# Conservative force

In physics , conservative forces are forces that do no work along any closed path . Energy used on sections of the route is recovered on other routes. This means that the kinetic energy of a specimen is retained in the end.

Examples of conservative forces are, on the one hand, those that, like the gravitational force or Coulomb force of the electric field, are mediated by conservative force fields (see below), on the other hand, forces such as e.g. B. spring forces that are not mediated by force fields in the real sense. Since a potential can be assigned to a conservative force, the force can only depend on the location and not, as for example dissipative forces, on the speed.

The best known example of a conservative force mediated by a force field is the homogeneous approximation of the earth's gravity near the earth's surface. The force is precisely the negative derivative of the potential energy with respect to the height h . Regardless of which way you get from a point at height to a point at height , the same work must always be done . The potential energy still relates to a sample mass m (or sample charge q in the case of the electric field), while the scalar field independent of the sample (or in the case of the electric field) is called the physical potential at the point in question and as such is an equivalent representation of the underlying vector field. ${\ displaystyle F = -mg}$${\ displaystyle W _ {\ mathrm {pot}} = mgh}$${\ displaystyle h_ {1}}$${\ displaystyle h_ {2}}$${\ displaystyle \ Delta W = mg (h_ {2} -h_ {1})}$${\ displaystyle \ Phi = W _ {\ mathrm {pot}} / m = g {\ text {·}} h}$${\ displaystyle \ Phi = W _ {\ mathrm {pot}} / q = E {\ text {·}} s}$

The opposite of conservative forces are non-conservative forces, i.e. those that do work along a self-contained path, and the more so, the longer the path covered. Examples of such non-conservative forces are on the one hand forces in non-conservative force fields such as (magnetic) vortex fields , on the other hand so-called dissipative forces (from the Latin dissipare = to disperse), e.g. B. Frictional Forces .

Most physical systems are, because they always lose energy through friction and / or non-conservative force fields (e.g. vortex fields), non-conservative. If, on the other hand, the perspective is broadened by also taking into account the energy content of coupled heat reservoirs when considering the energy losses through friction, for example, the energy is always retained in some form in the end.

## Conservative force fields

Conservative force fields are those in which a test specimen neither gains nor loses energy when passing through a self-contained path .

It can be shown that the following four characteristics of a conservative force field are equivalent to each other: ${\ displaystyle {\ vec {F}} ({\ vec {r}})}$

1. The work along any closed curve within the field is zero, so .${\ displaystyle C}$${\ displaystyle \ oint _ {C} {\ vec {F}} ({\ vec {r}}) \ cdot \ mathrm {d} {\ vec {r}} = 0}$
2. The work along any path through the force field depends only on the start and end point of the path, but not on its course.${\ displaystyle W = \ int _ {S} {\ vec {F}} ({\ vec {r}}) \ cdot \ mathrm {d} {\ vec {r}}}$${\ displaystyle S}$
3. There is a scalar field , which is called the associated potential of the force field, so that the force can also be described in terms of the form , i. H. as a gradient , with as the nabla operator , as the gradient of the potential and of the coupling constant , which in the case of the electric field electric charge q , in the case of the gravitational field of the test specimen its mass m is.${\ displaystyle \ Phi ({\ vec {r}})}$${\ displaystyle {\ vec {F}} ({\ vec {r}})}$${\ displaystyle {\ vec {F}} ({\ vec {r}}) = - k {\ vec {\ nabla}} \ Phi ({\ vec {r}})}$${\ displaystyle {\ vec {\ nabla}}}$${\ displaystyle {\ vec {\ nabla}} \ Phi ({\ vec {r}})}$ ${\ displaystyle k}$
4. The field is defined on a simply connected area and fulfills the integrability condition there . This means that the rotation disappears, i.e. or is.${\ displaystyle \ textstyle {\ frac {\ partial F_ {k}} {\ partial x_ {i}}} = {\ frac {\ partial F_ {i}} {\ partial x_ {k}}}}$${\ displaystyle {\ vec {\ nabla}} \ times {\ vec {F}} ({\ vec {r}}) = {\ vec {0}}}$${\ displaystyle \ operatorname {rot} \, {\ vec {F}} ({\ vec {r}}) = {\ vec {0}} \,}$

Analogous to what has just been said, vector fields, which can be described as gradients of scalar fields, are generally referred to as conservative in mathematics , composed of potential vectors that are opposed to the associated potentials on the side of the scalar output fields .

### Potentials and fields of potential

The concept of potential is sometimes used differently in physics and mathematics.

