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. F

    How Does Removing Degrees of Freedom Impact Mechanical Linkages?

    When a degree of freedom exists, the object is considered free to move in this direction, so that it is not possible to transmit a force onto the object in this direction (i.e. the object slips away when push against, like on ice or ball bearings). For the object to react to a force through the...
  2. P

    In a radiation dominated era, how do I show this?

    As above. In a radiation dominated how do I show this equation? http://i.imgur.com/nMwxvD5.png (It's easier for me to post the image to imgur than use latex. I know g is the relativistic degrees of freedom. I think that the scale factor a(t) is proportional to 1/t and that a(T) is proportional...
  3. mwspice

    Lagrange's Equation with Multiple Degrees of Freedom

    Hi, I'm currently trying to learn about finding equations of motion from the Lagrange equation, and I'm a little confused about how it applies to multiple degree of freedom systems. I am using the following form of the equation with T as total kinetic energy, V as total potential energy, R as...
  4. C

    Degrees of Freedom in Physics: Theory & Solutions

    Hi, in my previous course on basic Physics we learned to solve problems concerning simple mechanical systems like this: The method consists in analyzing separately the two degrees of freedom of the system, computing for each degree the acceleration of each body (or whathever) and the sum...
  5. Benevito

    What are Valley Degrees of Freedom in Semiconductor Bands?

    Could someone please explain what valley degrees of freedom are?
  6. B

    Do rotational degrees of freedom contribute to temperature?

    I cannot find a simple answer to this question anywhere. What degrees of freedom contribute to the temperature of a gas? Let's say we have a box of ideal gas. The temperature is the average kinetic energy of the particles and only includes translational degrees of freedom: velocity. Now...
  7. P

    Counting Degrees of Freedom in Tensor

    I've been thinking about the number of degrees of freedom in a tensor with n indices in 2-dimensions which is traceless and symmetric. Initially, there are 2^{n} degrees of freedom. The hypothesis of symmetry provides n!-1 number of conditions of the form: T_{i_{1}, \ldots i_{n}}-...
  8. throneoo

    Degrees of freedom of diatomic gas

    So I'm trying to figure out the average energy of diatomic gas molecules via the equipartition theorem at high temperature (such that it's translational , rotational and vibrational modes are activated) and to do that I need to know the degree of freedom. some websites claim it would be 6 , 3d...
  9. P

    Degrees of Freedom for a Lagrangian System

    Homework Statement A long light inflexible rod is free to rotate in a vertical plane about a fixed point O. A particle of mass m is fixed to the rod at a point P a distance l from O. A second particle of mass m is free to move along the rod, and is attracted to the point O by an elastic force...
  10. gfd43tg

    Gibb's phase rule degrees of freedom

    Hello, I am watching this video to try and better understand Gibb's Phase rule The part of interest starts at 4:30 I like that he is using a visual to explain, which is very helpful. When there is 1 phase and one component, there are 2 degrees of freedom. This means two variables can be...
  11. DOTDO

    Redundant degrees of freedom in EM fields?

    If we consider E and B individually, there are 6 total degrees of freedom. But they are actually related to each other by Maxwell's equations. So we can find potentials and reduce dof to 4; 3 in vector potential and 1 in scalar potential. Thus, there remain 2 redundant dof. This is the...
  12. ChrisVer

    Relativistic Degrees of freedom g(T)

    Hi, I have one question, The rel. dofs are: g_* (T) = \sum_{i=bosons} g_i \Big( \frac{T_i}{T} \Big)^4 + \frac{7}{8} \sum_{j=fermions} g_j \Big( \frac{T_j}{T} \Big)^4 Before the neutrinos decoupling, the relativistic degrees of freedom are: g_*=g_\gamma + \frac{7}{8} (g_e + g_\nu)= 10.75 This...
  13. E

    Molar heat capacity and Degrees of freedom

    Homework Statement The diagram shows the molar heat capacity of an ideal diatomic gas and the number of degrees of freedom at different temperatures. Explain why there are 3 discrete plateaus and why the curve is smooth and leaning between them. Homework Equations - The Attempt at a Solution...
  14. T

    Exploring the Vibrational Degrees of Freedom for Diatomic Molecules

    For diatomic molecule, why is the vibrational degree of freedom equal to two at high temperature? Why not just one? Thank you very much.
  15. S

    Determining degrees of freedom

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  16. G

    Modern Physics- Degrees of freedom

    Homework Statement i have a homework set to determine the degrees of freedom for certain objects. the professor never explained how to go about figuring these out and the book doesn't quite explain it well. two problems from the set are 1. a linear spring in 3space and 2. a bead constrained to...
  17. gfd43tg

    Degrees of freedom in a reacting system

    Hello I am confused in part b why they classify solid CaCO3 and solid CaO as two different phases. They are both solids, which is one phase
  18. J

    Find molar mass and number of degrees of freedom from heat capacities.

