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. K

    Degrees of freedom (question based on an interview I attended)

    I was asked by an interviewer the number of degrees of freedom (in both translation and rotational senses) this part has with respect to each axis. Indeed I can share what I think of here, but I want to start it fresh and correct. If you were the interviewee, what would have been your answer and...
  2. Alex8932

    Why does a wrist have 3 degrees of freedom?

    The other two movements make use of the wrist and can't be done without it so I see why they are considered to be degrees of freedom of the wrist.
  3. Fractal matter

    A Two degrees of freedom in GR after boundary conditions specified?

    I read there are 2 degrees of freedom in GR after boundary conditions specified. Does that mean 2 equations are enough for EFE equivalent? Those two seem like the amplitude and a phase.
  4. L

    Degrees of freedom and holonomic constraints

    Hi, I am not quite sure if I have solved task 2a and 2b correctly. For task 2a I would say, because of the constraints, that the system has only 1 degree of freedom. Since the vectors must always have an angle of Pi/4 to each other, this would mean that if one vector moves up, the other must...
  5. Ahmed1029

    I Constraints of a mechanical system

    I'm studying theoretical mechanics and I kind of find the notion of a "mechanical system" very slippery, especially when it comes to constraints. Take an example : I know that when a system consists of N particles and p constraints, it has 3N-p degrees of freedom; this is the definition. Then I...
  6. Ahmed1029

    I Degrees of freedom and constraints

    In case of P holonomic constraints and N particles, I have 3N-P degrees of freedom and I have to look for 3N-P generalized coordinates if I want them to vary independently, but what about non-holonomic constraints? I know if I have N particles and P non-holonomic constraints, I still need 3N...
  7. H

    I Dimensions vs. degrees of freedom

    Are dimensions the same thing as degrees of freedom? Would you say that a circle is a one dimensional object embedded in a two dimensional space?
  8. Alfredomaximun

    How can the degrees of freedom of a mechanism be obtained?

    Summary: ##M=3\left(n-1\right)-2j_1-j_2## Hi, I'm trying to get the degrees of freedom of a can crusher. So substituting I get ## \begin{array}{l}M=3\left(n-1\right)-2j_1-j_2\\ n=5\\ j_1=5\\ M=3\left(5-1\right)-2\cdot 5-0\\ M=2\end{array} ## And I would think it would be 1
  9. Like Tony Stark

    Degrees of freedom of "simple bicycle"

    Hi I thought that a bike could be simply modeled as two wheels attached by a rigid bar. If the wheels move without sliding, then there is one degree of freedom: one of the wheels moves and so does the other one since they are rigidly attached by the bar. Then, if the wheels can turn to the right...
  10. Kaguro

    Degrees of freedom with a particle and a rod

    The rod itself should have 3 translational+2 rotational DOF. The particle on top of the rod has one additional DOF. So total should be 6. But answer given is 4. What I'm thinking wrong?
  11. W

    I What do physicists mean by "local degrees of freedom"?

    When physicists talk about a theory having local degrees of freedom, what is exactly meant by that statement? What are examples of theories with local degrees of freedom and what are examples of theories with no local degrees of freedom?
  12. A

    A The chi square goodness-of-fit test with no degrees of freedom left

    I have an empirical frequency distribution as for example below: ##f_{2} = \, \, \, 21## ##f_{3} = 111## ##f_{4} = \, \, \, 24## The theoretical distribution is determined by two parameters. So for a chi-square goodness-of-fit test there are actually no degrees of freedom left. Yet the...
  13. S

    I What is the Dimension of SO(3) with Constraint det(O) = 1?

    The group ##\rm{O(3)}## is the group of orthogonal ##3 \times 3## matrices with nine elements and dimension three which is constrained by the condition, $$a_{ik}a_{kj} = \delta_{ij}$$ where ##a_{ik}## are elements of the matrix ##\rm{A} \in O(3)##. This condition gives six constraints (can be...
  14. LCSphysicist

    Degrees of freedom in a molecule

    I am not understand the solution of the letter b and c given by the author. Since Cv = NfK/2, we have Cv/N = fK/2. Now, the degree of freedom of a N linear molecule is 3(trans) + 2(rot) + x, where x is the degree of freedom due the vibrational motion. I am having trouble to calc x, could you...
  15. LCSphysicist

    Degrees of freedom of a water molecule

    Well, first of all is really good to say that we don't can appeal to quantum mechanics... So, i can see: Three degree of freedoms in translation on a space xyz +3 degree Three normal modes of vibration, but each normal mode has with it a potential and vibrational kinetic energy, so +6 degree...
  16. H

    A Degrees of freedom of thin rods

    (Note: I had this question posted at the intermediate level of difficulty for 11 days, but got only one, cryptic (to me) response that was rather quickly removed. So, I figured perhaps it's actually an advanced question, requiring more than a cursory understanding.) Assuming they've had an...
  17. S

    Thermodynamic state having 2 degrees of freedom (i.e., for properties)

    I'm trying to delve into the reason why this is so. It seems that there are 5 fundamental properties: P - Pressure V - Volume (specific) T - Temperature S - Entropy (specific) U - Internal Energy (Yes, there are other types of energy, but they are fully determinable from these 5 - e.g...
  18. tanaygupta2000

    Understanding Energy of Particles with 'x' Degrees of Freedom

    I've learned that for a particle having 3 degrees of freedom, its average energy is 3/2 kT. So for a particle having 'x' degrees of freedom, its energy should be xkT/2. So what is the use of given E = ax6 here? Please help!
  19. stephen weber

    I What are the degrees of freedom of a quark?

