Calculus of Variations

What is the Calculus of Variations?

A branch of calculus that deals with optimising functionals. It is mostly used to solve optimisation problems. Though the calculus of variations uses the same fundamental tools as conventional calculus, they are two domains that differ from one another. Whereas calculus focuses on changes regarding a function’s variables, the calculus of variations focuses on changes in an entire function - hence the use of functionals.

Why do we need the Calculus of Variations?

The calculus of variations is especially important for Lagrangian Mechanics, Optimal Control and Geodesics.

2. The Functional Analysis Definition of a Functional

In functional analysis, functionals differ in subtle ways to functions, taking functions as inputs to functionals, and outputting a scalar value - a number.

$$\mathcal{F}[f]\to \mathbb{R}$$ Link to original

See examples of functionals for more on functionals. In the case of Optimal Control and Lagrangian Mechanics, we are most interested in functionals who’s integrands involve $\frac{dy}{dx}$, $y(x)$ or $x$. Hence, a general functional form we are interested in is:

$$F(y)={\int }_a^{b}f(x,y(x),\frac{dy(x)}{dx})$$

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