What is Degrees of freedom: Definition and 174 Discussions

In various scientific fields, the word freedom is used to describe the limits to which physical movement or other physical processes are possible. This relates to the philosophical concept to the extent that people may be considered to have as much freedom as they are physically able to exercise. The number of independent variables or parameters for a system is described as its number of degrees of freedom. For example the movement of a vehicle along a road has two degrees of freedom; to go fast or slow, or to change direction by turning left or right. The movement of a ship sailing on the waves has four degrees of freedom since it can also pitch nose-to-tail and roll side-to-side. An aeroplane can also climb and sideslip, giving it six degrees of freedom.
Degrees of freedom in mechanics describes the number of independent motions that are allowed to a body, or, in case of a mechanism made of several bodies, the number of possible independent relative motions between the pieces of the mechanism. In the study of complex motor control, there may be so many degrees of freedom that a given action can be achieved in different ways by combining movements with different degrees of freedom. This issue is sometimes called the degrees of freedom problem.
In mathematics, this notion is formalized as the dimension of a manifold or an algebraic variety. When degrees of freedom is used instead of dimension, this usually means that the manifold or variety that models the system is only implicitly defined.
See:

Degrees of freedom (mechanics), number of independent motions that are allowed to the body or, in case of a mechanism made of several bodies, number of possible independent relative motions between the pieces of the mechanism
Degrees of freedom (physics and chemistry), a term used in explaining dependence on parameters, or the dimensions of a phase space
Degrees of freedom (statistics), the number of values in the final calculation of a statistic that are free to vary
Degrees of freedom problem, the problem of controlling motor movement given abundant degrees of freedom

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  1. U

    Electron and graviton degrees of freedom

    Hi dear all Please explain to a stupid dummy a very simple thing. Take an a photon in 1+3 dimensions. How DOF it has? We all know that 2. How we calculate it? a) 1) We have a spin 1 particle that should have 2s+1=3 spin state. So DOF=3. 2) We have 4 Aμ guys. One is out because of gauge...
  2. N

    Amount of data vs. degrees of freedom in fit

    Hi I am struggling with a problem here. I have 6 data points, and I have found the solution of a model which I believe should describe the behavior of the data. Now I am trying to fit the parameters of the solution to the 6 data points. The model contains 5 degrees of freedom, all of which...
  3. A

    How Do Holonomic Constraints Affect Degrees of Freedom in Lagrangian Mechanics?

    I'm starting on lagrangian mechanics and is a little puzzled by the use of generalized coordinates. Shortly, what is a degree of freedom? And what I find harder to understand, why is it that a holonomic constraint allows you to remove a degree of freedom? Consider for instance two particles...
  4. S

    Mixed symmetry property and degrees of freedom

    How can I calculate degrees of freedom of a rank (o,3) tensor, Aabc, that is mixed symmetry and antisymmetric in the first 2 indices? By mixed symmetry I mean this: Aabc+Acab+Abca=0.
  5. P

    Degrees of freedom of an oscillator in an Einstein solid

    I was reading through a book on statistical physics when i came across this sentence: "An Einstein solid has two degrees of freedom for every oscillator." How is this possible? I picture an oscillator (ex. mass on spring) to move only in one dimension, thus one degree of freedom. Where does...
  6. G

    Dimensions and Degrees of Freedom

    I had three questions that had to do with spacetime, digital physics, and the holographic principle: 1. Why are there dimensions in space? 2. Why isn't there just some sort of information matrix as suggested in digital physics? 3. How do these two ideas work with the holographic principle?
  7. N

    Symmetry breaking/ degrees of freedom

    Dear PF... Please help me with basic question more I think more I get confused... In O(n) space there are n(n-1)/2 generators... suppose I have symmetric tensor in O(n) space, it will have n(n+1)/2 independent components... and i am building invariant potential from it (quartic polynomial...
  8. C

    Thermo physics degrees of freedom question

    Homework Statement Consider a sample containing 2.35 mol of an ideal diatomic gas. (a) Assuming the molecules rotate but do not vibrate, find the total heat capacity of the sample at constant volume. nCv.= (b) Assuming the molecules rotate but do not vibrate, find the total heat...
  9. N

    Degrees of freedom - matrix diagonalization

    Hi, A symmetric 4x4 matrix has 10 independent components. Let's say that matrix describes graviton h_mu_nu. In general I can bring any symmetric matrix to diagonal form, so if can and I bring my h_mu_nu matrix to diagonal form where it has only 4 independent components ... than what happens...
  10. LarryS

    Do the degrees of freedom change for particles in a relativistic system?

