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