls rolling without slipping.
• mechanisms with blades or wheels.

Final goal of the topic

The final goal is to recognize nonholonomic constraints and set up the dynamics of the systems that contain them rigorously.