Conservative force

Conservative forces (from the Latin conservare = preserve ) are in physics forces who perform along a self-contained way no work - any time spent at a location of the road amount of energy is recovered at any other location again and vice versa, that is, all energy a specimen is received him at the end.

Examples of conservative forces are firstly those that like gravity or Coulomb force of the electric field by conservative force fields (see below) are taught, but on the other hand forces such as spring forces which are not mediated by force fields in the proper sense.

The opposite of conservative forces are non-conservative forces, ie those which do along a closed path in to work, and the more so the longer the case back on board. Examples of such non-conservative forces are the forces in a non-conservative force fields such as ( magnetic ) eddy fields, on the other so-called dissipative forces ( scatter of Latin dissipare =), such as frictional forces.

Most physical systems, since they always have energy goes (eg eddy fields) lost by friction and / or non-conservative force fields, non- conservative. If one extends the other hand, the perspective by the energy content of a coupled heat reservoirs into account, for example when considering the energy losses due to friction, the energy remains in the end always get in some form.

Conservative force fields

Conservative force fields are the foregoing, following those in which gains a specimen as it passes through a self-contained way neither energy nor loses.

It can be shown that the following four characteristics of a conservative force field equivalent to each other:

Are generally referred to in mathematics vector fields, which can be described as gradient of scalar fields as conservative, composed of potential vectors, which are compared with the corresponding potentials on the part of the scalar output fields Analogous to what has been said.

Potentials and potential fields

The term of the potential is used in physics and mathematics in different ways.

Thus, the potential in the mathematics generally referred to a class of scalar functions and local scalar with certain mathematical properties, while in the physics only the quotient of the potential energy of a body at the site and its electric charge q and mass m is defined as:

A potential in the physical sense is always also one in the mathematical sense, but not vice versa: Sun, however, both the gravitational and Coulomb potential and as well as the potential energy in a conservative force field its mathematical nature, potential, in the physical sense only the first two.

Similarly complicated is the case with the terminology of the gradient of potentials, ie those derived from the respective scalar vector fields: your mathematical nature, therefore, gradient, composed of Gradientvektoren, they are still often referred to as " potential fields ", composed of potential vectors.

The adjacent figure illustrates once again the relationships between the various concepts, and what is behind virtually hidden. As can be seen, the term diversity results from only two mathematical operations in reversed order: first, the division by charge or mass, on the other hand the derivative with respect to the location, that is, formation of the gradient using the nabla operator.

Example

The gradient of the potential energy at the point delivers the action at this point and the principle of least constraint, following always pointing in the direction of decreasing potential energy " restoring " force:

Near the earth's surface, the potential energy of a mass in height is just above the ground for assuming a small height changes approximately constant acceleration of gravity. If we replace, as it is the gravitational field of the earth around an at least locally radial field, the position vector by the height and the gradient by the derivative with respect to, this results for the gravity of the formula:

How to look at the sign of the result, the force of the direction of increasing height is opposite.

Local conservativeness

The last of the above four characteristics conservative force fields is mainly due to the criterion of " contiguous territory ", so make sure that the area, clearly spoken, contains no "holes" or similar definition gaps. Not " connected " in this sense, for example, the area around a current-carrying conductor, the magnetic field is indeed defined below outside the conductor, for the z- axis ( 0 | 0 | z ) itself but not yet exist its derivative:

Although the following applies outside the conductor, but does not vanish, a ring integral to the z-axis. By integrating, for example, along the unit circle by

Is parameterized, we obtain the path integral

Although the rotation with the exception of the definition of the gap in the z-axis disappears everywhere, the B-field is not characterized by continuously conservative. Since the energy still get on all paths remains that do not enclose the z- axis, this is called restrictive of local conservatism.

Proof of the equivalence of the criteria

As initially noted, the four definitions are equivalent to each other for a conservative force field. The first criterion is the very definition of a conservative force in the introduction, the others follow it.

1 Assuming that the work along a closed path disappears, the correctness of the second criterion can first be shown. Consider two paths to and between points 1 and 2 in a conservative force field as in the image on the right:

Runs from point 1 to point 2 on the way, then the way back to point 1, then the ring integral over this way results in order to

With

Is the then and exactly then zero when

Which corresponds exactly to the path independence and thus the second definition for a conservative force field.

2 If, as

3 If, then for the rotation

Where the last step because of the commutativity of the partial derivatives according to the set of black came about.

4 According to the Stokes' theorem applies to a closed curve C, which is surrounded by an area A

This integral vanishes for all curves C and then if and only if is.

Energy conservation

In classical mechanics applies to the kinetic energy of

Wherein the speed.

By Newton's second axiom

To constant mass, the energy to be written.

Then for the path from point 1 to point 2, the path integral

For the right-hand side of this equation

This means that the total work that is applied during the movement, corresponding to the change of the kinetic energy. However, applies to the left side using the properties of conservative forces

And thus

Or

Which corresponds exactly to the energy conservation law. The property of energy conservation is also the reason why conservative force fields were named - the energy is conserved.