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Field theory and weak Euler-Lagrange equation for classical particle-field systems.

Hong Qin1, Joshua W Burby2, Ronald C Davidson2

  • 1Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China and Plasma Physics Laboratory, Princeton University, P.O. Box 451, Princeton, New Jersey 08543, USA.

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Summary

This study formally establishes the link between space-time symmetry and energy-momentum conservation in classical particle-field systems. It introduces a generalized Euler-Lagrange equation to derive these conservation laws systematically.

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Area of Science:

  • Theoretical Physics
  • Mathematical Physics

Background:

  • Energy-momentum conservation is widely linked to space-time symmetry in physics.
  • This connection is not formally established for classical particle-field systems, such as charged particles interacting with electromagnetic fields.
  • A key challenge is that particle and field dynamics exist on different mathematical spaces.

Purpose of the Study:

  • To formally establish the connection between space-time symmetry and energy-momentum conservation for classical particle-field systems.
  • To overcome the challenge of differing dynamics manifolds for particles and fields.
  • To develop a systematic method for deriving these conservation laws.

Main Methods:

  • Generalizing the Euler-Lagrange equation to a weak form.
  • Introducing a novel concept: the weak Euler-Lagrange current.
  • Applying field theory in conjunction with the weak Euler-Lagrange equation.

Main Results:

  • A generalized Euler-Lagrange equation (weak form) is introduced.
  • A new flux, the weak Euler-Lagrange current, is defined and incorporated into conservation laws.
  • Energy-momentum conservation laws are systematically derived from space-time symmetry for particle-field systems.

Conclusions:

  • The study successfully bridges the gap between space-time symmetry and energy-momentum conservation in classical particle-field systems.
  • The developed weak Euler-Lagrange formalism provides a robust method for deriving conservation laws.
  • This work offers a new perspective on fundamental principles in theoretical physics.