    Hey having a bit of trouble with this one. I really don't know how to start. Think I just simply don't have the right equations or I'm using the wrong units. Hopefully you guys can help me. Homework Statement Find the molar mass and the number of degrees of freedom of molecules in a gas with...
  19. B

    Newton's Laws: Internal Degrees of Freedom

    Hello Everyone, I am currently re-reading Taylor's Classical Mechanics, in particular, section 4 in chapter 1. He is discussing Newton's Laws, and makes the statement: "A point mass, or particle, is a convenient fiction, an object with mass, but no size, that can move through space but has...
  20. K

    Counting degrees of freedom for Goldstone bosons

    I mean Goldstone bosons in the title. Sorry I don't know how to edit the title. Goldstone's Theorem says that there is a massless Goldstone mode for each breaking symmetry. For instance symmetry of a theory is broken from SU(N) to SU(N-1), the # of Goldstone bosons is (N^2-1)-((N-1)^2-1)=2N-1...
  21. R

    Degrees of Freedom: What's the Definition?

    Hello, I just want to make sure I am understanding how to define degrees of freedom of an object. If you have a rolling ball moving strictly in the x-direction, it has two degrees of freedom: one from its rotation, and one from its translation? Or is it just one DoF because its translation...
  22. E

    Degrees of Freedom: Can I Sum Individual Model DF?

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  23. A

    (Thermal) Vibrational degrees of freedom

    Let's say I have a molecule such as CO2, where there are three atoms and a linear structure. I understand that there are 3 translational degrees of freedom and 2 rotational degrees of freedom (since it's symmetric). However, the number of vibrational degrees of freedom (DoF) confuses me. My...
  24. C

    Bessel's correction and degrees of freedom

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  25. AbhiFromXtraZ

    Degrees of Freedom of Connected Compasses

    What will be the degrees of freedom of a ''Connected pair of compasses''?
  26. U

    Why Do Degrees of Freedom Differ in Chi-Squared Distributions?

    Hi, I am trying to understand the degrees of freedom parameter in the Chi_squared distribution and I have found two references from the same source that appear, to me, to contradict one-another. Can anyone explain what is going on ? In https://onlinecourses.science.psu.edu/stat414/node/171 it...
  27. B

    Equipartition Theorem Degrees of Freedom

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  28. binbagsss

    Degrees of Freedom Concepts - quick question.

    So the number of degrees of freedom of a system is defined as the number of dynamical variables required , where the definition of dynamical variables is to completely descibe the configuration -positions of parts of a mechanical system. The system I am to consider is a coin balanced on its...
  29. W

    How Does Infinite Freedom Affect Quantum Field Theory?

    Currently, I am working on Thermal Quantum Field Theory. In the introduction to that, many authors point out that infinitely many degrees of freedom and infinite volume are special. In one reference that I am reading said "The famous equivalence between the Heisenberg and the Schro ̈dinger...
  30. M

    Question about Degrees of Freedom

    Hi, I generally know the concept of degrees of freedom based on the commonly used explanation about how using the average of a sample reduces the the avaialble choices by 1. For example, I generally understand what is explained here. What is not clear to me is how having more choices...
  31. G

    Damping conditions for multi degrees of freedom

    I know that d^2<4mk for underdamped, d^2>4mk for overdamped and d^2=4mk for critically damped. This is true if there is only 1 mass and spring and damper. How to use these equations if I have 2 mass, 3 spring and 3 dampers. That is d,m,k are in 2x2 matrices. Please some one help me with this.
  32. K

    About degrees of freedom of fermions

    There are something I don't get about the degrees of freedom(dof). For massive dirac spinor, there are four complex components or 8 dofs. But for electron/position, there are only 4 dofs in total ( electron up &down, position up&down). Does it mean the equation of motion eliminate the other...
  33. JK423

    Coupling only to the center of mass degrees of freedom?

