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Aniket1
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What is the difference between Generalised co-ordinates and Degrees of freedom in classical mechanics? I know that they are not equal when we have non-holonomic equations of constraints. But I don't know why.
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.
Generalised coordinates are variables that describe the position and orientation of a system, while degrees of freedom refer to the number of independent parameters that are needed to fully specify the configuration of the system.
Generalised coordinates and degrees of freedom are closely related as the number of generalised coordinates needed to describe a system is equal to the number of degrees of freedom of that system.
No, a system cannot have more generalised coordinates than degrees of freedom. This is because each generalised coordinate represents an independent variable, and if there are more variables than degrees of freedom, the system would be overconstrained.
Generalised coordinates are independent of the coordinate system used, while degrees of freedom can vary depending on the choice of coordinate system. However, the total number of generalised coordinates and degrees of freedom will always remain the same.
Generalised coordinates and degrees of freedom are important in physics as they help in simplifying the mathematical description of complex systems. They allow for a more efficient analysis of the system's dynamics and help in finding the equations of motion for the system.