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Hetware
Posts:
148
Registered:
4/13/13


Re: Landau & Lifschitz, Mechanics, Principle of Least Action
Posted:
Jan 18, 2014 8:06 AM


On 1/17/2014 10:29 PM, William Elliot wrote:
> Ascii, when done well, is much easier to use than pdf., gif, etc. >
Technically, this is ASCII, but looks much better in Mathematica.
q[t]+\[Delta]q[t],Overscript[q, .][t]+\[DifferentialD]\[Delta]q[t]/\[DifferentialD]t
(q[t+\[CapitalDelta]\[InvisibleComma]t]+\[Delta]q[t+\[CapitalDelta]\[InvisibleComma]t]q[t]\[Delta]q[t])/(\[CapitalDelta]\[InvisibleComma]t)=(q[t+\[CapitalDelta]\[InvisibleComma]t]q[t]+\[Delta]q[t+\[CapitalDelta]\[InvisibleComma]t]\[Delta]q[t])/(\[CapitalDelta]\[InvisibleComma]t)
> 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.
https://ia601205.us.archive.org/11/items/Mechanics_541/LandauLifshitzMechanics.pdf
"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
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
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 qdot(t) is implicitly defined as d/dt (delta q(t)).
https://ia601205.us.archive.org/11/items/Mechanics_541/LandauLifshitzMechanics.pdf
\[Delta]\[InvisibleComma]S=\!\( \*SubsuperscriptBox[\(\[Integral]\), SubscriptBox[\(t\), \(1\)], SubscriptBox[\(t\), \(2\)]]\(\(L(q + \[Delta]\[InvisibleComma]q, \*OverscriptBox[\(q\), \(.\)] + \[Delta]\[InvisibleComma] \*OverscriptBox[\(q\), \(.\)], t)\) \[DifferentialD]t\)\)\!\( \*SubsuperscriptBox[\(\[Integral]\), SubscriptBox[\(t\), \(1\)], SubscriptBox[\(t\), \(2\)]]\(\(L(q, \*OverscriptBox[\(q\), \(.\)], t)\) \[DifferentialD]t\)\)



