# Degree of freedom

The six degrees of freedom of a body in free space (with the usual designations of the axes of rotation in vehicles ): forwards / backwards ( forward / back ), up / down ( up / down ), left / right ( left / right ), yaw ( yaw ) , nod ( pitch ), roll ( roll )

The degree of freedom describes in the narrow, mechanical sense the number of mutually independent (and in this sense " freely selectable") possibilities of movement, in the broader sense every independent variable internal or external parameter of a system . The system must have the following properties: ${\ displaystyle f}$

• It is clearly determined by the specification of the parameters.
• If a parameter is omitted, the system is no longer clearly defined.
• Each parameter can be changed without changing the other parameters.

In the case of kinematic chains , the degree of freedom is also called the degree of running . The individual possibilities of movement are also called freedoms . A rigid body in space has the degree of freedom , because you can move the body in three independent directions ( translation ) and rotate around three independent axes ( rotation ). ${\ displaystyle f = 6}$

In a somewhat different usage, each of the independent movement possibilities of a system, i.e. each of the mentioned freedoms , is called a degree of freedom. In this sense, a rigid body with no bonds has three degrees of translational freedom and three degrees of rotational freedom .

## mechanics

Double pendulum moving in one plane : The system has two degrees of freedom. His condition is by two rotation angles and fully described. Viewed in isolation, the mass point has only one degree of freedom. Its position is already described by the angle of rotation .${\ displaystyle \ theta _ {1}}$${\ displaystyle \ theta _ {2}}$${\ displaystyle m_ {1}}$${\ displaystyle \ theta _ {1}}$

Each degree of freedom of a physical system corresponds to an independent generalized coordinate with which the system can be described.

What is meant by the word “independent” can be seen in an example: Assuming a particle is in a plane (e.g. on a table) with a coordinate system and can only move in this plane along an “inclined” straight line move. The position of the particle can then be described by a single number. There are various ways of doing this, e.g. B.

• the x-coordinate of the particle (using the straight line equation, the y-coordinate can then also be calculated uniquely),
• the y-coordinate (from this the x-coordinate can be calculated in reverse),
• the angle coordinate in a polar coordinate system
• or the distance from a given fixed point on the straight line.

In each of these cases, however, it is always sufficient to specify a single value to determine the position. The particle therefore has only one degree of freedom.

The number of generalized coordinates is a system property . For example, a free mass point in space has three translational degrees of freedom that determine its position. However, since a point has no extent, it has no orientation. In contrast, a rigid body also has three degrees of freedom of rotation, each of which can be described by an angle of rotation .

Technical mechanics

According to Grübler's equation , the number of freedoms of a system that is formed from many sub-systems is equal to the sum of the freedoms of the sub-systems, provided this is not restricted by constraints . For example, a car has three degrees of freedom in the plane (change of position along the x and y coordinates and the direction of travel). A single-axle trailer has four degrees of freedom, as it can also tip forwards and backwards. If the trailer is attached to the car, the overall system still only has a total of four degrees of freedom (change of position along the x and y coordinates, rotation of the towing vehicle and change of the angle at which the trailer is to the towing vehicle), since tilting and the independent Movement of the trailer is prevented by the trailer coupling. See also degree of run .

Basically, the following cases can be distinguished:

• For the system can move (mechanism) ${\ displaystyle f \ geq 1}$
• For the system is inherently mobile, i. H. the movements of several elements must be specified (e.g. several drives ) so that the movements of all elements are defined.${\ displaystyle f> 1}$
• For is " compulsory ". If you specify the movement of an element (e.g. a drive), the movements of all remaining elements are also defined. Examples: A point moves along a line. In an (idealized) gear drive, the rotation of a gear always causes a precisely defined movement of all other gears.${\ displaystyle f = 1}$
• At can z. B. move two points of a system independently of each other along a line, a single point can move in a plane, or in a transmission, in addition to the rotary movement, another movement is possible, for example if it can be shifted into a second gear.${\ displaystyle f = 2}$
• For the system cannot move ${\ displaystyle f \ leq 0}$
• For there is a statically determined system that can only take up exactly one position.${\ displaystyle f = 0}$
• For there is a statically overdetermined system in which strong internal tensions can occur (it “jams”). If necessary, this can be remedied by additional conditions.${\ displaystyle f <0}$

