Why should degrees of freedom be independent?

Question

To define the position of a system of $N$ particles in space, it is necessary to specify $N$ radius vectors, i.e. $3N$ co-ordinates. The number of independent quantities which must be specified in order to define uniquely the position of any system is called the number of degrees of freedom; here, this number is $3N$. These quantities need not be the Cartesian co-ordinates of the particles, and the conditions of the problem may render some other choice of co-ordinates more convenient. Any $s$ quantities $q_1$, $q_2$, ..., $q_s$ which completely define the position of a system with $s$ degrees of freedom are called generalised co-ordinates of the system, and the derivatives $\dot q_i$ are called its generalised velocities.

The above paragraph is taken from MECHANICS book by L. D. LANDAU AND E. M. LIFSHITZ.

So, this is in general clear, but I would like to make sure I understand this correctly and that if there is more insight to gain or perspective, would be good.

Degrees of freedom should be independent, and this is beneficial for using less data to represent a physical problem or solution, so that equations could take less space and look nice. Is there another reason for it?

Answer 1

If you allow generalised co-ordinates to be dependent then you can create an infinite number of different parameters to describe a system. For example, if a system is described by two independent parameters $x$ and $y$ then you can add as many dependent parameters to the list as you like, such as $x+y$, $x+2y$, $2x+y$, $x-y$, $\frac 1 x$, $xy$ etc. etc. None of these dependent parameters give you any more information about the system than you already have if you know the values of $x$ and $y$.

On the other hand, if you restrict yourself to independent parameters, then you could replace the pair $x$ and $y$ with $x+y$ and $x-y$, or with $\frac 1 x$ and $\frac 1 y$, or with $x^3$ and $y^3$ etc., but you will always have exactly two independent parameters that fully describe the system, because this is the dimension of its configuration space.

Answer 2

Contrary to what you say, independent degrees of freedom are not

beneficial for using less data to represent a physical problem or solution

A priori, every quantity you can attribute to a system has to be safely assumed to be an independent degree of freedom (of which purely mechanical DOF are only a subset).

Only after you understand the equations that the system is subject to (also considering certain approximations), can you tell if some of the degrees of freedom are dependent of others. Hence, the term "independency" only makes sense if you have already learned that there are also "dependencies" among physical quantities.

Consider the two degrees of freedon $x$ and $z$ of a gravitational pendulum (a point mass hanging on a rigid rod that is fixed to some point, say the coordinate origin). If you are an alien coming from the Alpha Centauri system and look at a still photograph of the physical pendulum, you might say: "yeah, the mass clearly has two independent degrees of freedom, $x$ and $z$". But then you suddenly happen to find a movie of the pendulum in motion, and now you say: "oh, I was wrong, the coordinates are dependent, and also obey the equation $x^2+z^2=L^2$". You then notice that, instead of describing the state by the two variables $x$ and $z$, you could also describe it by a single variable, the angle $\phi$ between the rod and the vertical direction.

So, actually it's the additional constraint equations which are

beneficial for using less data to represent a physical problem or solution

in the case of dependent variables. The more dependencies you are able to find, the more economical becomes the description of the system (because you need less information to characterize its state). Contrary to that, maximum independency is always a nightmare for physics, because we could not predict anything in nature then. In fact, all we do in physics is find dependencies (algebraic or algebro-differential, or yet more generally, statistical ones).

Answer 3

It ought to be obvious from Newtonian Mechanics that dof (degree of freedom) are useful. And that each degree is neccessarily independent. For example:

• A particle on a line has just 1 dof to move in.

• A particle on a plane has just 2 dof to move in.

Etc etc.

dof capture an important physical aspect of the model at hand. It's not simply to reduce the number of equations or make them look nice. Though if that turned out to be the case, as it often is, is great.

The examples above shows dof is tied up with the dimensionality of space. Now, space is usually modelled as a vector space and there dimension is determined by the cardinality of a basis. And a basis is given by a set of spanning and linearly independent vectors.

This is the mathematical definition. The physical language of dof is intuitive and is obviously physical because it talks about motion (a particle is free to move to a certain degree).

We can go further and curve space. These are usually called manifolds. And here again we have dimension. But this can be referred back to the preceding because a small patch of space will be relatively flat and so can be modelled by a vector space amd its dimension will be the dimension and hence the dof of that space at that point.