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 *In physics, the degree of freedom (DOF) of a mechanical system is the number of independent parameters that define its configuration.

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