Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Physics
Classical Physics
Mechanics
Why are there 2s -1 independent integrals of motion?
Reply to thread
Message
[QUOTE="yucheng, post: 6801755, member: 683521"] I was reading Mechanics by Landau and Lifshitz and I am confused when it is stated in chapter 2 section 6 that one of the integrals of motion is not independent and it can be considered an additive constant of time. Hence I tried searching it up online. [URL]https://physics.stackexchange.com/questions/55861/constants-of-motion-vs-integrals-of-motion-vs-first-integrals?noredirect=1&lq=1[/URL] According to the OP in the link above (first paragraph second sentence), we need to specify 2N initial conditions, one of them is the initial time, the others the initial positions and velocity. [COLOR=rgb(184, 49, 47)]However, shouldn't it be N position and N velocities? Can it be shown to be equivalent? Plus aren't we working with an autonomous system of equations? Is this why we need ##t - t_0## so that it is independent of time?[/COLOR] [URL='https://physics.stackexchange.com/a/592205/259297']https://physics.stackexchange.com/questions/13832/integrals-of-motion[/URL] The answer provided above seems interesting. However, how correct is it? There are several points that I would like to verify... [COLOR=rgb(184, 49, 47)]Questions:[/COLOR] [LIST=1] [*]Because the ##\mathcal{L} (q, \dot q)##, the Lagrangian is independent of the acceleration. Hence $$\frac{d}{dt} \frac{\partial \mathcal{L} }{\partial \dot q} - \frac{\partial \mathcal{L}}{\partial q}$$ which only involved one time derivative, only introduces terms linear in ##\ddot q##. [*]According to the author (see a comment below the post as well), $$\ddot q_i =\frac{\text{d}\dot q_i}{\text{d} q_1} \frac{\text{d}q_1}{\text{d}t} =\dot q_1\frac{\text{d}\dot q_i}{\text{d}q_1}$$. How does a total time derivative become a total derivative in ##q_1##? Are we performing a change of variables by inverting ##q_1(t)## to get time as a function of ##q_1## then all coordinates become ##q_i(t(q_1))##? [*]Are there other proofs of this? [/LIST] [/QUOTE]
Insert quotes…
Post reply
Forums
Physics
Classical Physics
Mechanics
Why are there 2s -1 independent integrals of motion?
Back
Top