Quaternionic Differential Equations

Generalized field equations

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Generalized field equations hold for all basic fields. Generalized field equations fit best in a quaternionic setting.

Quaternions consist of a real number valued scalar part and a three-dimensional spatial vector that represents the imaginary part.

The multiplication rule of quaternions indicates that several independent parts constitute the product. In this comment, we use a suffix   to indicate the scalar real part of a quaternion, and we use   to indicate the imaginary vector part of quaternion  .

 

 

 

 

 

(1)

The   in equation (1) indicates that quaternions exist in right-handed and left-handed versions.

The formula can be used to check the completeness of a set of equations that follow from the application of the product rule.

The quaternionic conjugate of a is   From the product rule follows the formula for the norm   of quaternion  .

 

 

 

 

 

(2)

We define the quaternionic nabla as  .

The quaternionic nabla   acts like a multiplying operator. The (partial) differential   represents the full first-order change of field  . We assume that   exists in an enclosed region of the domain of  .

First order partial differential equation

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(3)

The equation is a quaternionic first order partial differential equation.

The five terms on the right side show the components that constitute the full first-order change.

They represent subfields of field  , and often they get special names and symbols.

Subfields

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  is the gradient of  

  is the divergence of  .

  is the curl of  

 

 

 

 

 

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(Equation (4) has no equivalent in Maxwell's equations!)

 

 

 

 

 

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Properties of the spatial nabla operator

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The nabla exists in several coordinate systems. This section shows the representation of the quaternionic nabla for Cartesian coordinate systems and for polar coordinate systems.

 

 

 

 

 

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Here   are the coordinates with   as coordinate axes.

 

 

 

 

 

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Special formulas
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The spatial nabla operator shows behavior that is valid for all quaternionic functions for which the first order partial differential equation exists.

Here the quaternionic field   obeys the requirement that the first order partial differential   exists.

 

 

 

 

 

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For constant   and parameter   holds

 

 

 

 

 

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The term   indicates the curvature of field  .

The term   indicates the stress of field  

Derivation of higher order equations
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With the help of the properties of the spatial nabla operator follows an interesting second-order partial differential equation.

 

 

 

 

 

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Most of the terms vanish. Further

 

 

 

 

 

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From the above formulas follows that the Maxwell equations do not form a complete set.

Physicists use gauge equations to make Maxwell equations more complete.

Second order partial differential equations

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We start with the quaternionic equivalent of the Maxwell-Faraday equation.

 

 

 

 

 

(46)

Two interesting second order quaternionic partial differential equations exist.

 

 

 

 

 

(47)

This is the quaternionic equivalent of the wave equation. It offers waves as part of the solutions of the homogeneous equation.

 

 

 

 

 

(48)

This equation can be split into two first order partial wave equations   and  .

This equation does not offer waves as part of the solutions of the homogeneous equation.

Differential operators

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(49)

This is the quaternionic equivalent of d'Alembert's operator.

 

 

 

 

 

(50)

This operator does not yet have a known name.

Operator   represents the main part of the Poisson equation. Together with   this operator configures the above operators.

As is shown above,   can be derived from the nabla operator. That cannot be said from the   operator.