Calculus of Variations: First Variation Definition?

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In summary, there is a difference in the way older and newer textbooks define the first variation of a functional in Calculus of Variations. The older definition is similar to the concept of a derivative in differential calculus, while the newer definition is closer to the gradient. The Gateaux derivative is considered to be the better and more commonly used definition.
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rdt2
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I put this question in the 'Calculus' forum but didn't really get a response. Maybe it's a silly question but I thought I'd try here anyway:

Older textbooks on the Calculus of Variations seem to define the first variation of a functional [tex] \Pi [/tex] as:

[tex] \delta \Pi = \Pi(f + \delta f) - \Pi (f) [/tex]

which looks analogous to:

[tex] \delta f = \frac {df} {dx} \delta x = lim_{\delta x \rightarrow 0} (f(x+ \delta x) -f(x)) [/tex]

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

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

which looks more like the gradient [tex]\frac {df} {dx} [/tex] than the difference [tex]\delta x [/tex]. Which is the better 'basic' definition?
 
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  • #2
If you know vector calculus, of course you should go on with Gateaux derivative.
 
  • #3
I've never seen anyone call the first one called the first variation. Everyone calls the Gateaux derivative the first variation.
 

FAQ: Calculus of Variations: First Variation Definition?

What is the first variation definition in the calculus of variations?

The first variation definition in the calculus of variations is a mathematical principle that is used to find the optimal solution for a given functional. It involves taking the derivative of the functional with respect to a small variation in the dependent variable and setting it equal to zero.

How is the first variation used in the calculus of variations?

The first variation is used to find the optimal solution for a given functional by minimizing the value of the first variation. This leads to the Euler-Lagrange equation, which provides a necessary condition for the optimal solution.

What is the significance of the first variation definition in the calculus of variations?

The first variation definition is significant because it allows for the determination of the optimal solution for a wide range of problems in physics, economics, and engineering. It is also the basis for the calculus of variations, which has numerous applications in mathematics and science.

Can the first variation definition be applied to any functional?

Yes, the first variation definition can be applied to any functional as long as it satisfies certain conditions, such as differentiability and continuity. This allows for the application of the calculus of variations to a wide range of problems.

Are there any limitations to the first variation definition in the calculus of variations?

While the first variation definition is a powerful tool for finding optimal solutions, it does have some limitations. It may not always lead to a unique solution, and it may not provide a solution at all for some problems. It is important to carefully consider the assumptions and limitations when applying the first variation in the calculus of variations.

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