Scalar potential

The gravitational potential of a homogeneous sphere

The scalar potential , often simply called potential , is a scalar field in mathematics - in contrast to the vector potential - whose gradient is according to the following formula ${\displaystyle \Phi ({\vec {r}})\,}$

${\displaystyle {\vec {F}}({\vec {r}})=\operatorname {grad} \ \Phi ({\vec {r}})={\vec {\nabla }}\Phi ({\vec {r}})}$

supplies a vector field called a “ gradient field ” . ${\displaystyle {\vec {F}}({\vec {r}})\ }$

If a conservative force field , in which the force is always directed against the direction of the maximum increase in potential , following the principle of the smallest constraint , the definition applies alternatively ${\displaystyle {\vec {F}}({\vec {r}})\ }$ ${\displaystyle {\vec {F}}\ }$ ${\displaystyle \Phi \ }$

${\displaystyle {\vec {F}}({\vec {r}})=-\operatorname {grad} \ \Phi ({\vec {r}})=-{\vec {\nabla }}\Phi ({\vec {r}}).}$

Scalar potentials form the mathematical basis for the investigation of conservative force fields such as the electric and gravitational fields , but also of eddy - free so-called potential flows .

Formal definition and properties

A scalar field is a scalar potential if and only if it is in a simply connected area ${\displaystyle \Phi \colon {\vec {r}}\mapsto \Phi ({\vec {r}})}$

1. is twice continuously differentiable , that is, does not contain any “jumps”, steps or other points of discontinuity;
2. a vector field exists for it, so that: ${\displaystyle {\vec {F}}\colon {\vec {r}}\mapsto {\vec {F}}({\vec {r}})}$
   ${\displaystyle {\vec {F}}({\vec {r}})=\operatorname {grad} \,\Phi ({\vec {r}})={\vec {\nabla }}\Phi ({\vec {r}})}$

${\displaystyle {\vec {F}}}$The associated gradient field is therefore often called, which, as a gradient of the scalar potential, always fulfills the following conditions: ${\displaystyle \Phi \ }$

1. Path independence of the curve integral : The value of the curve integral along any curve S within the field only depends on its start and end point, but not on its length .
2. Disappearance of the closed curve integral for any boundary curve S :
   ${\displaystyle \oint _{S}\operatorname {grad} \,\Phi ({\vec {r}})\,\mathrm {d} {\vec {r}}=\oint _{S}{\vec {F}}({\vec {r}})\,\mathrm {d} {\vec {r}}=0}$
3. General freedom of rotation or freedom from vortices of the field :
   ${\displaystyle \operatorname {rot} \,(\operatorname {grad} \,\Phi ({\vec {r}}))=\operatorname {rot} \,{\vec {F}}({\vec {r}})={\vec {\nabla }}\times {\vec {F}}({\vec {r}})={\vec {0}}}$

It can be shown that the last-mentioned three characteristics of a gradient field are mathematically equivalent to each other, that is, the fulfillment of one of the three conditions is sufficient for the other two to also apply.

Potential functions and harmonic functions

If one forms the sum of the second partial derivatives of a scalar potential with the help of the Laplace operator ${\displaystyle \Delta \ }$

${\displaystyle \Delta \Phi ({\vec {r}})=\operatorname {div} \,(\operatorname {grad} \,\Phi ({\vec {r}}))=\operatorname {div} \,{\vec {F}}({\vec {r}})={\vec {\nabla }}\cdot {\vec {F}}({\vec {r}})={\frac {\partial ^{2}\Phi ({\vec {r}})}{\partial x^{2}}}+{\frac {\partial ^{2}\Phi ({\vec {r}})}{\partial y^{2}}}+{\frac {\partial ^{2}\Phi ({\vec {r}})}{\partial z^{2}}},}$

In principle, two results are possible:

1. The sum is a non-zero function , or else ${\displaystyle f({\vec {r}})}$
2. As a special case of 1), the sum is always zero.

