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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 8:11 PM
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On 1/17/2014 5:52 PM, Hetware wrote:
> 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).


Sorry for changing notation again. It's hard to communicate in ASCII.

It seems the piece that I'm missing is the definition of delta(q'[t]).
L&L only explicitly define delta_q[t]. So taking the limit as
Delta_t->0 of the following gives dq/dt + d/dt(delta_q)

(q[t+Delta_t] + delta_q[t+Delta_t] - q[t] - delta_q[t])/Delta_t
= (q[t+Delta_t]- q[t])/Delta_t
+ (delta_q[t+Delta_t] - delta_q[t])/Delta_t

So d/dt(delta_q) appears to be implicitly defined as delta(dq/dt).

It appears to be implied by the first expression on page 2. This is the
2nd edition. Mine is the the 3rd, but the content under discussion is
the same.

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



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