    I am interested in the following scenario. Suppose that you have a composite system, e.g. ion, atom or whatever you want. This system will have internal degrees of freedom (d.o.f.), i.e. relative positions and momenta, and also center of mass d.o.f..What i am looking for is to find an...
  34. shounakbhatta

    Lagrangian and degrees of freedom

    Hello, I have a very basic question: Degrees of freedom for a particle describes the formal state of a physical system. Like a particle in 3 dimension space has 3 co-ordinates and if it moves in 3 velocity components, then it has 6 degrees of freedom. Lagrangian also measures this, right?
  35. ajayguhan

    Degrees of Freedom for Rigid Body of n Particles

    How is for rigid body made up of n particle, the degree of freedom is 3n-nC2.i can understand that degree freedom of n particle is 3n and suppose you have 2 particle whose distance between them is fixed, then degree of freedom is n 3(2)-1=5.if we have three particles then it is 7. Therefore...
  36. ajayguhan

    Why does a fixed distance reduce degrees of freedom?

    A particle has 3 degrees of freedom, therefore N particles have 3N degrees of freedom. But a two particle whose distance between them is constant has 5 degrees of freedom instead of 6. I know that the fixed distance is constarin so it reduced the degrees of freedom, but why does the constarin...
  37. L

    Gauge fields - how many physical degrees of freedom?

    The photon field has two physical degrees of freedom (dof): its two transverse polarization directions. But what about non-abelian gauge theories? What about N massless spin-1 particles that transfom under SU(N), how do I count their degrees of freedom? Gluons, for example, are massless spin-1...
  38. marellasunny

    Number of degrees of freedom of a double toggle mechanism for punching

    This is a mechanism used for crushing rocks/punching holes(when horizontal). When I use the Kutzbach criterion to find the number of degrees of freedom,I get 2. Is this because of the failure of Kutzbach criterion to recognize the geometry of the mechanism or did I go wrong somewhere? Kutzbach...
  39. S

    Temperature changes and degrees of freedom and intermolecular forces

    Hi I have a couple of questions relating to these topic hope you guys can help :) Comparing ethanol and water, ethanol has weaker intermolecular forces than water. However, it has a greater specific heat capacity by kJ/mol.K than water. Meaning more energy would have to be absorbed per mole of...
  40. S

    Degrees of freedom of an EM field?

    Homework Statement Can anyone point me in the right direction (i.e. an explanation or resource) that shows why an EM field has two degrees of freedom (attributable to the KE and PE, due to the harmonic oscillator description of the field)? The matter is mentioned in passing in a chapter...
  41. N

    Degrees of Freedom in Linearized Gravity (and beyond)

    Dear all, This is a continuation of my previous thread, but I figure it's cleaner to start afresh with this topic. I'm trying to understand why GR has only two polarizations. I've only seen treatments of this in linearized terms, so I'll start there. The reasoning is often as follows (e.g...
  42. Z

    Understanding the Six Degrees of Freedom in Crystal Structures

    It seems to me that there should only be three degrees of freedom for each atom in a crystal, one for each direction of vibration; but apparently there are six? Can someone explain? Thanks.
  43. C

    Degrees of Freedom: Why 3/2*RT for Kinetic Energy in a Solid?

    Homework Statement I want to know why the average kinetic energy FOR A SOLID is 3/2*RT. Homework Equations For every degree of freedom = 1/2*RT Possible Degrees of freedom are: Translation, rotation, vibration The Attempt at a Solution In a solid, I am certain there is vibrational energy...
  44. nomadreid

    Confusion on application of definition of degrees of freedom

    I am confused about the counting of degrees of freedom. Yes, I know that it is the number of vectors which are free to vary. But that definition gives way to different interpretations: (1) the number of data points minus the number of independent variables. This seems to be the basis of the...
  45. J

    Degrees of Freedom: Square & Triangular Lamina

    What are the number of degrees of freedom of 1)a square sheet moving in XY plane 2)a triangular lamina moving freely in XY plane
  46. A

    Difference between Generalised co-ordinates and degrees of freedom

    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.
  47. C

    Quadratic form and degrees of freedom- fixed title

    My question is: Is there any gain in intuitive mathematical understanding of degrees of freedom from learning their expression using the 'quadratic form' and matrix algebra techniques? This sort of explanation is at least understandable and self-consistent, if not rigorous mathematically...
  48. H

    Thermodynamics of a adiabatic system with degrees of freedom

    In an adiabatic process 16 J of work are done on each mole of a gas. If the gas has 5 degrees of freedom, how much does its temperature change? Answer in terms of R I know the answer is 6.4/R (K), but don't quite sure understand the the theory or math behind it.
  49. S

    Question about counting degrees of freedom

    Suppose that for some application it is mathematically convenient to represent certain objects of interest (e.g., lines or conics) as n-dimensional vectors. That such a representation exists let's us conclude that in order to specify such an object, no more than n values are necessary. That is...
  50. D

    What Are the Degrees of Freedom for an Electron?

    HI PF, I wanted to know what are the degrees of freedom of an electron? How should it behave when it is left with its least energy in vacuum?
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