Based on this, scalar potentials can be classified again as follows:

• Solutions of the potential equation called Poisson's differential equation or Poisson's equation are called potential  functions (or simply potentials). ${\displaystyle \Delta \Phi ({\vec {r}})=f({\vec {r}})}$
• Solutions to the potential equation known  as Laplace's differential equation or Laplace's equation (as a special case of Poisson's equation) are also called harmonic functions . Harmonic functions are accordingly special cases of potential functions that also satisfy the Laplace equation. Some authors, however, use both terms synonymously , so that the terms “potential” or “potential function” only mean solutions to the Laplace equation, that is, “any function that is twice continuously differentiable according to all three variables and at the same time to a certain extent Area of ​​space the equation is fulfilled, a potential function or harmonic function in this area is called "and the definition of the potential theory in this case only takes Laplace potentials into account: " The potential theory is the theory of the solutions to the potential equation . " ${\displaystyle \Delta \Phi ({\vec {r}})=0}$
  ${\displaystyle \Phi ({\vec {r}})}$ ${\displaystyle \Delta \Phi ({\vec {r}})=0}$ ${\displaystyle \Delta U=0}$

Poisson and Laplace fields

The vector fields resulting as gradients of a scalar potential are always eddy-free and are therefore - in contrast to "eddy fields " - often summarized under the umbrella term " source fields ", which does not mean that they therefore cannot be source-free anyway .

Depending on whether the underlying potentials are simply solutions of a Poisson equation or also the Laplace equation, the gradient fields obtained from them can be classified as follows:

• Gradient fields that result from solutions of a Poisson equation are also called “Poisson fields” or “Newton fields” and are simply free of eddies. In other words: If a scalar potential is based on a space (charge) density and is therefore a particular solution of a corresponding inhomogeneous (Poisson's) differential equation , the gradient field derived from the potential is called a “Poisson field” or “Newton field” . Examples of such fields are the gravitational field or the electric field in the absence of an opposing second charge, the effect of which is therefore always spatially unlimited. ${\displaystyle f({\vec {r}})\neq 0}$ ${\displaystyle \rho ({\vec {r}})}$ ${\displaystyle \Delta \Phi ({\vec {r}})=\rho ({\vec {r}})}$
• Gradient  fields of harmonic functions, on the other hand, which result from solutions to the Laplace equation (or a Poisson equation with ) are called "Laplace fields" and are also source-free . In other words: If a scalar potential is based on a surface (charge) density on the surface of charged bodies and if it is the homogeneous solution of a homogeneous (Laplace) differential equation for the correspondingly selected boundary conditions, the gradient field derived from the potential becomes "Laplace field " called. Examples of such fields are, for example, the electric field in the presence of an opposite second charge, on which the field lines emanating from the first charge end. So “Laplace fields” always have an “edge” in the finite, while in the case of “Poisson” or “Newton fields” this is almost infinite. ${\displaystyle f({\vec {r}})=0}$ ${\displaystyle \sigma ({\vec {r}})}$ ${\displaystyle \Delta \Phi ({\vec {r}})=0\,}$

For the superposition of both field types, a so-called total potential function can usually be formulated, which is the sum of a particular and a homogeneous solution of the above-mentioned differential equations.

Examples

By far the best known scalar potential is the so-called. "Newtonian potential"

${\displaystyle \Phi ({\vec {r}})={\frac {1}{|{\vec {r}}|}},}$

that is only in the three-dimensional, i.e. for , a harmonic function that fulfills the Laplace condition. Conversely, the "logarithmic potential" comparable to the "Newtonian potential" in two-dimensional ${\displaystyle r^{2}=x^{2}+y^{2}+z^{2}}$

${\displaystyle \Phi ({\vec {r}})=\operatorname {ln} (|{\vec {r}}|)}$

just like the function ln (1 / r) = -ln (r) only there, i.e. for , a harmonic function, in the three-dimensional, on the other hand, an ordinary potential with ΔΦ = 1 / r² or ΔΦ = −1 / r². The functions and are also only defined for R² . ${\displaystyle r^{2}=x^{2}+y^{2}}$ ${\displaystyle \Phi (x,y)=e^{x}\cdot \sin(y)}$ ${\displaystyle \Phi (x,y)=e^{x}\cdot \cos(y)}$

history

The term potential in its current mathematical meaning goes back to the French mathematician Joseph-Louis Lagrange , who was investigating Newton's law of gravitation

