Why Lagrangian only contain q and dq/dt?

[huishui]

L.D.Landau's book <Mechanics> first page have below word:
if all the co-ordinates and velocities are simultneously specified,it is known from experience that the state of system is completely determined and that its subsequent motion can,in principle,can be calculated.Mathematically,this means that,if all the co-ordinates q and velocites dq/dt aregiven at some instant,the accelerations d^2q/dt^2 at that instant are uniquely defined.

so how to from q and dq/dt ==> d^2q/dt^2 ?
equipollent problem
why Lagrangian cannot contain d^2q/dt^2 or high-term ?

excuse me for bad english

 

[masudr]

The equations of motion are second order differential equations, so only two constants are required; these are the positions and velocities.

 

[Meir Achuz]

"why Lagrangian cannot contain d^2q/dt^2 or high-term ?"

That is a basic assumption of mechanics. Other assumptions are possible.
For instance, some theories of radiation reaction (so far, all wrong) include higher time derivatives.

 

[lalbatros]

This is indeed an experimental fact, as far as the known "radiation reaction" question is not involved. (not really a problem for CM) Note that the Lagrangian formulation extends also to classical fields theory (see Landau too), and also with first order derivatives only.

For classical mechanics, looking at the transition from QM to CM explains the Lagrangian structure of CM. So the question can be transposed to QM. What surpises me however is that -if I am not mistaken- the Lagrangian formalism/structure goes really much further than the Schroedinger to Newton tale. Am I wrong to say that it underlies the whole physics? Therefore, -I believe- "why the least action" is maybe the deepest question in physics.

The structure of the lagrangian for CM as well as for classical fields, q and dq/dt, seems to me to be more like an experimental fact.

Note, in addition, that high order differential equations can always be reformulated as systems of first order differential equations. Therefore, maybe, the real question is not so much " why q 's and dq/dt 's " in the Lagrangian, but more likely "why does it -in the end- fit in a least action principle".

There is maybe first a mathematical question to ask: could higher order derivatives in a least action principle be eliminated by a reformulation?

Michel

 

[huishui]

thanks

i think my question is my understand on least action and lagrangian not deeply enough, would you give me some suggest?

 

[lalbatros]

huishui,

I think that Landau and Lifchitz is indeed an excellent reference.
It may be hard to read if this is your first reading in theoretical physics.
You may need to read it twice: try reading "Mechanics", "Fields Theory" and "Quantum Mechanics".
In "Fields Theory" you will have most of the fundamental laws derived from a least action principle: motion of relativistic charges, electromagnetic fields, general relativity.
In addition, some math exrcices may help, for exemple on variational calculus.

Michel

 

[jaredkipe]

Remember all the baseball thrown at an angle problems? All you need is the coordinate and initial velocity, then you can calculate the rest.

The reason is because when you know a position and velocity, it is assumed you can sample the fields to know its accelerations or any higher derivatives. There are different Lagrangians for different circumstances. It is these circumstances that define the motion based on its initial conditions.

 

[eljose]

This is not completely true...in "Mathematical methods for Physicist III" in my Ph D. degree we studied Hamiltonians of the form:

$$L= \dot (q^{2})-\ddot q$$ or something similar...

A good argument against..higher order terms different from firs order and yielding to dq/dt expressions..is found in EM if the "derivative" of acceleration is present some "weird" effect can occur..for example the particle could move from rest before the force acts on it...

 

[SeniorTotor]

Classical mechanics reference

Dear Huishui,

First let me answer in a simple manner your question about q and dot q: Basically speaking, a lagrangian is usually kinetic energy, that is squared velocity, minus potentiel energy, which is usually a function of the coordinates (could be different, but this is the base case).

Indeed, Landau's CM book is really excellent, but a bit concise and obscure sometimes, with a lot of shortcomings. Really, this book is not intended for beginners (forget about the other volumes, field theory and quantum mechanics, also excellent but not designed for a first reading, and surely not designed for someone who is trying to understand classical mechanics...). Perhaps you should first try Goldstein's book (analytical mechanics or classical mechanics, something like that). For a mathematical treatment, look at Arnold's book (mathematical methods of classical mechanics).

 

[masudr]

@SeniorTotor:

You do realize that this conversation ended almost 20 months ago?

 

[SeniorTotor]

@Masudr: Now I do.

 

[Count Iblis]

Dear Huishui,

First let me answer in a simple manner your question about q and dot q: Basically speaking, a lagrangian is usually kinetic energy, that is squared velocity, minus potentiel energy, which is usually a function of the coordinates (could be different, but this is the base case).

Indeed, Landau's CM book is really excellent, but a bit concise and obscure sometimes, with a lot of shortcomings. Really, this book is not intended for beginners (forget about the other volumes, field theory and quantum mechanics, also excellent but not designed for a first reading, and surely not designed for someone who is trying to understand classical mechanics...). Perhaps you should first try Goldstein's book (analytical mechanics or classical mechanics, something like that). For a mathematical treatment, look at Arnold's book (mathematical methods of classical mechanics).

You can read Landau and Lifschitz in your second year at university if you just take your time. The advantage of the L&F series is that there are a lot of good exercises. Most other books have too simple problems in their problems sections.

 

[Count Iblis]

@SeniorTotor:

You do realize that this conversation ended almost 20 months ago?

Some conversations never end.

 

[SeniorTotor]

@ Count Iblis:

I totally agree with you. LL series is outstanding, I really love it, and everybody would agree on that point. I was just trying to give some less straightforward references which could help. Goldstein and Arnold books really don't have simple problems, and are by far deeper and more complete than Landau on analytical mechanics topics (Lanczos book is also excellent). I was also reacting because I didn't see the point with LL QM and class. field theory (which are also outstanding, but this is not the point here). Well, well, well...