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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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Hetware

Posts: 148
Registered: 4/13/13
Re: Landau & Lifschitz, Mechanics, Principle of Least Action
Posted: Jan 18, 2014 8:06 AM
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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/LandauLifshitz-Mechanics.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 q-dot(t) is implicitly defined as d/dt (delta q(t)).

https://ia601205.us.archive.org/11/items/Mechanics_541/LandauLifshitz-Mechanics.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\)\)






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