%0 Journal Article
%T Cauchy, Cosserat, Clausius, Einstein, Maxwell, Weyl Equations Revisited
%A J.-F. Pommaret
%J Journal of Modern Physics
%P 2365-2397
%@ 2153-120X
%D 2024
%I Scientific Research Publishing
%R 10.4236/jmp.2024.1513097
%X The Cauchy stress equations (1823), the Cosserat couple-stress equations (1909), the Clausius virial equation (1870) and the Maxwell/Weyl equations (1873, 1918) are among the most famous partial differential equations that can be found today in any textbook dealing with elasticity theory, continuum mechanics, thermodynamics or electromagnetism. Over a manifold of dimension <i>n</i>, their respective numbers are <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>n</mi><mo>,</mo><mi>n</mi><mrow><mrow>
   <mrow><mo>(</mo>
    <mrow>
     <mi>n</mi><mo>&#x2212;</mo><mn>1</mn></mrow>
   <mo>)</mo></mrow></mrow><mo>/</mo><mn>2</mn></mrow>
  <mo>,</mo><mn>1</mn><mo>,</mo><mi>n</mi></mrow>
</math>

 with a total of <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>N</mi><mo>=</mo><mrow><mo>(</mo>
   <mrow>
    <mi>n</mi><mo>+</mo><mn>1</mn></mrow>
  <mo>)</mo></mrow><mrow><mrow>
   <mrow><mo>(</mo>
    <mrow>
     <mi>n</mi><mo>+</mo><mn>2</mn></mrow>
   <mo>)</mo></mrow></mrow><mo>/</mo><mn>2</mn></mrow>
  </mrow>
</math>

, that is 15 when <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>n</mi><mo>=</mo><mn>4</mn></mrow>
</math>

 for space-time. This is also just the number of parameters of the Lie group of conformal transformations with <i>n</i> translations, <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>n</mi><mrow><mrow>
   <mrow><mo>(</mo>
    <mrow>
     <mi>n</mi><mo>&#x2212;</mo><mn>1</mn></mrow>
   <mo>)</mo></mrow></mrow><mo>/</mo><mn>2</mn></mrow>
  </mrow>
</math>

 rotations, 1 dilatation and <i>n</i> highly non-linear elations introduced by E. Cartan in 1922. The purpose of this paper is to prove that the form of these equations only depends on the structure of the conformal group for an arbitrary <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>n</mi><mo>&#x2265;</mo><mn>1</mn></mrow>
</math>

 because they are described <i>as a whole</i> by the (formal) adjoint of the first Spencer operator existing in the Spencer differential sequence. Such a group theoretical implication is obtained by applying totally new differential geometric methods in field theory. In particular, when <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>n</mi><mo>=</mo><mn>4</mn></mrow>
</math>

, the main idea is not to <i>shrink</i> the group from 10 down to 4 or 2 parameters by using the Schwarzschild or Kerr metrics instead of the Minkowski metric, but to <i>enlarge</i> the group from 10 up to 11 or 15 parameters by using the Weyl or conformal group instead of the Poincar&#233;
%K Janet Sequence
%K Spencer Sequence
%K Poincar&#233
%K Sequence
%K Gauge Sequence
%K Lie Group of Transformations
%K Lie Pseudogroup of Transformations
%K Conformal Group of Transformations
%K Adjoint Representation
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=138353