${\displaystyle F=-G\ {\frac {m_{0}\cdot m_{1}}{r^{2}}}}$

it was already established in 1773 that the component decomposition of the force F , to which a point mass in the gravitational field of a second point mass is exposed, amounts to three partial forces F _x , F _y and F _z , all of which are partial derivatives of a common scalar "antiderivative" U ( x ₀ ; y ₀ ; z ₀ ) could be interpreted: ${\displaystyle m_{0}}$ ${\displaystyle m_{1}}$

${\displaystyle {\begin{aligned}{\vec {F}}({\vec {r}}_{0}|{\vec {r}}_{1})&={-G\,m_{0}}\ {\frac {m_{1}}{r^{2}}}\ {\hat {\vec {r}}}_{10}\\&={-G\,m_{0}}\ {\frac {m_{1}}{r^{3}}}{\begin{pmatrix}x_{0}-x_{1}\\y_{0}-y_{1}\\z_{0}-z_{1}\end{pmatrix}}={\vec {F}}_{x}({\vec {r}}_{0}|{\vec {r}}_{1})+{\vec {F}}_{y}({\vec {r}}_{0}|{\vec {r}}_{1})+{\vec {F}}_{z}({\vec {r}}_{0}|{\vec {r}}_{1})\\&={-G\,m_{0}\,m_{1}}\,{\frac {x_{0}-x_{1}}{r^{3}}}\,{\hat {\vec {x}}}\ {-G\,m_{0}\,m_{1}}\,{\frac {y_{0}-y_{1}}{r^{3}}}\,{\hat {\vec {y}}}\ {-G\,m_{0}\,m_{1}}\,{\frac {z_{0}-z_{1}}{r^{3}}}\,{\hat {\vec {z}}}\\&={\frac {\partial }{\partial x}}({G\,m_{0}}\,{\frac {m_{1}}{r}})\,{\hat {\vec {x}}}+{\frac {\partial }{\partial y}}({G\,m_{0}}\,{\frac {m_{1}}{r}})\,{\hat {\vec {y}}}+{\frac {\partial }{\partial z}}({G\,m_{0}}\,{\frac {m_{1}}{r}})\,{\hat {\vec {z}}}\\&\quad \ {\text{mit}}\ \ r=((x_{0}-x_{1})^{2}+(y_{0}-y_{1})^{2}+(z_{0}-z_{1})^{2})^{\frac {1}{2}}\end{aligned}}}$

As you can see, the found "antiderivative" U (x ₀ ; y ₀ ; z ₀ ) is defined for all points in space except (x ₁ | y ₁ | z ₁ ), and it is also a measure of the (negative) potential energy of m ₀ in the force field of mass m ₁ :

${\displaystyle W_{pot}({\vec {r}}_{0}|{\vec {r}}_{1})=-{G\,m_{0}}\,{\frac {m_{1}}{r}}}$

A little later, summarized under the term potential, this discovery was continued in the work of the English mathematician and physicist George Green , who coined the term potential function in his 1828 essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism . First and foremost, however, it was Carl Friedrich Gauss who in 1840 (according to other sources as early as 1836) further deepened and popularized the concept of potential and its theory.

Definition of terms

Unfortunately, for historical reasons, the use of the concept of potential is often inconsistent. For example, it is often unclear whether the word “potential” refers to the relevant scalar field, that is to say the relevant position function, or one of its function values.

Mathematical and physical concept of potential

The concept of potential in its mathematical meaning - as a scalar field with certain, initially purely abstract properties - must not be confused with the physical “ potential ” concept from which it originally emerged.

A term that primarily means the ability of a conservative force field to make a body exposed to it do a work, usually expressed in terms of the ratio of its potential energy and charge or mass. However, this can mean that one is dealing with the scalar field in the given context , which reflects this relationship in the form of its function values, or that the "potential" means the individual function values ​​of the field at the relevant point are, for example, the electrical or gravitational potential , measured in volts (= J / C) or J / kg.

In addition, as far as its mathematical properties are concerned, the potential energy of a body in a conservative force field can itself be described as scalar potential, not to mention the velocity potential of fluid dynamics, which is only mathematically a potential .

In general, (almost) every physical potential can be modeled by a mathematical one, while conversely not every mathematical potential is also one in the sense of physics.

