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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
Landau & Lifschitz, Mechanics, Principle of Least Action
Posted: Jan 15, 2014 10:47 AM
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On page 3 of Landau & Lifschitz's Mechanics it is stated that

delta q-dot = d(delta q)/dt.

This fact is not demonstrated, it is asserted. q-dot is a vector
/tangent to/ the particle trajectory. delta-q is a displacement /of/
that trajectory. I see no a priori reason to believe the two are
interchangeable.

Another way of saying this is that {q(t), q-dot(t),t} is a phase space
in which q and q-dot are ostensibly independent variables which can only
be correlated by introducing further conditions. The given equation
seems tantamount to constructing some parametrized curve {x(t), y(t)},
and claiming

delta y = d(delta x)/dt.

I understand that q-dot and q are more intertwined since one is the time
derivative of the other.

Are Landau & Lifschitz justified in reversing the order of
differentiation and variation as shown above? Can this be shown rigorously?



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