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Generating true minima in constrained variational formulations via modified Lagrange multipliers.

Francisco J Solis1, Vikram Jadhao2, Monica Olvera de la Cruz2

  • 1School of Mathematical and Natural Sciences, Arizona State University, Glendale, Arizona 85306, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
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This study introduces a novel method to ensure variational principles yield true minima, crucial for computational physics. By modifying Lagrange multipliers, researchers can generate stable formulations for systems like charged objects and magnetic particles.

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Area of Science:

  • Physics
  • Computational Science
  • Applied Mathematics

Background:

  • Variational principles are fundamental in physics for analyzing systems and developing computational methods.
  • Constrained variational formulations, often using Lagrange multipliers, are essential for systems like charged objects in dielectrics and magnetic particles.
  • Ensuring these formulations yield positive-definite functionals (true minima) is critical for simulation stability and accuracy.

Purpose of the Study:

  • To present a general approach for identifying equivalent variational functionals that guarantee true minima.
  • To develop a method for modifying Lagrange multipliers to achieve stable variational formulations.
  • To demonstrate the applicability of this method to key physical systems, including Poisson and Poisson-Boltzmann equations.

Main Methods:

  • A general approach based on modifying Lagrange multipliers within constrained variational principles.
  • Generation of families of equivalent variational formulations from a single original principle.
  • Application and validation of the method on specific examples, notably Poisson and Poisson-Boltzmann equations.

Main Results:

  • A systematic method to derive variational formulations with guaranteed minima.
  • Demonstration of the method's effectiveness for systems described by Poisson and Poisson-Boltzmann equations.
  • The ability to generate multiple stable variational formulations from a single constrained principle.

Conclusions:

  • The proposed method provides a robust way to find stable variational formulations for complex physical systems.
  • This approach enhances the reliability of computational techniques based on variational principles.
  • The findings are particularly relevant for simulations involving electrostatic and electrodynamic phenomena.