    In computer science we simplify lots of things down to arrays. Tensor equations just show the symmetries between these multidimensional arrays. And any model of a quark in our ordered world must have N degrees of freedom. The term called Degrees of Freedom is simple enough. For example a XYZ...
  20. LarryS

    I Why is position just a label in QFT?

    At the beginning of every course in QFT we are told that, unlike in ordinary QM in which the position variable is a physical observable , the position variable in QFT is just a label. Yet there are areas within QFT where the position variable is treated like a real physical degree of freedom...
  21. S

    A Do Holographic Screens eliminate holographic dualities?

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  22. J

    The association of degrees of freedom with temperature

    I delved a bit into the kinetic theory of gases and it got me wondered how it is discovered that the temperature, and thus heat capacity, is dependent on the number of degrees of freedom of a molecule or atom. I know that from the piston experiment a certain constant value can be found for the...
  23. S

    I Explanation of spin degrees of freedom with respect to the x axis

    Hey Guys/Gals i understand the general premise of this question and can calculate the solution but i am a bit confused. I am supposed to represent a generic state as a linear combination of the |-,x> , |+,x> basis vectors. However i don't know why, is the question actually asking for the...
  24. Vital

    I Degrees of freedom for t-test for 2 samples, 2 variances

    Hello. I will be grateful for your help in finding the logical meaning of each part of the formula of degrees of freedom, which are computed for a t-test when variances are unknown and are assumed to be unequal. Please, take a look at the formula, the way I managed to understand some parts of...
  25. e2m2a

    Degrees of freedom confusion

    I see a lot of ambiguous explanations of degrees of freedom on the web and I need clarification. Suppose there is an object in space that can move freely along either the x,y, or z axis. Do we say it has six degrees of freedom because it can move along the x-axis one way or the opposite way...
  26. Roger Dodger

    B Standard Deviation as Function of Sample Size

    In high school, I was taught that the standard deviation drops as you increase the sample size. For this reason, larger sample sizes produce less fluctuation. At the time, I didn't question this because it made sense. Then, I was taught that the standard deviation does not drop as you increase...
  27. JuanC97

    I Minimum requisite to generalize Proca action

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  28. Pushoam

    Degrees of freedom of a diatomic molecule

    Homework Statement Homework EquationsThe Attempt at a Solution Considering the molecule as collection of two spherical atoms whose centers are joined such that they touch each other, Then, as a rigid body the system has 6 degrees of freedom and around the axis joining the two centers, the...
  29. F

    A Implementing Higher Degrees of Freedom

    In practical terms, what quantum theories can be applied to the mediating forces of interacting particles so as to permit degrees of freedom in excess of 3D space? For example, could an inner product of scattering theory be extended to a higher dimension of Hillbert space so as to define...
  30. Erland

    I Why does a quantum system with many degrees of freedom imply orthogonality?

    Quantum decoherence means that when a quantum system interacts with its environment, coherence is lost, which means that all the density matrix becomes diagonal after the interaction. I never understood why it is so, but I get a clue here...
  31. A

    Determining the number of degrees of freedom

    Good day I have issue to understand why in the second case we still have 1 degree of freedom, because according to my understanding ( the circles for me represent the trajectory of the rotation of the two segment, and according to it the two segments can't rotate simultaneously. but according to...
  32. S

    Good coordinates and degrees of freedom

    I have to present a topic "Good coordinates and degree of freedom" I know what are good coordinate and degree of freedom. but I will have to explain examples/question given below(from Liboff's text) I know the answer to all of them but I really do not know how to explain these how will I explain...
  33. P

    I How are degrees of freedom understood in QM?

    I'm having a hard time understanding 'degrees of freedom'. Could someone please provide an example in terms of Quantum Mechanics about what a 'degree of freedom' could be represented as? Is it simply a number of observations of a physical system to determine the arrangement of particles within...
  34. J

    A Do the equations of motion simply tell us which degrees of freedom apply?

