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The definition I come up with is $3M - N$ ...where $N$ is the number of constraints. I assume $M$ is the number of distinct points. In what context is it used?

According to Wiki it says "an independent parameter" So then $f(x) =y$ has no degrees of freedom since $y$ depends on $x $? What is wiki trying to say?

Is the concept useful in building objects or theoretical work? Perhaps something Lagrange invented? Maybe a useful problem solved by this concept would help?

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Degrees of freedom can be defined as the number of independent ways in which the space configuration of a mechanical system may change.

Suppose I place an ant on a table with the restriction that the ant can move only through a tube on a line along x-axis. Then the ant will have only one degree of freedom in three dimensional space.

However if I allow the ant to move freely on the table, then it can be at any point on the surface at any time $t$ and it can change its $x$ as well as $y$ coordinate as time evolves. Then one can say the ant has two degrees of freedom. Thus the number of independent coordinates does define the configuration, and the degree of freedom is the count of that number.

However if I place an ant which has wings, then it can travel independently in $x$, $y$ and $z$ direction, and its position can be located at a point $P(x,y,z,t)$ at any instant, so now it has three degrees of freedom as it can be located by three independent variables.

To sum up:

  • A material particle confined to a line in space can be displaced only along the line, and therefore has one degree of freedom.

  • A particle confined to a surface can be displaced in two perpendicular directions and accordingly has two degrees of freedom.

  • A free particle in physical space has three degrees of freedom corresponding to three possible perpendicular displacements.

Now suppose I have two ants with wings then this system has three coordinate each and can be located by six independent variables. In this case, the degree of freedom = no. of particles x3 = 3N degree of freedom.

A system composed of two free particles has six degrees of freedom, and one composed of $N$ free particles has $3N$ degrees.

If a system of two particles is subject to a requirement that the particles remain a constant distance apart, the number of degrees of freedom becomes five.

Imagine those two ants bound by a string such that their distance apart is constant $L$ . An equation of the type $F(x,y,z,x',y',z',t) =L$ governing this condition will hold. This equation is called equation of constraint and each constraint can reduce the degrees of freedom by one. Therefore the constraint system of $N$ particles will have a no. of degrees of freedom = $3N -m$, where $m is the number of constraining equations operating on the system.

Any requirement which diminishes by one the degrees of freedom of a system is called a holonomic constraint.

Each such constraint is expressible by an equation of condition which relates the system's coordinates to a constant, and may also involve the time.

When applied to systems of particles, a holonomic constraint frequently has the geometrical significance of confining a particle to a specified surface, which may be time-dependent.

Constraints are defined as restrictions on the natural degrees of freedom of a system.

If $n$ and $k$ are the numbers of the natural and actual degrees of freedom, the difference $n − k$ is the number of constraints.

A pendulum ball is a body having 3 degrees of freedom. When one hangs the bob by a string of length $l$, then it can only move in a plane with the condition that $x^2 + y^2 = l^2$ . This is one constraining equation, and another one is that z = constant, or $z-c =0$. Now the bob has only one degree of freedom and can be defined with only one coordinate, say $\theta$, the angle made by the string with the vertical.

So the simple pendulum has only one degree of freedom. The advantage of the above description is that one can go to an independent set of coordinates to describe the motion of the system. Such sets are called generalized coordinates.

Is the concept useful in building objects or theoretical work? Perhaps something Lagrange invented ?? Maybe a useful problem solved by this concept would help ?

The new description in terms of 'generalised coordinates' and momenta leads to further development in Lagrangian mechanics by using the 'principle of virtual work' and developing a set of equations of motion in generalised coordinates and velocities, which are an independent set and free from the constraining forces, in terms of the evolution of kinetic and potential energies only.

One can check:

Reference:http://theory.tifr.res.in/~sgupta/courses/cm2011/hand2.pdf

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"the minimum number of independent coordinates that can specify the position of the system completely." -- wiki.

In your example, with $y=f(x)$, if we specify either $x$ or $y$ than we can deduce the other (assuming $f$ is invertible), so we say the system has one degree of freedom. This might be a useful abstraction if $x$ and $y$ are the coordinates of a roller coaster on a track, for example, and the curve $y=f(x)$ describes the shape of the track. Then we could define a single parameter (let's call it $\gamma$), to signify distance traveled along the curve, and create the functions $X$ and $Y$ such that $x = X(\gamma)$ and $y=Y(\gamma)$. Note that we can completely describe the state of the system with any one of the three variables $x$, $y$, and $\gamma$. Another way of putting this is that the state space of the roller coaster is a one dimensional surface in a three dimensional space (where the dimensions correspond to $x$, $y$, and $\gamma$).

Degrees of freedom frequently come up in statistical mechanics. As a first example, the macroscopic description of an ideal gas requires two variables, usually chosen from the three canonical state variables (temperature, volume, and pressure). Those three variables are related by the equation $PV = Nk_BT$, where $N$ is the number of molecules and $k_B$ is Bolzmann's constant. So the abstract ideal gas, considered as a thermodynamical system at equilibrium, has three degrees of freedom. Here is a visualization of the two-dimensional ideal gas surface in a three dimensional space:enter image description here

On a microscopic level, a gas has many more degrees of freedom. Each molecule has a position which is specified by three coordinates and a velocity specified by three additional coordinates, so there are at least six degrees of freedom per molecule. Furthermore, the molecules can have angular moments and vibrational modes, adding additional degrees of freedom. In a way the object of statistical mechanics is to take a system with a very large number of degrees of freedom and give a statistical description of it with few degrees of freedom. Note that the ideal gas abstraction is only able to predict aggregated statistical properties of a gas, and cannot tell us about the individual molecules, as it contains nowhere near enough information in its two-dimensional state space.

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    $\begingroup$ If I could pick both answers I would. I had to pick one or the other and up voted both but in reality both are excellent and are deserving of acknowledgement ...each has a perspective and merit. The thermodynamic aspect I would not have guessed. $\endgroup$ – Sedumjoy Jul 10 '18 at 4:19

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