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Quantifying Variational Solutions: Historical Perspectives and Modern Applications

This research explores the development and application of variational methods throughout history, from early Greek mathematicians to modern computational techniques. The paper presents the derivation of variational corrections, including both linear and nonlinear examples. Highlights include pivotal contributions from figures such as Fermat, Euler, and Lagrange, as well as the evolution of variational methods in quantum mechanics and solid-state physics. The importance of variational approximations for solving complex problems is also discussed, emphasizing their relevance in contemporary mathematical and engineering applications.

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Quantifying Variational Solutions: Historical Perspectives and Modern Applications

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  1. University of Central Florida Institute for Simulation & Training and Department of Mathematics and CREOL D.J. Kaupand Thomas K. Vogel Quantifying Variational Solutions† (Preprint available at http://gauss.math.ucf.edu/~kaup/) † Research supported in part by NSF and AFOSR

  2. OUTLINE • History of Variation Methods • Uses and Variational Approach • Derivation of Variational Corrections • Linear Example • Nonlinear Example • Summary

  3. History of Variational Methods • Early Greeks – Max. area/perimeter • Hero of Alexandria – Equal angles of incidence /reflection • Fermat - least time principle (Early 17th Century) • Newton and Leibniz – Calculus (Mid 17th Century) • Johann Bernoulli - brachistochrone problem (1696) • Euler - calculus of variations (1744) • Joseph-Louis Lagrange – Euler-Lagrange Equations (??) • William Hamilton – Hamilton's Principle (1835) • Raleigh-Ritz method – VA for linear eigenvalue problems (late 19th Century) • Quantum Mechanics - Computational methods – (early 20th Century) • Morse & Feshbach – technology of variational methods (1953) • Solid State Physics, Chemistry, Engineering – (mid-late 20th Century) • Personal computers – new computational power (1980’s) • Technology of variational methods essentially lost (1980-2000) • D. Anderson – VA for perturbations of solitons (1979) • Malomed, Kaup – VA for solitary wave solutions (1994 – present)

  4. Why Use Variational Methods? • Linear problems are very well understood. • Nonlinear problems are very different. • Nonlinear waves have solitary wave (soliton) solutions. • They exist in a limited parameter space. • Where should one look? • Amplitude=?, width=?, phase=?, etc. • Equation coefficients for solitons=? • These Q’s mostly irrelevant for linear systems. • VA for nonlinear system is same as for linear system. • Simple ansatzes point to regions where solitons are. • Basic functional relations found from ansatzes. • No need to search entire parameter space. • Each parameter in ansatz reduces parameter space. • Cascading knowledge.

  5. Variational Approximations • Is based on a Minimization Principle • Solution = path that extremizes an “Action” • Action = time-integral over a Lagrangian • Lagrangian is specified by the system • By freezing out specific modes, one can • obtain reduced systems • The method will still find the path which • is closest to the actual solution • Definite need for quantitative measure

  6. Variational Corrections Definition of Action is: Definition of variational derivative is: Euler – Lagrange Equations are: Now consider Variational Perturbations about an ansatz: Ansatz Corrections e = ? Variational Parameters

  7. Expansion Calculate Action and Expand: Zeroth order is the VA: Next order is (vary u1): • eis determined by E-L Eq. • R is thereby defined 0

  8. Equation for Correction • Perturbed Euler-Lagrange Equation with Source • SUMMARY: • Drop Ansatz into Action • Calculate new E-L equations to determine q’s • Drop Ansatz plus correction into the full E-L equations • Solve for u1 • Determine quantitative accuracy

  9. Vibrating String Eigenmodes Examples -- two different Ansatzes: • Only need fundamental mode • Will normalize intensity to unity

  10. Variational Eigenvalues ``Action” for eigenvalue problems is eigenvalue itself. Variation of l and u results in Euler- Lagrange equation. For our models:

  11. Ansatzes and Corrections where:

  12. Quantitative Estimates Eigenvalues and corrections: RMS measure: which gives:

  13. KdV Example Look for soliton solution and integrate once: Take the Lagrangian and Ansatz to be: Then the action is: With the variational solution:

  14. KdV correction The correction equation can be scaled: In which case, it reduces to: where

  15. Ansatz and Correction Erms = 0.038

  16. Soliton separation Want soliton separation such that tails = 0.001 Ansatz = 1.56; plus correction = 2.1 Ratio = 2.1 / 1.56 = 1.35; whence 35% error

  17. Quantitative Variational • Can calculate variational corrections • Can quantify variational approximations • Do not need exact solutions • Only need to solve linear equation • Quantitative estimate depends on what use is • Most VA’s will be poor/excellent depending on use

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