    A massless spin 1 particle has 2 degrees of freedom. However, we usually describe it using four-vectors, which have four components. Hence, somehow we must get rid of the superfluous degrees of freedom. This job is done by the Maxwell equations. To quote from Gilmore's "Lie Groups, Physics, and...
  35. M

    A Degrees of freedom and dependent sample variances

    How can I calculate the effective degrees of freedom when linearly combining dependent sample variances? I know that the Welch–Satterthwaite equation exists, but that is for combining independent sample variances. Is there an equivalent expression for dependent sample variances? . Example...
  36. L

    I Calculating Degrees of Freedom for Chi-Squared & P Value

    I am trying to understand how to decide the number of degrees of freedom when calculating a chi-squared and p value. I have the data: England: people with no pets = 665 people with 1 pet = 976 people with 2+ pets = 913 Scotland people with no pets = 313 people with 1 pet = 527 people...
  37. F

    Physical degrees of freedom of an Electromagnetic field

    As I understand it, the classical source-free electric, ##\mathbf{E}## and magnetic, ##\mathbf{B}## wave equations are solved by solutions for the electric and magnetic fields of the following form: $$\mathbf{E}=\mathbf{E}_{0}e^{i (\mathbf{k}\cdot\mathbf{x}-\omega t)}$$...
  38. M

    I How many degrees of freedom does a swan's neck have

    how many degrees of freedom swan's neck have
  39. F

    Counting degrees of freedom in field theory

    I'm having a bit of trouble with counting the number of physical ("propagating") degrees of freedom (dof) in field theories. In particular I've been looking at general relativity (GR) and classical electromagnetism (EM). Starting with EM: Naively, given the 4-potential ##A^{\mu}## has four...
  40. S

    A Local degrees of freedom and correlator

    In quantum field theory, the degrees of freedom ##\phi({\bf{x}},t)## are local. This means that the the dynamics of the field in a given region of spacetime is not governed by events outside its lightcone.Is the local/non-local nature of degrees of freedom in a quantum field theory independent...
  41. grandpa2390

    Why does Water Vapor have 12 degrees of freedom?

    Homework Statement How many degrees of freedom does water vapor have Homework Equations Translational up to 3 rotational up to 3 Vibration up to 6 The Attempt at a Solution Well I said water vapor had 3 translational. It can move along the x, y, or z axis I said it had 2 rotational (the...
  42. L

    I Degrees of freedom of elementary particles

    The EM wave and the photon have two degrees of freedom. Their polarization directions and spin states, respectively. But they move in space, too. I mean light has the freedom to go in all directions in space. Like a macroscopic ball in 3-D space, which can go all three directions, if there are...
  43. Q

    Conservation of Momentum Degrees of Freedom

    Hi I have been dealing with a fluid mechanics pressure gradient problem and from a statistical view point I can see how it resolves itself but am puzzled as to how it can occur at the molecular scale from a conservation of linear momentum perspective if Momentum is a conserved quantity While...
  44. M

    Degrees of freedom of harmonic oscillator

    Homework Statement A three-dimensional harmonic oscillator is in thermal equilibrium with a temperature reservoir at temperature T. The average total energy of oscillator is A. ½kT B. kT C. ³⁄₂kT D. 3kT E. 6kT Homework Equations Equipartition theorem The Attempt at a Solution So I know the...
  45. Schwarzschild90

    Canonical partition function for N ideal gasses

    Homework Statement Exercise 4 in the upload titled Dok1.pdf. Write down an expression for the canonical partition function for N ideal Na2 gas molecules, when the rotational contribution is treated classically, and all inner degrees of freedom are treated quantum mechanically. Use this and...
  46. R

    Temperature for "melting" vibrational degrees of freedom

    Hello everyone! I recently read some information about the equipartition theorem and degrees of freedom in thermodinamics. I read that for the linear N-atomic and non-linear N-atomic molecules in order to allow the vibrational degrees of freedom to appear we need a really high temperature. I...
  47. S

    Canonical Transformation (two degrees of freedom)

    Homework Statement Point transformation in a system with 2 degrees of freedom is: $$Q_1=q_1^2\\Q_2=q_q+q_2$$ a) find the most general $P_1$ and $P_2$ such that overall transformation is canonical b) Show that for some $P_1$ and $P_2$ the hamiltonain...
  48. N

    Normal modes and degrees of freedom in coupled oscillators

    Not a textbook/homework problem so I'm not using the format (hopefully that's ok). Can someone offer an explanation of normal modes and how to calculate the degrees of freedom in a system of coupled oscillators? From what I've seen the degrees of freedom seems to be equal to the number of...
  49. 0

    Lack of Orientation of a Point Mass

    The point mass (aka particle) is a fictional but useful concept. However, I have yet been able to find a definition of what exactly a point mass is. It is commonly accepted that a point mass does not have an orientation, and thus only 3 coordinates to determine its position are required (as...
  50. Dale

    Could you express GR in terms of a density of spacetime?

    Edit: this thread was split off from another where it was off topic https://www.physicsforums.com/threads/could-it-be-meaningful-to-speak-of-the-density-of-space.843442/ . The question in the original thread was if you could express GR in terms of a density of spacetime. I made a mistake in the...
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