AdvancedTopic 103 of 104Engineering Mathematics
Calculus of variations
Finds functions that minimise integrals, such as shortest paths and energy functionals.
In depth
Find a function y(x) that makes ∫F(x,y,y')dx stationary. The Euler-Lagrange equation d/dx(∂F/∂y')=∂F/∂y is the condition. It yields shortest paths, minimal surfaces, beam deflection from energy, and Lagrangian mechanics.
Key points
- Functional maps functions to numbers
- Boundary conditions fix constants
- Hamilton's principle of least action
ExampleThe shortest curve between two points in a plane is a straight line.
What is Calculus of variations?
Finds functions that minimise integrals, such as shortest paths and energy functionals.
Why is it important to understand calculus of variations?
Variational methods find the function that makes an energy or time integral stationary, and they underpin finite elements and optimal control.
What is the Euler-Lagrange equation?
It is the condition a function must satisfy to make an integral stationary. Solving it gives the optimal path or shape.
How does this relate to finite elements?
Finite elements minimise the energy functional approximately over a set of trial functions, so the energy principle gives the method its justification.
How is Calculus of variations used in practice?
Typical applications include finding the shape of a hanging cable that minimises potential energy, deriving the equations of motion from Hamilton's principle and finding the path of least time for light or a robot in a plane.
What should I learn before Calculus of variations?
Study "Tensors" first (topic 102 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Calculus of variations?
Functional maps functions to numbers. Boundary conditions fix constants. Hamilton's principle of least action.
Can you give an example of Calculus of variations?
The shortest curve between two points in a plane is a straight line.
What comes after Calculus of variations?
The next topic is "Stochastic processes", which builds directly on this one.
How difficult is Calculus of variations?
It is classed as Advanced in Engineering Mathematics (topic 103 of 104). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Engineering Mathematics.