Calculus of Variations: The Gateaux vs First Variation Debate

[rdt2]

Older textbooks on the Calculus of Variations seem to define the first variation of a functional $$\Pi$$ as:

$$\delta \Pi = \Pi(f + \delta f) - \Pi (f)$$

which looks analogous to:

$$\delta f = \frac {df} {dx} \delta x = lim_{\delta x \rightarrow 0} (f(x+ \delta x) -f(x))$$

from differential calculus. However, newer books seem to define the first variation as the Gateaux derivative:

$$\left[ \frac {d} {d \epsilon} \Pi (f+ \epsilon h) \right]_{\epsilon = 0 }$$

which looks more like the gradient $$\frac {df} {dx}$$ than the difference $$\delta x$$. Which is the better 'basic' definition?

 

[EnumaElish]

The first definition is similar to the discrete $\Delta$ operator in real analysis. The 2nd def. is similar to the continuous d operator.

 

[tacman]

The debate between the first variation and Gateaux derivative as the "basic" definition in the Calculus of Variations is a topic of ongoing discussion in the field. Both definitions have their strengths and weaknesses, and ultimately, the choice between them may depend on the specific application or problem being considered.

On one hand, the first variation definition has the advantage of being more intuitive and closely related to the concept of a derivative in differential calculus. It also allows for a more direct connection to the Euler-Lagrange equation, which is a fundamental tool in the Calculus of Variations. However, it can be limiting in some cases where the functional is not differentiable, as the first variation requires differentiability in order to be well-defined.

On the other hand, the Gateaux derivative definition has the advantage of being more general and applicable to a wider range of problems. It does not require differentiability of the functional, making it a more robust tool in certain situations. Additionally, it has a more elegant mathematical formulation and can be extended to infinite-dimensional spaces, which is important in many applications. However, it may be less intuitive for those familiar with the traditional first variation definition, and it may require more advanced mathematical techniques to work with.

In the end, both definitions have their merits and are widely used in the Calculus of Variations. It is important for students and practitioners to be familiar with both and understand their respective strengths and limitations. Ultimately, the choice between them may depend on the specific problem at hand and the preferences of the person using them.