### Example: double pendulum

Two free point masses and each have three degrees of translational freedom in three-dimensional space, a total of six. A double pendulum , which is connected via swivel joints (in contrast to ball joints ), can only swing in one plane, so that its mobility is restricted by the following constraints (see fig.): ${\ displaystyle m_ {1}}$${\ displaystyle m_ {2}}$

• ${\ displaystyle m_ {1}}$is in the level ( ), also ( ).${\ displaystyle xy}$${\ displaystyle z_ {1} = 0}$${\ displaystyle m_ {2}}$${\ displaystyle z_ {2} = 0}$
• The rods of the two pendulums are rigid ( and ). Each point mass can therefore only move on an arc around the center of the circle.${\ displaystyle L_ {1} = \ mathrm {const.}}$${\ displaystyle L_ {2} = {\ text {const.}}}$

These four constraints reduce the number of degrees of freedom . For the description of the system, the two angles and as independent generalized coordinates are sufficient . ${\ displaystyle f = 6-4 = 2}$${\ displaystyle \ theta _ {1}}$${\ displaystyle \ theta _ {2}}$

### Example: joints

In the joint of a mechanism, two parts are movably connected to each other. The degree of freedom is the number of possible movements the joint can make. In principle, the six freedoms of the rigid body are available for this. At least one of them is prevented in the joint, so a maximum of five are available for a technical application. More than three freedoms are achieved with multiple joints. Multiple joints can be viewed as a combination of several simple joints. ${\ displaystyle f}$

Joints always have a degree of freedom greater than zero. Otherwise it is not a joint, but a restraint .

Joint type Degree of freedom Fig.
Swivel joint ${\ displaystyle f = 1}$   → rotation Fig. 2.
Screw joint ${\ displaystyle f = 1}$   → rotation Fig. 3.
Rotary thrust, plate joint ${\ displaystyle f = 3}$   → Longitudinal, transverse movement (over a short distance), rotation Fig. 5.
Swivel joint ${\ displaystyle f = 2}$   → Longitudinal movement, rotation Fig. 6.
Ball joint ${\ displaystyle f = 3}$   → Movement in a plane (spherical surface), rotation Fig. 7.

## Thermodynamics and Statistical Mechanics

### Degrees of freedom of the molecules

Complex molecules have many degrees of freedom

Every molecule with atoms has general ${\ displaystyle n}$

${\ displaystyle f = 3n}$

Degrees of freedom, because you need three coordinates for each atom to define its position. These can be formally divided into translational, rotational and internal vibrational degrees of freedom:

{\ displaystyle {\ begin {aligned} f & = f _ {\ mathrm {trans}} + f _ {\ mathrm {red}} + f _ {\ mathrm {vib}} \\\ Rightarrow f _ {\ mathrm {vib}} & = 3n-f _ {\ mathrm {trans}} -f _ {\ mathrm {red}} \ end {aligned}}}

For atomic molecules: ${\ displaystyle n}$

linear molecules non-linear molecules
${\ displaystyle f _ {\ mathrm {trans}}}$ ${\ displaystyle 3}$ ${\ displaystyle 3}$
${\ displaystyle f _ {\ mathrm {red}}}$ ${\ displaystyle 2}$ ${\ displaystyle 3}$
${\ displaystyle f _ {\ mathrm {vib}}}$ ${\ displaystyle 3n-5}$ ${\ displaystyle 3n-6}$
total ${\ displaystyle f}$ ${\ displaystyle 3n}$ ${\ displaystyle 3n}$

Complex molecules with many atoms therefore have many degrees of freedom of vibration (see Molecular Vibration ) and thus make a high contribution to entropy .

