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Topic: Landau & Lifschitz, Mechanics, Principle of Least Action
Replies: 10   Last Post: Jan 19, 2014 8:24 PM

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Posts: 148
Registered: 4/13/13
Re: Landau & Lifschitz, Mechanics, Principle of Least Action
Posted: Jan 19, 2014 11:05 AM
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On 1/18/2014 10:59 PM, William Elliot wrote:
>>> It comes of the general definition in use, namely.
>>> delta f(x) = f(x + delta x) - f(x)

>> That is NOT the definition of delta f(x) in use. This is variational calculus
>> not traditional differential calculus.

> What? It's a calculus of variation problem? Not by what you wrote.

Even if you are unfamiliar with "Landau & Lifschitz, Mechanics", the
part about "Principle of Least Action" clearly frames the discussion as
pertaining to variational dynamics.

>> https://ia601205.us.archive.org/11/items/Mechanics_541/LandauLifshitz-Mechanics.pdf

> References are useless.

A link to the text under discussion is useless?

>> "Let q=q(t) be the function for which S is a minimum. This means that S is
>> increased when q(t) is replaced by any function of the form

> What's S?

>> q(t) + delta q(t), (2.2)
>> where delta q(t) is a function which is small everywhere in the interval of
>> time from t_1 to t_2; delta q(t) is called a /variation/ of the function q(t).

>> Since for t=t_1 and for t=t_2, all function (2.2) must take values q^(1) and
>> q^(2) respectively, it follows that

> What does "^(1)" mean.

It appears the authors assumed its meaning to be clear from context.
It's obvious to me that it is a designation for q[t_1]. I don't know
why they chose that notation. I merely reproduced it.

>> delta q(t_1) = delta q(t_2) = 0."
>> To be pedantic, they also need some caveats about differentiability
>> The first equation on page 3 (not page 2, as I originally indicated) is where
>> delta q-dot(t) is implicitly defined as d/dt (delta q(t)).
>> https://ia601205.us.archive.org/11/items/Mechanics_541/LandauLifshitz-Mechanics.pdf

> References are useless.

In your experience, I trust that they are useless.

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