The discussion focuses on the differences between generalized coordinates and degrees of freedom in classical mechanics, particularly in the context of holonomic and non-holonomic constraints. Participants explore the implications of these concepts for systems of particles and the equations of motion (EOM).
Participants express differing views on the relationship between generalized coordinates and degrees of freedom, particularly in non-holonomic contexts. The discussion remains unresolved regarding the specific reasons for the differences in these concepts.
Some participants reference the need for solvable equations to determine the number of coordinates required, while others highlight the role of constraint forces and the implications of rigid body assumptions. The discussion includes references to specific examples and previous threads that elaborate on non-holonomic systems and integrability.
Studiot said:In general for holonomic constraints, g = 3N-c
Where g is the number of independent generalised coordinates sufficient to specify (mechanically) the system of N particles and c is the number of independent constraint equations.
If the constraints are not specified as equations then the above is non holonomic and the above does not hold. Further conditions of constraint are needed to specify the system.
A simple example is a system of particles constrained to the surface of a sphere
R2=x2+y2+z2
This is holonomic.
But if the particle is constrained to the atmosphere and space above the sphere the system is not.
R2<x2+y2+z2
ZealScience said:Firstly, degrees of freedom is the NUMBER of general coordinates required to describe the EOM. Simply saying "Generalised coordinates" is a bit vague.
Secondly, not all the coordinates are required. Under most instances, we assume that bodies are rigid, meaning that particles are constrained to each other, and we only need to consider the body as a whole. Otherwise we would need a set of coordinates to describe each molecule (if the molecules are rigid of course).
In addition to that, there are constraint forces which furthermore eliminates degree of freedom. (e.g if a particle is on the table, with no extra vertical forces other than gravity, then you only need two coordinates rather than three, when the particle is free to move everywhere).
Usually, just try to count how many general coordinates you need at least to describe the free body, then take away the holonomic constraints, as you have mentioned, so that you don't have to deal with number on the order of Avogadro constant.
Studiot said:That is a slightly different question from the one you originally asked. I did wonder if that was what you really meant.
Start with the 3N degress of freedom (in 3D), which I assume you understand.
That means you need 3N pieces of information (the value of 3N variables) to fully determine the system.
If you can write 3N simultaneous equations or otherwise specify these values eg by the old standby x3 = 0 (given) then you can solve the system.
If you can introduce other equations from other sources then you can reduce the numberrequired from 3N.
But they have to be solvable equations, simultaneous with the rest of the set.
An inequality, such as I gave in my second previous example, does not satisfy this requirement and so does not qualify in the reducing set.
It is, nevertheless, a constraint condition.
So the equation that I originally offered could be rephrased:
The number of coordinates required = 3N minus the number of other equations you can dig up.
andrien said:here in this thread I have explained some issues regarding non holonomicity and non integrable relation
https://www.physicsforums.com/showthread.php?t=647529&highlight=holonomic
non holonomic system contain atleast one non integrable relation.These relations reduces the degree of freedom owing to non integrability.The example of a sphere on rough surface in the thread provided above will illustrate it.