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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 17, 2014 5:52 PM
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On 1/17/2014 2:28 AM, William Elliot wrote:
> On Wed, 15 Jan 2014, Hetware wrote:
>

>> 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.

>
> pho = position vector = q
> tau = tangent vector = q-dot
> tau = dpho/dt
>
> delta tau(t) = tau(t + delta t) - tau(t)
> = dpho(t + delta t)/dt - dpho(t)/dt
> = d(pho(t + delta t) - pho(t))/dt
> = d(delta pho(t))/dt
>


deltaq (same as delta q) is a variation of the trajectory, not an
infinitesimal displacement along the trajectory. The variation of the
Lagrangian is written as

deltaL = L(q(t) + deltaq(t), q'(t) + deltaq'(t), t)-L(q(t), q'(t), t)

where q-dot has been re-witten as q'(t), etc. Note specifically that t
is NOT varied. That is to say, q(t), deltaq(t), q'(t) and deltaq'(t)
are all evaluated at t.

deltaq'(t) = d(deltaq(t))/dt = lim[(deltaq(t+h) - deltaq(t))/h, h->0].

deltaq(t) is an arbitrary smooth vector field parametrized which is
added to q(t).



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