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

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

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

4. 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.

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

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

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

8. Lagrangian Mechanics Kinetic energy Lagrangian Evaluating
Simplifies to