In mathematics , the potential denotes a class of scalar position functions or scalar fields with certain mathematical properties, while in physics it only defines the quotient of the potential energy of a body at the point and its electrical charge q or mass m : ${\ displaystyle W _ {\ mathrm {pot}}}$${\ displaystyle {\ vec {r}}}$

${\ displaystyle V_ {C} ({\ vec {r}}) = {\ frac {W _ {\ mathrm {pot}} ({\ vec {r}})} {q}} \ quad {\ text {or .}} \ quad V_ {G} ({\ vec {r}}) = {\ frac {W _ {\ mathrm {pot}} ({\ vec {r}})} {m}}}$

A potential in the physical sense is always also a potential in the mathematical sense, but not the other way round: Thus, both the gravitational and Coulomb potential and also the potential energy in a conservative force field are potentials according to their mathematical nature, in the physical sense however only those the first two. ${\ displaystyle V ({\ vec {r}})}$${\ displaystyle V_ {G}}$${\ displaystyle V_ {C}}$${\ displaystyle W _ {\ mathrm {pot}}}$

It is similarly complicated with the terminology of the gradients of potentials, i.e. the vector fields derived from the respective scalar fields : According to their mathematical nature, gradient fields , composed of gradient vectors, are nevertheless often referred to as " potential fields ", composed of potential vectors. ${\ displaystyle \ Phi ({\ vec {r}})}$ ${\ displaystyle {\ vec {F}} ({\ vec {r}})}$

The illustration on the right illustrates the relationships between the various terms and what is practically hidden behind them. As can be seen, the variety of terms results from just two mathematical operations in reversed order: on the one hand, division by charge or mass, on the other hand, derivation according to location, i.e. H. Formation of the gradient with the help of the Nabla operator.

### example

The gradient of the potential energy at the point provides the "pushing back" force acting at this point and, following the principle of the smallest constraint, always pointing in the direction of decreasing potential energy : ${\ displaystyle W _ {\ mathrm {pot}} \}$${\ displaystyle {\ vec {r}}}$${\ displaystyle - {\ vec {F}} ({\ vec {r}})}$

${\ displaystyle {\ vec {F}} ({\ vec {r}}) = - {\ vec {\ nabla}} W _ {\ mathrm {pot}} ({\ vec {r}})}$

In the vicinity of the earth's surface, the potential energy of a mass at a height above the ground, assuming an approximately constant acceleration due to gravity, is the same for small changes in height . If, since the earth's gravitational field is at least locally radial, the position vector is replaced by the height and the gradient by the derivative , the formula for gravity is: ${\ displaystyle W _ {\ mathrm {pot}}}$${\ displaystyle m}$${\ displaystyle h}$ ${\ displaystyle g}$${\ displaystyle mgh}$${\ displaystyle {\ vec {r}}}$${\ displaystyle h}$${\ displaystyle h}$

${\ displaystyle F (h) = - {\ frac {\ mathrm {d}} {\ mathrm {d} h}} W _ {\ mathrm {pot}} (h) = - {\ frac {\ mathrm {d} } {\ mathrm {d} h}} (mgh) = - mg}$

As can be seen from the sign of the result, the force is opposite to the direction of increasing altitude. ${\ displaystyle F (h)}$

## Local conservatism

Magnetic field of a current-carrying conductor (view against the direction of the
current )

In the case of the last of the above four characteristics of conservative force fields, the criterion of the “simply connected area” must be taken into account, ie that the area, clearly speaking, does not contain any “holes” or similar definition gaps. For example, the area around a current-carrying conductor is not "simply connected" in this sense, the magnetic field of which is defined outside the conductor as follows, but neither for the z -axis (0 | 0 | z) nor its derivative exist: ${\ displaystyle {\ vec {B}}}$

${\ displaystyle {\ vec {B}} (x, y, z) = {\ frac {\ mu _ {0} \, I} {2 \ pi}} \, {\ frac {1} {x ^ { 2} + y ^ {2}}} {\ begin {pmatrix} -y \\ x \\ 0 \ end {pmatrix}}}$

This is true outside of the head . Nevertheless, a ring integral around the z -axis does not vanish . If one integrates, for example, along the unit circle that runs through ${\ displaystyle \ operatorname {red} \, {\ vec {B}} = 0 \}$

