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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


On page 3 of Landau & Lifschitz's Mechanics it is stated that
delta qdot = d(delta q)/dt.
This fact is not demonstrated, it is asserted. qdot is a vector /tangent to/ the particle trajectory. deltaq 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), qdot(t),t} is a phase space in which q and qdot 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 qdot 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?



