A Gateaux vs. Frechet in Calc.Variations

[Trying2Learn]

Good Morning

Could someone summarize the distinction between the Frechet and the Gateaux derivative with specific regard to the Calculus of Variations?

Why use one or the other? Or both? Is one easier than the other? More precise? Or Accessible?

I feel I understand the process, but with all things mathematical, I could be assuming I understand when I really don't.

 

[fresh_42]

Good Morning

Could someone summarize the distinction between the Frechet and the Gateaux derivative with specific regard to the Calculus of Variations?

Why use one or the other? Or both? Is one easier than the other? More precise? Or Accessible?

I feel I understand the process, but with all things mathematical, I could be assuming I understand when I really don't.

I once tried to answer the same question and found what I summarized here (at the beginning):
https://www.physicsforums.com/insights/pantheon-derivatives-part-ii/

What I see is, that it depends on how the Gâteaux derivative is defined, how variations are affected, namely: continuity and additivity. Fréchet seems to set more strict conditions, which are useful for variations, whereas Gâteaux is most general in order to define a derivative if anyway possible.

 

[Trying2Learn]

I once tried to answer the same question and found what I summarized here (at the beginning):
https://www.physicsforums.com/insights/pantheon-derivatives-part-ii/

What I see is, that it depends on how the Gâteaux derivative is defined, how variations are affected, namely: continuity and additivity. Fréchet seems to set more strict conditions, which are useful for variations, whereas Gâteaux is most general in order to define a derivative if anyway possible.

Hi Fresh_42

That is what I was thinking. I got that from the same document. However, (and forgive me for the vague writing), the words reside in my head as tap dancing (for me and please do not take that as a criticism of your response, as it is an admission of my ignorance).

Is there any way you can make this more concrete? I am not a mathematician. I am using this for dynamics and Hamilton's Principle.

I am just searching for enough of an explanation so I can be happy, but not so much that I get inundated. I don't know where to draw the line.

 

[fresh_42]

Fréchet implies Gâteaux in all directions. So if you have nice topological spaces, e.g. $\mathbb{R}^n$, and your derivatives are linear, there is no need to deal with Gâteaux. So ordinary Jacobi matrices would be fine. You said dynamics, but dynamics of what? Dynamics in QFT is probably different from dynamics in mechanics.

If you only have convex, normed vector spaces, possibly infinite dimensional, and your derivatives are neither required to be continuous nor additive, then you have to carefully manage the properties and the theorems you can apply.

So it is more like what do we have, than what shall be required. Moreover, Gâteaux doesn't seem to be used the same way across different textbooks, which is why condition management is necessary.

 

[Trying2Learn]

Fréchet implies Gâteaux in all directions. So if you have nice topological spaces, e.g. $\mathbb{R}^n$, and your derivatives are linear, there is no need to deal with Gâteaux. So ordinary Jacobi matrices would be fine. You said dynamics, but dynamics of what? Dynamics in QFT is probably different from dynamics in mechanics.

If you only have convex, normed vector spaces, possibly infinite dimensional, and your derivatives are neither required to be continuous nor additive, then you have to carefully manage the properties and the theorems you can apply.

So it is more like what do we have, than what shall be required. Moreover, Gâteaux doesn't seem to be used the same way across different textbooks, which is why condition management is necessary.

Thanks again... and...

Could you repeat what you just wrote but with a focus on rigid mutli-body dynamics?

 

[fresh_42]

Thanks again... and...

Could you repeat what you just wrote but with a focus on rigid mutli-body dynamics?

The shortest way I can think of: Show me the equations (and spaces) and I'll tell you what it is.

The second shortest, which I assume apply to rigid bodies:
finite dimensional, Euclidean space + Jacobi matrix $\Longrightarrow$ Fréchet $\Longrightarrow$ Gâteaux

In case you have it the other way around, i.e. given the requirement that something has a Fréchet, resp. Gâteaux derivative, then it should also be said somewhere, what this means. Not necessarily in the Fréchet case, which is unambiguous, but in the Gâteaux case as it depends to some extend on the definition.