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6 THE LAGRANGIAN FORMULATION OF ELECTRODYNAMICS

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Electrodynamics
This chapter is in the book Electrodynamics
6 THE LAGRANGIAN FORMULATION OF ELECTRODYNAMICS 6.1 ACTION PRINCIPLES IN CLASSICAL FIELD THEORIES Our reformulation of electrodynamics in terms of 4D quantities in four-dimen-sional spacetime was motivated by a desire to produce a self-consistent de-scription of this theory with the "new" (special relativistic) mechanics. The application of Galilean transformations to the Maxwell equations clearly pro-duces relationships between the field components that are wrong in the case of relativistic systems with v -t c. In contrast, our interest in a Lagrangian formulation of Maxwell's equations, which by the way builds upon the special relativistic treatment, has nothing to do with a need to remove deficiencies in the theory. As we shall see, the sole motivation for using action principles is to "improve" our understanding of the underlying physics, with a goal of extracting additional laws that might not otherwise be apparent. As an example of how a different perspective has served this purpose before, consider that the Newtonian formulation of classical mechanics is a description of the particle dynamics in terms of forces. But now look at the advantage of also introducing the concept of a potential energy (6.1) from which the forces are derivable, i.e., F = -VU. (6.2) For one thing, a description in terms of cP allows us to define a conserved energy. 145
© 2020 University of Chicago Press

6 THE LAGRANGIAN FORMULATION OF ELECTRODYNAMICS 6.1 ACTION PRINCIPLES IN CLASSICAL FIELD THEORIES Our reformulation of electrodynamics in terms of 4D quantities in four-dimen-sional spacetime was motivated by a desire to produce a self-consistent de-scription of this theory with the "new" (special relativistic) mechanics. The application of Galilean transformations to the Maxwell equations clearly pro-duces relationships between the field components that are wrong in the case of relativistic systems with v -t c. In contrast, our interest in a Lagrangian formulation of Maxwell's equations, which by the way builds upon the special relativistic treatment, has nothing to do with a need to remove deficiencies in the theory. As we shall see, the sole motivation for using action principles is to "improve" our understanding of the underlying physics, with a goal of extracting additional laws that might not otherwise be apparent. As an example of how a different perspective has served this purpose before, consider that the Newtonian formulation of classical mechanics is a description of the particle dynamics in terms of forces. But now look at the advantage of also introducing the concept of a potential energy (6.1) from which the forces are derivable, i.e., F = -VU. (6.2) For one thing, a description in terms of cP allows us to define a conserved energy. 145
© 2020 University of Chicago Press

Chapters in this book

  1. Frontmatter i
  2. CONTENTS v
  3. PREFACE ix
  4. 1 INTRODUCTION 1
  5. 2 TIME-INDEPENDENT FIELDS 25
  6. 3 GENERAL PROPERTIES OF MAX"WELL'S EQUATIONS 61
  7. 4 ELECTROMAGNETIC WAVES AND RADIATION 77
  8. 5 A NEED FOR THE SPECIAL THEORY OF RELATIVITY 115
  9. 6 THE LAGRANGIAN FORMULATION OF ELECTRODYNAMICS 145
  10. 7 RELATIVISTIC TREATMENT OF RADIATION 161
  11. 8 SPECIAL TOPICS 195
  12. REFERENCES 239
  13. INDEX 245
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https://doi.org/10.7208/9780226788449-007

Chapters in this book

  1. Frontmatter i
  2. CONTENTS v
  3. PREFACE ix
  4. 1 INTRODUCTION 1
  5. 2 TIME-INDEPENDENT FIELDS 25
  6. 3 GENERAL PROPERTIES OF MAX"WELL'S EQUATIONS 61
  7. 4 ELECTROMAGNETIC WAVES AND RADIATION 77
  8. 5 A NEED FOR THE SPECIAL THEORY OF RELATIVITY 115
  9. 6 THE LAGRANGIAN FORMULATION OF ELECTRODYNAMICS 145
  10. 7 RELATIVISTIC TREATMENT OF RADIATION 161
  11. 8 SPECIAL TOPICS 195
  12. REFERENCES 239
  13. INDEX 245
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