I Difference between δ and ∆ variation?

[pallab]

what is the difference between δ- variation and ∆-variation in variational principle, used in classical mechanics?

 

[dextercioby]

Can you point to a reference (book, website) for the „∆-variation”?

 

[muscaria]

what is the difference between δ- variation and ∆-variation in variational principle, used in classical mechanics?

Typically, $\Delta$ is not a variation but an $\textit{actual}$ difference, e.g. $\Delta f= f(x_2)-f(x_1)$.
Lagrange introduced a special symbol for the process of variation, which he called $\delta$. Although variation is an infinitesimal change in a similar manner to the $d$ in $dy$ from calculus, it is not the same. It is not an actual infinitesimal change but a virtual change, like a mathematical experiment of some kind, where you're saying to yourself: suppose i were to move "so and so" (some object say) a little bit in that direction, how would "such and such" change. The object isn't actually moving there but you're asking yourself what if it was to move there. Do you see the difference?

 

[jtbell]

what is the difference between δ- variation and ∆-variation

Typically,

Which is why it is important to give the sourc(es) of where you saw δ-variation and ∆-variation. You can't depend on all textbooks and web sites using the same standard definition.

 

[PhysicsExplorer]

what is the difference between δ- variation and ∆-variation in variational principle, used in classical mechanics?

Can't speak for anyone else, but I reserve \Delta for changes in the uncertainty, say between time and energy

\Delta E \Delta t

Technically speaking, there is no difference between this above and

\delta E \delta t

You could reserve the small delta notation only for small/infinitesimal changes in a system.

 

[Kulkarni Sourabh]

what is the difference between δ- variation and ∆-variation in variational principle, used in classical mechanics?

δ- VARIATION :-
In δ- variation process, we are compared all imaginary paths connecting two given points A & B at to given times t1 & t2 . The system must be travel from one end A to another end point B in the same time. The system point is separated up or slow down in order to make the total travel time along the path.
∆- VARIATION :-
In ∆- variation the process we shall restrict the comparison to all paths involving conservation of energy. Thus , in ∆- variation , the system point is separated up or slow down in order to make Hamiltonian constant along actual & varied path.

 

[vanhees71]

So obviously it's the difference between the action principle in the form of Lagrange, where you vary trajectories in configuration space between two fixed times $t_1$ and $t_2$ with fixed endpoints of that trajectory and in the form of Maupertuis, where you vary the trajectories in configuration space keeping the energy fixed. A very good discussion of the different variational principles of classical mechanics can be found in A. Sommerfeld, Lectures on Theoretical Physics, vol. 1.

 

[SammyS]

δ- VARIATION :-
In δ- variation process, we are compared all imaginary paths connecting two given points A & B at to given times t1 & t2 . The system must be travel from one end A to another end point B in the same time. The system point is separated up or slow down in order to make the total travel time along the path.
∆- VARIATION :-
In ∆- variation the process we shall restrict the comparison to all paths involving conservation of energy. Thus , in ∆- variation , the system point is separated up or slow down in order to make Hamiltonian constant along actual & varied path.
Reference: https://www.physicsforums.com/threads/difference-between-d-and-variation.923212/

You are replying to a thread which is more than 5 years old.

Furthermore: The reference you gave refers back to this very thread !

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