The phrase "degrees of freedom" is used in three different branches of science: in physics and physical chemistry, in mechanical and aerospace engineering, and in statistics. The three usages are linked historically and through the underlying mathematics through the concept of dimensionality, but they are not identical. Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), Nature) is the science of Nature in the broadest sense. ... Physical chemistry is the study of the physical basis of chemical systems and processes. ... Mechanical engineering is the application of physical principles to the creation of useful devices, objects and machines. ... Aerospace engineering is the branch of engineering concerning aircraft, spacecraft and related topics. ... Engineering is the application of science to the needs of humanity. ... For Wikipedia statistics, see m:Statistics Statistics is the science and practice of developing human knowledge through the use of empirical data expressed in quantitative form. ... Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... There is an ontological procession of dimensionality. ...

Physics and chemistry

In physics and chemistry, each independent mode in which a particle or system may move or be oriented is one degree of freedom. For a roughly dumbbell-shaped hydrogen molecule, three such modes would be rotation (twirling), translation (hurtling through space) and vibration (the two dumbbell "balls" bouncing together and apart). According to the Equipartition Theorem of thermodynamics, in case of thermal equilibrium each degree of freedom in every particle of a system will contain the same energy on average (equal to kT, the temperature of the system multiplied by the fundamental Boltzmann constant). However, thermal equilibrium can only be reached among interacting particles, a process called thermalization. According to quantum mechanics and more specifically Heisenberg's uncertainty principle, the amount of energy within any degree of freedom is never zero, but is always at least equal to the zero-point energy for that mode. The Equipartition Theorem is a principle of classical (non-quantum) statistical mechanics which states that the internal energy of a system composed of a large number of particles will distribute itself evenly among each of the degrees of freedom allowed to the particles of the system. ... Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ... In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann-distribution. ... Temperature is the physical property of a system which underlies the common notions of hot and cold; the material with the higher temperature is said to be hotter. ... Ludwig Boltzmann Ludwig Boltzmann (February 20, 1844 – September 5, 1906) was an Austrian physicist famous for the invention of statistical mechanics. ... Fig. ... Werner Heisenberg Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics. ... In quantum physics, the Heisenberg uncertainty principle, sometimes called the Heisenberg indeterminacy principle, expresses a limitation on accuracy of (nearly) simultaneous measurement of observables such as the position and the momentum of a particle. ... In a quantum mechanical system such as the particle in a box or the quantum harmonic oscillator, the lowest possible energy is called the zero-point energy. ...

Engineering

In mechanical engineering, aeronautical engineering and robotics, degrees of freedom (DOF) describes flexibility of motion. A mechanism or linkage that has complete freedom of motion (even if only in a limited area, or envelope) has six degrees of freedom. Three modes are translation - the ability to move in each of three dimensions. Three are rotation, or the ability to change angle around three perpendicular axes. Mechanical engineering is the application of physical principles to the creation of useful devices, objects and machines. ... Aerospace engineering is the branch of engineering concerning aircraft, spacecraft and related topics. ... In practical usage, a robot is a mechanical device which performs automated tasks, either according to direct human supervision, a pre-defined program or, a set of general guidelines, using artificial intelligence techniques. ... The locking pliers is an example of a four-bar, one degree of freedom mechanical linkage; or a five-bar, two DOF linkage when the adjustment screw is considered. ...

This is a typical robot arm which has 6 DOF. Only 3 DOF are necessary to get it anywhere in space, but 6 gives it more versatility.

To put it in simpler terms, each of the following is one degree of freedom: This image comes from [1] . This robot has 7 DOF. Their copyright policy is I did e-mail them. ...

1. Moving up and down (heaving);
2. moving left and right (swaying);
3. moving forward and back (surging);
4. tilting up and down (pitching);
5. turning left and right (yawing);
6. tilting side to side (rolling).

See also: Euler angles. In robotics Degrees of Freedom is the number of directions that a robot can pivot or move a joint. A human arm is considered to have 7 DOF. A shoulder gives pitch, yaw and roll, an elbow allows for pitch, and a wrist allows for pitch yaw and roll. Only 3 of those movements would be necessary to move the hand to any point in space, but people would lack the ability to grasp things from different angles or directions. This article is about the soft drink. ... Flight dynamics is the study of orientation of air and space vehicles and how to control the critical flight parameters, typically named pitch, roll and yaw. ... Flight dynamics is the study of orientation of air and space vehicles and how to control the critical flight parameters, typically named pitch, roll and yaw. ... Flight dynamics is the study of orientation of air and space vehicles and how to control the critical flight parameters, typically named pitch, roll and yaw. ... Euler angles are the classical way of representing rotations in 3-dimensional Euclidean space, named after Leonhard Euler. ...

Statistics

In statistics, degrees of freedom are the number of values in probability distributions that are free to be varied. Examples of this statistical parameter include the chi-square distribution, the F-distribution, Student's t-distribution, and the beta distribution that underlies them. See Pearson's chi-square test and analysis of variance for more information. For Wikipedia statistics, see m:Statistics Statistics is the science and practice of developing human knowledge through the use of empirical data expressed in quantitative form. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... A statistical parameter is a parameter that indexes a family of probability distributions. ... For any positive integer , the chi-square distribution with k degrees of freedom is the probability distribution of the random variable where Z1, ..., Zk are independent normal variables, each having expected value 0 and variance 1. ... In statistics and probability, the F-distribution is a continuous probability distribution. ... In probability and statistics, the t-distribution or Students distribution arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ... In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function defined on the interval [0, 1]: where a and b are parameters that must be greater than zero. ... Pearsons chi-square test (χ2) is one of a variety of chi-square tests – statistical procedures whose results are evaluated by reference to the chi-square distribution. ... In statistics, analysis of variance (ANOVA) is a collection of statistical models and their associated procedures which compare means by splitting the overall observed variance into different parts. ...

In the familiar uses of these distributions, the degrees of freedom takes only integer values (usually low ones). The underlying mathematics do allow for fractional degrees of freedom, which can arise in more sophisticated uses. The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ...