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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 10:04 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On 1/18/2014 8:06 AM, Hetware wrote:

> 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\)\)
>
>


Technically the above expression is true only to the first order. The
exact equation is:

\[Delta]\[InvisibleComma]S=\!\(
\*SubsuperscriptBox[\(\[Integral]\),
SubscriptBox[\(t\), \(1\)],
SubscriptBox[\(t\), \(2\)]]
\*FractionBox[\(\[PartialD]L\),
\(\[PartialD]q\)]\)\[InvisibleComma]\[Delta]\[InvisibleComma]q+\[PartialD]L/\[PartialD]Overscript[q,
.]\[InvisibleComma]\[Delta]\[InvisibleComma]Overscript[q,
.]\[DifferentialD]t

See the discussion leading to (2.5)

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





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