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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


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 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. >> >> pho = position vector = q >> tau = tangent vector = qdot >> 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 qdot has been rewitten 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/LandauLifshitzMechanics.pdf



