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Euler-Lagrange equations for variational problems on space curves.

Peter Hornung¹

¹Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom. p.hornung@bath.ac.uk

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|September 28, 2010

Summary

This study derives generalized Euler-Lagrange equations for variational problems on curves. The findings simplify and unify existing methods, yielding equilibrium equations for internal forces and moments.

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Area of Science:

Mathematics
Physics
Engineering

Background:

Variational problems are fundamental in describing physical systems.
Existing methods for deriving Euler-Lagrange equations can be complex.
Generalization of these equations is needed for broader applications.

Purpose of the Study:

To derive a generalized form of the Euler-Lagrange equations.
To simplify the process of obtaining these equations for curves.
To provide a unified framework for analyzing internal forces and moments.

Main Methods:

Derivation of Euler-Lagrange equations using calculus of variations.
Application to a broad class of variational problems defined on curves.
Demonstration of the method's simplicity and self-contained nature.

Main Results:

A generalized set of Euler-Lagrange equations for variational problems on curves.
The derived equations directly represent equilibrium conditions for internal force and moment.
The result offers a significant improvement over recent literature findings.

Conclusions:

The generalized Euler-Lagrange equations provide a powerful and simplified tool.
This work unifies and extends existing theoretical frameworks.
The direct link to internal force and moment equilibrium has practical implications.