    Consider one particle traveling at a relativistic velocity in 3-space. Then the configuration space of the system consisting of that one particle would have 3 degrees of freedom – 1 particle times 3 dimensions. Because of its high energy, the particle decays into, say, 2 particles. Now the...
  11. H

    E ffective relativistic degrees of freedom of the early universe

    Homework Statement Calculate the number g* of effective relativistic degrees of freedom as the universe cools through the temperature ranges (i) T > 103 GeV, (ii) 1 MeV < T < 100 MeV, and (iii) T < 0:1 MeV. Homework Equations for the equation that is required to be used look the...
  12. Z

    Degrees of freedom for a metal crystal

    Recently did an exercise calculating the theoretic heat capacity of a metal. In the solutions manual they put the degrees of freedom f=6. Why is that? I know that a rigid body has 6 degrees of freedom but does that apply to each atom in the crystal?
  13. L

    Degrees of freedom of quantum fields and elementary particles

    They say that a photon has two degrees of freedom, its two polarization states. Does that also mean that the electron has only two degrees of freedom, its two spin states? What about the frequency of a photon, is that not a degree of freedom? Or the three space directions that a electron can...
  14. A

    Degrees of freedom an constraints

    I'm not quite sure I get the idea of a degree of freedom for a system. First of all: Is there freedom in characterizing the DOF for a system - i.e. will specifying the DOF for a system relative to any coordinate system always be the same? Next let me do an example: If we have 2 particles free...
  15. A

    Degrees of freedom and triatomic gas

    i wonder how can a non linear triatomic gas molecule have 6 degrees of freedom.As per me three of those will come from translational motion,two from rotational and two from vibrational modes.so it should be 7. could you please tell me where am i going wrong. note:the triatomic gas molecule has...
  16. G

    Degrees of freedom in the energy density formula for black body radiation

    I'm looking for a conceptual explanation of a formula in Quantum Mechanics Demystified introduction. They introduce you to the theoretical black body radiation experiment, where demonstrated how a classical approach leads to the ultraviolet catastrophe. In the explanation they have the...
  17. K

    Spontaneous decrease of degrees of freedom using magnetism?

    What happens then if one has applied a static magnetic field to a heat conductor? Could the temperature increase as a result of reducing the degrees of freedom, increasing the ability for heat to be transferred from that body to others?
  18. O

    Klein Gordon eqn, decoupling degrees of freedom

    Having some trouble following my notes in QFT. Any help greatly appreciated. We have the Klein Gordon equation for a real scalar field \phi\left(\overline{x},t\right); \partial_{\mu}\partial^{\mu}\phi + m^{2}\phi = 0. To exhibit the coordinates in which the degrees of freedom decouple...
  19. C

    Degrees of Freedom Help: Comparing Fits to Data with Unknown X Values

    I'm trying to compare fits to data in a certain way. I start with the initial data and fit and compute the \chi^2 value for example: x: 1 2 3 4 y: 2.1 3.9 6.0 10.1 I get the linear fit, \chi^2 value and degrees of freedom=3. Now with what I'm working with, I don't actually know the...
  20. A

    Degrees of freedom and conversion of a system of O.D.E.'s into 1 O.D.E.

    I am wrighting a short introduction to dynamical systems, and I can't seem to understand when are we allowed to talk about the "degrees of freedom" of a dynamical system. A system of one degree of freedom can be described by a 2nd order ordinary differential equation of the form...
  21. L

    Two Degrees of Freedom for Photons - Energy & Momentum

    They say that the two degrees of freedom of a photon are its two helicity states. Why are the energy or the momentum of a photon not degrees of freedom of a photon? They can differ and they are Lorentz invariant. thanks
  22. Rasalhague

    Degrees of freedom and falling chalk

    In lecture 2 of his series on Classical Physics, Prof. Balakrishnan asks how many degrees of freedom there are in a system consisting of a falling piece of chalk. Looking at my notes, I'm not sure I understand the concept. Is the answer 6 (3 of position, 3 of orientation), or 12 (3 of position...
  23. M

