A practical proposal to obtain solutions of certain variational problems avoiding Euler formalism

Area of Science:

• Variational calculus
• Mathematical physics
• Optimization problems

Background:

• Traditional methods for variational problems with moving boundaries often rely on Euler formalism and transversality conditions.
• These established approaches can be complex and computationally intensive.

Purpose of the Study:

• To introduce novel methodologies for solving variational problems with moving boundaries.
• To provide exact and analytical approximate solutions.
• To circumvent the necessity of Euler formalism and complex transversality conditions.

Main Methods:

• Development of the Moving Boundary Conditions Without Employing Transversality Conditions (MWTC) method.
• Development of the Moving Boundary Condition Employing Transversality Conditions (METC) method.
• Utilization of the Generalized Bernoulli Method (GBM) as a foundational tool.

Main Results:

• The MWTC method successfully avoids the use of transversality conditions.
• The METC method incorporates transversality conditions through a simplified, systematic procedure.
• Both methods yield exact and analytical approximate solutions for the addressed problems.

Conclusions:

• The proposed MWTC and METC methods offer efficient alternatives for solving variational problems with moving boundaries.
• The Generalized Bernoulli Method is a key component for achieving these solutions.
• These new approaches simplify the solution process without compromising accuracy.