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

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

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

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

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

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

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

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

Variational Calculus: Euler’s Equation Substituting Gives And finally

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

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

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

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

Lagrangian Mechanics Kinetic energy Lagrangian Evaluating Simplifies to

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

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

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