Potential vectors and potential fields

Another problem arises from the fact that the term "potential ..." is also used in some word formations, in which it is not clear whether it means scalar or vector quantities or fields, for example in terms such as " vector potential ", " Potential vector " or " potential field ". So you could just accept the latter that allow the scalar field potential is meant itself - the vast majority of authors but this expression used not for it, but for the derived from the respective potential vector of potential or gradient field vectors .

Similarly, some authors refer to the vectors that make up gradient fields to better differentiate between the gradient as a mathematical operator and the result of its use as gradient vectors , while others refer to the (scalar) potentials from which they are derived as potential vectors .

Relations between scalar and vector potential

Eddy fields that are rotations of another vector field are always source-free - source-free vector fields can therefore always be interpreted as a rotation of another vector field, which in this case is called the “ vector potential ” of the relevant source-free vector field.

According to the fundamental principle of vector analysis, also Helmholtz Theorem mentioned here can (almost) every vector field as a superposition of two components and are considered, the first of the gradient of a scalar potential , the second on the other hand, the rotation of a vector potential : ${\displaystyle {\vec {H}}({\vec {r}})}$ ${\displaystyle {\vec {F}}({\vec {r}})\,}$ ${\displaystyle {\vec {G}}({\vec {r}})}$ ${\displaystyle \Phi ({\vec {r}})\,}$ ${\displaystyle {\vec {\Gamma }}({\vec {r}})}$

${\displaystyle {\vec {H}}({\vec {r}})={\vec {F}}({\vec {r}})+{\vec {G}}({\vec {r}})=\operatorname {grad} \,\Phi ({\vec {r}})+\operatorname {rot} \,{\vec {\Gamma }}({\vec {r}})={\vec {\nabla }}\Phi ({\vec {r}})+{\vec {\nabla }}\times {\vec {\Gamma }}({\vec {r}})}$

If there is a conservative force field in which the force, following the principle of least constraint, is always directed in the opposite direction to the direction of the maximum increase in potential , the notation applies as an alternative ${\displaystyle {\vec {F}}({\vec {r}})\,}$ ${\displaystyle {\vec {F}}\,}$ ${\displaystyle \Phi \ }$

${\displaystyle {\vec {H}}({\vec {r}})={\vec {F}}({\vec {r}})+{\vec {G}}({\vec {r}})=-\operatorname {grad} \,\Phi ({\vec {r}})+\operatorname {rot} \,{\vec {\Gamma }}({\vec {r}})=-{\vec {\nabla }}\Phi ({\vec {r}})+{\vec {\nabla }}\times {\vec {\Gamma }}({\vec {r}}).}$

Individual evidence

1. ^ ^{A b} Walter Gellert, Herbert Küstner, Manfred Hellwich, Herbert Kästner (eds.): Small encyclopedia of mathematics. Leipzig 1970, pp. 547-548.
2. ↑ ^{a b c} Lothar Papula: Mathematics for engineers and natural scientists: vector analysis, probability calculation, mathematical statistics, error and compensation calculation, Volume 3 ; Vieweg + Teubner, 2008, pp. 85-92.
3. ^ ^{A b} Walter Gellert, Herbert Küstner, Manfred Hellwich, Herbert Kästner (eds.): Small encyclopedia of mathematics. Leipzig 1970, pp. 743-746.
4. ↑ ^{a b c} Adolf J. Schwab; Conceptual world of field theory ; Springer, 2002, pp. 18-20.
5. ^ W. Gellert, H. Küstner, M. Hellwich, H. Kästner (eds.): Small encyclopedia of mathematics. Leipzig 1970, p. 746.
6. ↑ ^{a b c} W. Gellert, H. Küstner, M. Hellwich, H. Kästner (eds.): Small encyclopedia of mathematics. Leipzig 1970, pp. 741-742.
7. ↑ Grimsehl: Textbook of Physics, Vol. I ; Leipzig 1954, p. 160.
8. ↑ §4 potential fields. (PDF; 1.9 MB) In: Mathematics for Engineers III. WS 2009/2010, University of Kiel.
9. ^ Albert Fetzer, Heiner Fränkel: Mathematics 2: Textbook for engineering courses. Springer, Berlin / Heidelberg, p. 322.
10. ↑ Grimsehl: Textbook of Physics, Vol. I. Leipzig 1954, p. 579.