In the case of molecules that are adsorbed on solid surfaces, the number of degrees of freedom can be reduced. For example, instead of three degrees of freedom of rotation for a molecule in the gas phase, only one can be possible for the adsorbed molecule. The same applies to degrees of translational freedom that z. B. from three (gas phase) to only two in the case of adsorption . Due to the discrete energy levels of quantum mechanics , not all degrees of freedom can usually be excited at low energies , since the first excited state already has too high an energy. This allows a system to effectively have fewer degrees of freedom at a given temperature: ${\ displaystyle f}$

${\ displaystyle f _ {\ mathrm {eff}} \ leq f}$

For example, an atom at room temperature effectively only has the three translational degrees of freedom, since the mean energy is so low that atomic excitations practically do not occur.

The concept of degrees of freedom from mechanics also appears in statistical mechanics and thermodynamics : the energy of a thermodynamic system is evenly distributed over the individual degrees of freedom according to the equipartition theorem . The number of degrees of freedom is included in the entropy , which is a measure of the number of achievable states . Thermodynamic systems generally have a large number of degrees of freedom, for example in the order of 10 23 , the order of magnitude of the Avogadro constant , since they usually contain amounts of substance on the order of a mole . However, many similar systems with only a few degrees of freedom can come into being, e.g. B. 10 23 atoms with effectively (see below) three degrees of freedom each.

The internal energy of an ideal gas with particles can be given as a function of the temperature and the number of degrees of freedom of a gas particle: ${\ displaystyle U}$${\ displaystyle N}$ ${\ displaystyle T}$${\ displaystyle f_ {U}}$

${\ displaystyle U = N \ cdot {\ frac {f_ {U}} {2}} \ cdot k _ {\ mathrm {B}} \ cdot T}$

with the Boltzmann constant . ${\ displaystyle k _ {\ mathrm {B}}}$

It is important that vibrations are counted twice when determining , because they have both kinetic and potential energy (see below): ${\ displaystyle f_ {U}}$

${\ displaystyle f_ {U} = f _ {\ mathrm {trans}} + f _ {\ mathrm {rot}} +2 \ cdot f _ {\ mathrm {vib}}}$
material Degrees of freedom
${\ displaystyle f _ {\ mathrm {trans}}}$ ${\ displaystyle f _ {\ mathrm {red}}}$ ${\ displaystyle f _ {\ mathrm {vib}}}$ ${\ displaystyle f}$ ${\ displaystyle f _ {\ mathrm {eff}}}$ ${\ displaystyle f_ {U}}$
Gas molecule, 1 atom 3 0 0 3 3 03
Gas molecule, 2 atoms 3 2 1 6th 5 07th
Gas molecule, 3-atom linear 3 2 4th 9 13
Gas molecule, angled 3 atoms 3 3 3 9 12
1 atom in the solid 0 0 3 3 06th

A diatomic molecule like molecular hydrogen has - besides the electronic excitations - six degrees of freedom: three of translation , two of rotation , and one degree of freedom of vibration . Rotation and oscillation are quantized and if the total energy of a molecule is low, higher degrees of freedom of rotation and oscillation cannot be excited; they are said to be "frozen". Rotation is already stimulated at medium temperatures, and vibration only at higher temperatures. This is how most diatomic gases behave, e.g. B. hydrogen, oxygen or nitrogen under normal conditions effectively as if the individual molecules had only five degrees of freedom, which can be read off the adiabatic exponent . At high temperatures, the system has access to all degrees of freedom.

### Degrees of freedom of the state variables

The thermodynamic degrees of freedom of the state variables on the macroscopic level result for any systems in thermodynamic equilibrium via Gibbs' phase rule .