${\ displaystyle \ quad C: {\ vec {r}} (\ varphi) = {\ begin {pmatrix} \ cos (\ varphi) \\\ sin (\ varphi) \ end {pmatrix}} \ quad}$ With ${\ displaystyle \ quad 0 \ leq \ varphi <2 \ pi}$

is parameterized , one obtains as a path integral

{\ displaystyle {\ begin {aligned} \ int _ {C} {\ vec {B}} \, \ mathrm {d} {\ vec {r}} & = \ int {\ vec {B}} ({\ vec {r}} (\ varphi)) \ cdot {\ frac {\ partial {\ vec {r}} (\ varphi)} {\ partial \ varphi}} \ mathrm {d} \ varphi \\ & = {\ frac {\ mu _ {0} \, I} {2 \ pi}} \ int _ {0} ^ {2 \ pi} {\ begin {pmatrix} - \ sin (\ varphi) \\\ cos (\ varphi ) \ end {pmatrix}} \ cdot {\ begin {pmatrix} - \ sin (\ varphi) \\\ cos (\ varphi) \ quad \ end {pmatrix}} d \ varphi \\ & = \ mu _ {0 } \, I \ neq 0 \ \ end {aligned}}}

Although the rotation disappears everywhere with the exception of the definition gap on the z -axis, the B-field is not consistently conservative. Since the energy is still retained on all paths that do not enclose the z- axis, one speaks here of local conservativity . ${\ displaystyle \ operatorname {red} \, {\ vec {B}}}$

## Proof of the equivalence of the criteria

As stated at the beginning, the four definitions for a conservative force field are synonymous with each other. The first criterion is precisely the definition of a conservative force from the introduction, the others follow from it.

Any two paths in a conservative force field

1. Assuming that the work disappears along a closed path, the correctness of the second criterion can first be shown. Consider two ways and between points 1 and 2 in a conservative force field as in the picture on the right: ${\ displaystyle S_ {1}}$${\ displaystyle S_ {2}}$

If it runs from point 1 over path to point 2, then over the path back to point 1, the ring integral over this path results in to ${\ displaystyle C}$${\ displaystyle S_ {1}}$${\ displaystyle S_ {2}}$

${\ displaystyle 0 = \ oint _ {C} {\ vec {F}} ({\ vec {r}}) \ cdot \ mathrm {d} {\ vec {r}} = \ int _ {1, S_ { 1}} ^ {2} {\ vec {F}} ({\ vec {r}}) \ cdot \ mathrm {d} {\ vec {r}} + \ int _ {2, -S_ {2}} ^ {1} {\ vec {F}} ({\ vec {r}}) \ cdot \ mathrm {d} {\ vec {r}}}$

With

${\ displaystyle \ int _ {1, S_ {1}} ^ {2} {\ vec {F}} ({\ vec {r}}) \ cdot \ mathrm {d} {\ vec {r}} = - \ int _ {2, -S_ {2}} ^ {1} {\ vec {F}} ({\ vec {r}}) \ cdot \ mathrm {d} {\ vec {r}} = \ int _ {1, S_ {2}} ^ {2} {\ vec {F}} ({\ vec {r}}) \ cdot \ mathrm {d} {\ vec {r}}}$

is that then and precisely zero if

${\ displaystyle \ int _ {1, S_ {1}} ^ {2} {\ vec {F}} ({\ vec {r}}) \ cdot \ mathrm {d} {\ vec {r}} = \ int _ {1, S_ {2}} ^ {2} {\ vec {F}} ({\ vec {r}}) \ cdot \ mathrm {d} {\ vec {r}}}$

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

2. If so is ${\ displaystyle {\ vec {F}} ({\ vec {r}}) = - {\ vec {\ nabla}} V ({\ vec {r}})}$

${\ displaystyle \ int _ {1} ^ {2} {\ vec {F}} ({\ vec {r}}) \ cdot \ mathrm {d} {\ vec {r}} = - \ int _ {1 } ^ {2} {\ vec {\ nabla}} V ({\ vec {r}}) \ cdot \ mathrm {d} {\ vec {r}} = V (1) -V (2)}$Regardless of the way south .

