Analyzing Riemann-Liouville constraints in second-order Lagrangian fractional electrodynamic models

Area of Science:

• Theoretical Physics
• Fractional Calculus
• Electrodynamics

Background:

• Singular Lagrangians present challenges in classical field theory.
• Fractional derivatives offer potential for modeling complex systems but face difficulties with non-locality and memory effects.
• Existing models inadequately address second-order fractional derivatives in electrodynamics.

Purpose of the Study:

• To develop a methodology for constraining singular Lagrangians using second-order fractional derivatives.
• To extend the Hamilton-Jacobi formalism within Podolsky's electrodynamics.
• To establish fractional equations linking Coulomb's law and the superposition principle.

Main Methods:

• Application of second-order fractional derivatives to construct Hamilton-Dirac equations.
• Extension of the Hamilton-Jacobi formalism to include second-order derivatives.
• Development of a systematic strategy for handling non-local and non-differentiable fractional derivatives.

Main Results:

• Successfully constrained singular Lagrangians and constructed comprehensive Hamilton-Dirac equations.
• Established fractional equations connecting Coulomb's law with the superposition principle.
• Provided a framework for overcoming limitations in singular Lagrangians by integrating fractional calculus with classical field theory.

Conclusions:

• The developed methodology effectively addresses challenges associated with second-order fractional derivatives, including non-locality and memory effects.
• This research expands classical field theory by incorporating fractional calculus, offering new insights into electrodynamic systems.
• The findings pave the way for future investigations into fractional special relativity and advanced electrodynamic theories.