Drexel dragonThe Math ForumDonate to the Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math.independent

Topic: Landau & Lifschitz, Mechanics, Principle of Least Action
Replies: 10   Last Post: Jan 19, 2014 8:24 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]

Posts: 148
Registered: 4/13/13
Re: Landau & Lifschitz, Mechanics, Principle of Least Action
Posted: Jan 17, 2014 5:52 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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).

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.