Degrees of freedom (physics and chemistry)

With degree of freedom, the number of independent ( and in that sense " arbitrary " ) movement possibilities of a system is called. A rigid body in space has therefore the degree of freedom, because you can move the body in three independent directions and rotate in three independent levels. In this meaning, as the total number of independent movement possibilities of the concept of degree of freedom is found only in the singular. The individual movement possibilities are then also called freedoms. In literature and in common usage, but also each of the independent movement possibilities of a system is called a degree of freedom. A rigid body without bindings accordingly has three translational degrees of freedom and three rotational degrees of freedom.


Each degree of freedom of a physical system corresponds to an independent generalized coordinate, at which the system can be described. What is meant by the word "independent", you can see an example: Suppose a particle is located in a plane ( eg on a table ), and can move in this plane only along a straight line. For example, the x coordinate of the particle is known, then there's almost always the corresponding y -coordinate directions and vice versa. Therefore, the particle has only one degree of freedom, because an independent coordinate ( such as the distance on the line ), the position fully describes.

The number of generalized coordinates is a system property. For example, a free mass point three translational degrees of freedom. A rigid body, however, has in addition to the three translational, three rotational degrees of freedom, the latter described by its angle of rotation.

The number of freedom of a system, which is made up of many sub- systems, is the sum of the freedoms of the subsystems, provided these do not by constraints: is restricted (eg, vehicle clutch The trailer can not move independently of the towing vehicle ).

  • A degree of freedom > 0 describes a self- moving system (mechanism).
  • A degree of freedom = 0 describes a statically determinate system.
  • A degree of freedom <0 stands for a statically over-determined system may occur in the high internal voltages ( " clamped ").

Example: double pendulum

Two point masses and have in three-dimensional space a total of six degrees of freedom. However, these are limited by several constraints in the double pendulum ( see figure): located in the plane ( ), as (). In addition, the staffs of the two oscillating rigid (and). These four constraints reduce the number of degrees of freedom. Therefore satisfy the two angles and as independent coordinates for the description of the system.

Example: Joints

In a joint mechanism comprises two parts are movably connected to each other. The degree of freedom is the number of possible movements that can perform the joint. The freedom of the rigid body are available. Within a joint at least one freedom is suppressed, a maximum of five freedoms are for a technical application. More than three freedoms are achieved with multiple joints.

  • Figure 2: swivel
  • Figure 3: screw joint
  • Figure 5: sliding pivot, hinge plate
  • Figure 6: sliding pivot
  • Figure 7: Ball Joint

Thermodynamics and statistical mechanics

The same concept of degrees of freedom of the mechanism also appears in statistical mechanics and thermodynamics: The energy of a thermodynamic system is evenly distributed according to the equipartition theorem to the individual degrees of freedom. The number of degrees of freedom is included in the entropy is a measure of the number of reachable states. Thermodynamic systems generally have many degrees of freedom, about on the order of 1023. It can, however, many similar systems with only few degrees of freedom about, for example 1023 atoms with effective ( see below), each with three degrees of freedom.

Can be specified, the internal energy of an ideal gas with the particles as a function of the temperature and the number of degrees of freedom of a gas particle. In general, the Boltzmann constant:

In the case of a monatomic ideal gas with three degrees of freedom per particle results

This is important for the determination of oscillations are counted twice, since they have both kinetic and potential energy (see below).

Due to the discrete energy levels of the quantum mechanics, all degrees of freedom at low energies usually not be excited, because the first excited state already has a too high energy. This allows a system at a given energy have effectively fewer degrees of freedom. For example, an atom at ambient temperature has effectively only the three translational degrees of freedom, since the average energy is so low that atomic excitations practically not occur.

A diatomic molecule, such as molecular hydrogen has - in addition to the electronic excitations - six degrees of freedom: three translational, two of the rotation, and an oscillation degree of freedom ( although the double- counts to calculate the internal energy ). Rotation and vibration are quantized and at low total energy of a molecule can not be excited higher-energy rotational and vibrational degrees of freedom; they say that they are " frozen. " So the most diatomic gases such as hydrogen, oxygen or nitrogen behave under normal conditions effectively so as if the individual molecules only five degrees of freedom, which is reflected by the adiabatic exponent. At high temperatures the system all degrees of freedom are available.

More complex molecules have more vibrational degrees of freedom, and thus provide a greater contribution to entropy.

Each molecule with atoms has degrees of freedom, because one for each atom needs three coordinates to define its position. This can be formally divided into translational, rotational and internal vibrational degrees of freedom.

The following applies

  • For diatomic linear molecules:
  • For diatomic non-linear molecules:

A molecule with atoms generally has for the internal vibrational energy

Vibrational degrees of freedom.

The thermodynamic degrees of freedom of the state variables at the macroscopic level are obtained for any system in equilibrium via the Gibbs phase rule.