Understanding the Meaning of \delta W

[gulsen]

I know $$\Delta W$$, $$dW$$, $$\partial W$$. But what does $$\delta W$$ exactly mean? How does it differ from the previous ones?

 

[berkeman]

Is that the Dirac delta? Where have you seen the symbol?

 

[gulsen]

Of course it's not Dirac delta.
Classical mechanics book. It's somekind of infintesimal change in work.

 

[berkeman]

Of course it's not Dirac delta.
Classical mechanics book. It's somekind of infintesimal change in work.

Okay, whatever. What does this classical mechanics book say about it? How is it used differently compared to the first delta symbol in your original question?

 

[HallsofIvy]

Of course it's not Dirac delta.
Classical mechanics book. It's somekind of infintesimal change in work.

Then ask in a physics forum, not mathematics.

 

[gulsen]

Then ask in a physics forum, not mathematics.

It's mathematical operator.
Replace the W with something else if you like. Satisfied?

 

[berkeman]

Okay, whatever. What does this classical mechanics book say about it? How is it used differently compared to the first delta symbol in your original question?

Bump...

 

[arildno]

The VARIATION of a quantity W is very often written as $\delta{W}$

 

[gulsen]

Bump...

Whoops! Sorry, I didn't notice your post. Really sorry!
That notation pops-up in many places, especially in Lagrangian. I've had a quick look at wikipedia, and found http://en.wikipedia.org/wiki/Lagrangian" (at the bottom of the page). I just though this a famous-to-all notation.

I had this somewhere in my after reading https://www.physicsforums.com/showpost.php?p=949279&postcount=15" of Clasius2. I was actually looking for a rigorous definition of it, from the standpoint of a mathematician.

 

[Integral]

That notation falls into the category of symbolism which is defined as you go. Different authors will use it in different ways. It does not have a standard mathematical definition, being a parameter of a physical system which should be defined or clear from context when it is used.

In the link to the Lagrangian you provided it is used to state the "Least action principle". To understand it's meaning you will need to research that term since the author of the article assumes you know about that. So when you understand least action principle then you will understand how it is used in that context.

 

[dextercioby]

It stands for the G^ateaux derivative of the action integral.

Daniel.

 

[Rach3]

It's the differential of a functional.

You have a "functional", a function defined on a space of functions (paths in configuration space). This function assigns to each path, a real scalar, the work associated with that path. Remember with ordinary derivatives, you look at the displacement dy resulting from an infinitesimal dx. There's an analagous concept with functionals, the infinitesimal $\delta W$ associated with an infinitesimal change in path $\,\delta J$ (or whatever notation you use). Look it up in whatever resource you have available on Calculus of Variations.

In this case I do not think it refers to the action integral, that is usually notated by S and the differential $\delta S$.