    Getting from (3D gravity + local degrees of freedom) to 4D gravity

    Haelfix pointed out the paper http://arxiv.org/abs/1105.4733" , and Witten 2007 (discussed in that thread) expresses doubt that 4D gravity could be exactly solved, precisely because it has local excitations. And yet here Maloney et al have done it in 3 dimensions. Can something about their...
  24. V

    Degrees of Freedom of the Theory of Everything

    I just want to know if there is any theorem that can forbid certain degrees of freedom in the Theory of Everything. Supposed quantum mechanics and generel Relativity are just classical limits of a third theory and the theory of everything. Is it possible for applications of this Theory of...
  25. K

    Confused about the intuitive explanation of degrees of freedom

    One common explanation of the concept of D.F. is this: Suppose you have n numbers (a, b, c,...) that make up a sample of a population. You want to estimate the variance of the population with the sample variance. But the sample mean m is being calculated from these numbers, so when...
  26. K

    Components involved in degrees of freedom?

    Hi, I'm a total layperson, haven't taken a physics class since AP in high school. Trying to figure some stuff out for some writing I'm doing (poetry, bizarrely enough). So excuse if this question is super-simple or if the terms of the question are nonsensical somehow. "Degree of freedom" in...
  27. A

    Solving a Construction with 3 Degrees of Freedom

    Hey, For a school project we have a construction as indicated in the attached figure. The idea is that point A is able to move in 3 degrees of freedom, but the position, velocity and acceleration of this point is known. The length ||A-P1|| is fixed and is connected to a bell crank that is...
  28. U

    Problem understanding number of degrees of freedom

    Hi, I do a classical mechanics course, and part of it relates to degrees of freedom. For a system M of point masses with j constraints, there is N=3M-j DoF For a rigid body, I know there is 6 DoF (3 translational, 3 rotational). However, I've tried using drawing the constraints on a 4 point...
  29. M

    Degrees of freedom in the SM / MSSM?

    I'm trying to work out the number of effective degrees of freedom in the LSS (Light Stop Scenario) within the MSSM (Minimal Supersymmetric Standard Model). To make sure that I understand the concept I am trying to reproduce the number of dofs in the Standard Model. The way I count them the...
  30. P

    Question on Number of Degrees of Freedom in a Simple Structure

    I have attached the problem drawing. Please refer to it. This is a lumped mass structure and all elements have the same Modulus of Elasticity and Moment of Inertia. Determine the number of degrees of freedom of the structure So I am thinking between 2 different scenarios. 1 scenario...
  31. Z

    Degrees of Freedom of a Diatomic Gas: 5 or 7?

    The number of degrees of freedom of a diatomic gas is 5 if vibrational energy is not considered However, if we consider their vibration, total number of degrees of freedom is 7. What independent quantities do we need to specify for vibrational motion of the pair of atoms? If it is the...
  32. inflector

    What does spin degrees of freedom mean?

    What does "spin degrees of freedom" mean? In the http://en.wikipedia.org/wiki/Spin_ice" , I find: I've searched on the web for "spin degrees of freedom" and gather from the comment that this has something to do with magnetism since it appears that the sentence defines a magnet as a...
  33. pellman

    Field Lagrangians as systems with infinte degrees of freedom?

    There is nothing particular quantum about this question but I'm posting it here because I think the quantum folks are likely more familiar with the topic. Hope that's ok. There are two ways of looking at field Lagrangian densities in relation to particle Lagrangians. (1) A particle (one...
  34. V

    How Do You Calculate Degrees of Freedom in Diatomic Gases Like Oxygen?

    How do i figure out the degrees of freedom of a certain object/gas for example oxygen gas i know it's 02 two oxygen molecules - the degrees of freedom is supposibly 5 but how do you figure that out? thanks
  35. E

    Padmanabhan and spacetime microscopic degrees of freedom

    In his latest paper below, he discusses the microscopic degrees of spacetime and entropy, and how the emergent thermodynamic features of spacetime do not require details of the microscopic degrees of freedom. Still, I wonder what sort of microscopic degrees of freedom of space time would be...
  36. J

    How Many Physical Degrees of Freedom Exist in Gauge Theories?