3. If so, then applies to the rotation ${\ displaystyle {\ vec {F}} ({\ vec {r}}) = - {\ vec {\ nabla}} V ({\ vec {r}})}$

${\ displaystyle {\ vec {\ nabla}} \ times {\ vec {F}} ({\ vec {r}}) = - {\ vec {\ nabla}} \ times {\ vec {\ nabla}} V ({\ vec {r}}) = {\ begin {pmatrix} {\ frac {\ partial} {\ partial x}} \\ {\ frac {\ partial} {\ partial y}} \\ {\ frac { \ partial} {\ partial z}} \ end {pmatrix}} \ times {\ begin {pmatrix} {\ frac {\ partial V} {\ partial x}} \\ {\ frac {\ partial V} {\ partial y}} \\ {\ frac {\ partial V} {\ partial z}} \ end {pmatrix}} = {\ begin {pmatrix} {\ frac {\ partial ^ {2} V} {\ partial y \ partial z}} - {\ frac {\ partial ^ {2} V} {\ partial z \ partial y}} \\ {\ frac {\ partial ^ {2} V} {\ partial z \ partial x}} - { \ frac {\ partial ^ {2} V} {\ partial x \ partial z}} \\ {\ frac {\ partial ^ {2} V} {\ partial x \ partial y}} - {\ frac {\ partial ^ {2} V} {\ partial y \ partial x}} \ end {pmatrix}} = {\ vec {0}}}$,

where the last step came about because of the interchangeability of the partial derivatives according to Schwarz's theorem .

4. According to Stokes' theorem , for an area A, which is enclosed by a closed curve C, applies

${\ displaystyle \ iint _ {A} {\ vec {\ nabla}} \ times {\ vec {F}} \ cdot \ mathrm {d} {\ vec {A}} = \ oint _ {C} {\ vec {F}} \ mathrm {\ cdot} \ mathrm {d} {\ vec {r}}}$.

This integral vanishes for all curves C if and if and only if is. ${\ displaystyle {\ vec {\ nabla}} \ times {\ vec {F}} ({\ vec {r}}) = {\ vec {0}} \}$

## Energy conservation

In classical mechanics applies to the kinetic energy

${\ displaystyle T = {\ frac {1} {2}} m {\ vec {v}} ^ {2}}$,

where is the speed . ${\ displaystyle {\ vec {v}}}$

With Newton's second axiom

${\ displaystyle {\ vec {F}} = m {\ dot {\ vec {v}}}}$

the energy can be written for constant masses . ${\ displaystyle m}$

${\ displaystyle E = \ int _ {t_ {1}} ^ {t_ {2}} {\ vec {F}} (t) \ cdot {\ vec {v}} (t) \, \ mathrm {d} t = \ int _ {t_ {1}} ^ {t_ {2}} m {\ dot {\ vec {v}}} (t) \ cdot {\ vec {v}} (t) \, \ mathrm { d} t}$.

Then the path integral applies to the path from point 1 to point 2

${\ displaystyle \ int _ {1, S1} ^ {2} {\ vec {F}} \ cdot \ mathrm {d} {\ vec {r}} = m \ int _ {t_ {1}} ^ {t_ {2}} {\ dot {\ vec {v}}} (t) \ cdot {\ vec {v}} (t) \, \ mathrm {d} t}$.

For the right side of this equation applies

${\ displaystyle \ int _ {t_ {1}} ^ {t_ {2}} {\ frac {\ mathrm {d}} {\ mathrm {d} t}} {\ frac {1} {2}} m { \ vec {v}} ^ {2} (t) \ mathrm {d} t = {\ frac {1} {2}} m {\ vec {v}} ^ {2} (t_ {2}) - { \ frac {1} {2}} m {\ vec {v}} ^ {2} (t_ {1}) = T (t_ {2}) - T (t_ {1}) = T_ {2} -T_ {1}}$.

This means that all of the work that is done in motion corresponds to the change in kinetic energy. For the left side, however, the following applies using the properties of conservative forces

${\ displaystyle \ int _ {1, S1} ^ {2} {\ vec {F}} \ cdot \ mathrm {d} {\ vec {r}} = - \ int _ {1, S1} ^ {2} \ nabla V \ cdot \ mathrm {d} {\ vec {r}} = - V (r_ {2}) + V (r_ {1}) = - V_ {2} + V_ {1}}$

and thus

${\ displaystyle T_ {2} -T_ {1} = - V_ {2} + V_ {1} \}$

or.

${\ displaystyle T_ {1} + V_ {1} = T_ {2} + V_ {2} \}$

which exactly corresponds to the law of conservation of energy. The property of energy conservation is also the reason why conservative force fields got their name - the energy is conserved.

## Individual evidence

1. David Halliday, Robert Resnick, Jearl Walker : Physik. = Halliday physics. Bachelor edition. Wiley-VCH, Weinheim 2007, ISBN 978-3-527-40746-0 , pp. 143-145.
2. ^ A b Walter Gellert, H. Küstner, M. Hellwich, Herbert Kästner (eds.): Small encyclopedia of mathematics. Verlag Enzyklopädie, Leipzig 1970, p. 547.