Variational Methods

1. Variational Methods Seminar on Computational Engineering 19.4.2001 by Jukka-Pekka Onnela

2. Variational Methods • Brief history of calculus of variations • Variational calculus: Euler’s equation • Derivation of the equation • Example • Lagrangian mechanics • Generalised co-ordinates and Lagrange’s equation • Examples • Soap bubbles and the Plateau Problem

3. Brief History of Calculus of Variations • Calculus of variations: A branch of mathematics concerned with the problem of finding a function for which the value of a certain integral is either the largest or the smallest possible • Calculus: from Latin calx for stone; used as pebbles or beads on the countingboard and abacus; to calculate • Isoperimetricproblem known to Greek mathematicians in the 2nd century BC • Euler developed a general method to find a function for which a given integral assumes a max or min value • Introduced isoperimetric problems as a separate mathematical discipline: calculus of variations

4. Variational Calculus: Euler’s Equation • We seek a function that minimises the distance between the two points • Minimising • Generally:

5. Variational Calculus: Euler’s Equation • Introduce test function • Required property • Extremum at • At extremum • Differentiating

6. Variational Calculus: Euler’s Equation • Integrating the second term by parts • Noticing • Gives • By the fundamental lemma of calculus of variations we obtain Euler’s Equation

7. Variational Calculus: Euler’s Equation • Example: Surface of revolution for a soap film • Film minimises its area <=> minimises surface energy • Infinitesimal area • Total area

8. Variational Calculus: Euler’s Equation • This function satisfies • Derivatives • Substituting • Integrating • Integrating again

9. Variational Calculus: Euler’s Equation • Substituting • Gives • And finally

10. Lagrangian Mechanics • Incorporation of constraints as generalised co-ordinates • Minimising the number of independent degrees of freedom

11. Lagrangian Mechanics • For conservative forces Lagrange’s equation can be derived as • Lagrangian defined as kinetic energy - potential energy

12. Lagrangian Mechanics • Example 1: Pendulum • The generalised co-ordinate is • Kinetic energy • Potential energy • Lagrangian • Pendulum equation

13. Lagrangian Mechanics • Example 2: Bead on a Hoop • The generalised co-ordinate is • Cartesian co-ordinates of the bead • Velocities obtained by differentiation

14. Lagrangian Mechanics • Kinetic energy • Lagrangian • Evaluating • Simplifies to

15. Soap Bubbles and The Plateau Problem • Physicist Joseph Plateau started experimenting with soap bubbles to examine their configurations - Plateau problem • Accurately modelled by minimal surfaces • Why bubbles are spherical? • Poisson: Surface of separation between two media in equilibrium is the surface of constant mean curvature • Pressure = [surface tension][mean curvature] => For bubbles and films the pressure on two sides of the surface is a constant function • Soap film enclosing a space with pressure inside greater than outside => constant positive mean curvature • Soap film spanning a wire frame => zero mean curvature

16. Soap Bubbles and The Plateau Problem • Response to perturbations depends on the nature of extremum point: • Local minimum => Film is stable and resists small perturbations • Saddle point => Film is unstable and small perturbations decrease its surface area • New configuration lower in energy and topologically different • Example: Two coaxial circles

17. Soap Bubbles and The Plateau Problem • Examples of minimal surfaces - Soap Bubbles!