    "physical" degrees of freedom Starting with the Lagrangian for EM, it looks like there are four degrees of freedom for the four-vector potential. But one term is not physical in that it can be expressed completely in terms of the other degrees of freedom (so it is not a freedom itself), and...
  37. T

    What is the degrees of freedom of helium gas ?

    what is the degrees of freedom of helium gas ? shouldn't helium gas be diatomic ? If so , it will be 5 . i have problems differentiating monoatomic and diatomic . Though its obvious that mono is one atom and diatomic is 2 atoms . Is it possible to know whether its mono or diatomic from...
  38. R

    Number of degrees of freedom in the sigma estimate

    Homework Statement I have estimated the standard deviation of the population of my samples from the standard deviations from each of the samples with the equation found below. And I am to construct a confidence interval for a contrast, thus I will need the number of degrees of freedom for which...
  39. S

    Degrees of Freedom total confusion

    What are degrees of Freedom in the Lagrangian/Hamiltonian formulation of classical mechanics? I've been getting very confused trying to understand this concept. Would six degrees of freedom mean up, down, left, right, back & forward? I've seen it said that 3 degrees of freedom mean the...
  40. Z

    Degrees of Freedom in Chi-square Test: n=Observed Values?

    In the Chi-square test, my textbook says that degrees of freedom are the number of independent variables minus one so df = n - 1 does this mean that that n is equal to the number of observed values from the equation aka the number of times I've added together the numbers? sum [(O-E)^2]/E Is...
  41. M

    What Are the Degrees of Freedom in Lorenz and Columb Gauges?

    My teacher told me that in columb gauge we have div A = 0 along with d/dt(Ao) = 0 where A is the vector potential and Ao is its time component , and time & space are not on equal footing , so these are two degrees of freedom and in lorenz gauge we have 4 dimensional divA = 0 where space & time...
  42. T

    Degrees of freedom and RMS-speed of CO2?

    Are the RMS-speed of gas particles related to degrees of freedom? I give up! I must consult help no matter how embarrassing it is! Any help is greeted with a big smile! :smile: Does the formula for root mean square speed of particles in a gas (below) apply for all particles...
  43. R

    Exploring the Four Degrees of Freedom of the Higgs Field

    Hi, I'm doing an essay on the Higgs boson and I'm confused about what the "four degrees of freedom" of the Higgs field are. I know that the W and Z bosons become massive by absorbing 3 of the degrees of freedom and the remaining one becomes the Higgs boson but I don't really understand what...
  44. N

    Helicity states degrees of freedom

    Hello, 1)Why does the number of helicity states depend on space-time dimension ? Masselss graviton in 4 dimensional space has 2 helicity states ( 2 degrees of freedom). In 5 dimensional space it has (still massless in 5 D) 5 helicity states (5 degrees of freedom) ... In 6 dimesional...
  45. J

    Degrees of freedom affect on heat capacity ratio

    Homework Statement Why would vibrational and rotational degrees of freedom affect the heat capacity ratio more for N2 and CO2 than He or Ar? Would this occur at room temperatures? Homework Equations d.o.f.: 3N vibrational d.o.f., linear molecule: 3N - 5 nonlinear: 3N - 6 The...
  46. P

    Over-determined systems, too many degrees of freedom. What's moving?

    Hello, consider this: http://www.alice-dsl.net/l.hansen/overdetermination.png See this image? Every point in space can be reached for the mass by the springs motion plus the springs rotation and then there's the pendulums rotation additionally. Not solvable analytically but numerically...
  47. T

    Mechanics: Degrees of freedom (extremely simple)

    Homework Statement A particle slides along a wire which is rotating at a constant angular velocity w. In polar coordinates the equation of constraint is : theta' - w = 0 OR theta - wt = 0. * What is the equation of constraint in cartesian coordinates? Homework Equations The...
  48. A

    Rotational degrees of freedom I don't get it.

    "Rotational degrees of freedom"...I don't get it. How can you determine how many rotational degrees of freedom an object or system has? For example, how many rotational degrees of freedom does a diatomic molecule have, and why? I've never been good at figuring this out.
  49. C

    Degrees of Freedom: Time an Atom/Molecule Factor?

    when we talk about degrees of freedom in atoms and molecules should time be a degree of freedom.
  50. O

    Degrees of freedom in the curvature tensor

    The Einstein field equations (EFE) in 4 dimensions have 10 degrees of freedom; The Riemannian curvature tensor in 4 dimensions has 20. If I understood this correctly, one can split up the curvature tensor and describe the remaining degrees of freedom by its traceless part, which is called the...
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