# Degrees of freedom and constraints

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• Ahmed1029
In summary, when dealing with P holonomic constraints and N particles, there are 3N-P degrees of freedom and 3N-P generalized coordinates are needed for independent variation. However, in the case of non-holonomic constraints, which require more coordinates than degrees of freedom, the coordinates must adhere to the constraint and do not vary independently. This is still an area of ongoing research, with the term "Flannery nonholonomic" yielding relevant results when searched on Google.
Ahmed1029
In case of P holonomic constraints and N particles, I have 3N-P degrees of freedom and I have to look for 3N-P generalized coordinates if I want them to vary independently, but what about non-holonomic constraints? I know if I have N particles and P non-holonomic constraints, I still need 3N coordinates to completely describe the mechanical system, but do those 3N coordinates vary independently?(assume I'm talking about any set of coordinates, not just cartesian)

In systems with non-holonomic constraints, we always need more coordinates than degrees of freedom. Of course, those coordinates do not vary independently, they have to obey the non-holonomic constraint.

vanhees71 and Ahmed1029
FYI: the handling of non-holonomic constraints is still an active research area. Google for "Flannery nonholonomic" to see some relevant references.

vanhees71, andresB and Ahmed1029

## What is the concept of degrees of freedom?

Degrees of freedom refers to the number of independent variables or parameters that can vary in a system without violating any constraints. In other words, it is the number of ways a system can move or change without being restricted by its environment or other factors.

## How are degrees of freedom related to constraints?

Constraints limit the number of degrees of freedom in a system. For example, a system with a fixed length constraint can only move in two dimensions, while a system with no constraints can move in three dimensions.

## What is an example of degrees of freedom in a physical system?

An example of degrees of freedom in a physical system is a pendulum. The pendulum has two degrees of freedom, the angle at which it swings and the length of the string. These degrees of freedom determine the motion of the pendulum.

## How do degrees of freedom affect the stability of a system?

The more degrees of freedom a system has, the less stable it tends to be. This is because more degrees of freedom allow for more potential movements and interactions, making it more difficult to control and predict the behavior of the system.

## Can degrees of freedom be changed or controlled?

Degrees of freedom are inherent to a system and cannot be changed or controlled. However, constraints can be added or removed to alter the number of degrees of freedom in a system.

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