How Does Dirac Challenge Classical Mechanics Assumptions in Quantum Theory?

[snoopies622]

Dirac's Lectures on Quantum Mechanics begins with a big chapter on classical mechanics called "The Hamilton Method". Within the first ten pages he says,

"Now in the usual dynamical theory, one makes the assumption that the momenta are independent functions of the velocities, but that assumption is too restrictive for the applications which we are going to make. We want to allow for the possibility of these momenta not being independent functions of the velocities. In that case, there exist certain relations connecting the momentum variables, of the type \phi (q,p) = 0."

What is this \phi function?

 

[snoopies622]

Looking ahead it appears to have the same dimensions as the Hamiltonian (energy). I don't know why it's introduced, though, except for generality. I must say the whole chapter seems pretty abstract.

 

[dextercioby]

If it's about his famous Yeshiva University lectures, then he's speaking of Hamiltonian constraints. I suggest you keep reading him and be prepared to learn some very rare things not described in many books at all.

 

[Studiot]

I think you will find that the potential, \phi is a general potential introduced so that we can say:

In Newtonian mechanics force is the gradient of some potential function, say \phi.

Then d\phi is a 1form

Now in order to say rate of change of momentum = force,

either p is covarient

or

d\phi

is contravarient.

That is either

\frac{{d(mv)}}{{dt}} = G\left( {d\phi } \right)

(Newtonian)

or

\frac{{d(G(mv))}}{{dt}} = d\phi

Which is the alternative I think Dirac was discussing.

Edit 1
This is rubbish. It is nothing like what I wrote.
Latex is screwed again. I will try to correct it some other time.

Edit 2
The above is now correct. Why couldn't LaTex display this first time round?

 

[dgOnPhys]

Hi snoopies622

I think Dirac is simply saying that his book will deal with non-holonomic constraints... if you have Goldstein's book you can look up some examples OR http://en.wikipedia.org/wiki/Nonholonomic_system"

 

[snoopies622]

So then, a holonomic constraint is a function of position and time, while a non-holonomic constraint is a function of position and momentum?

 

[dgOnPhys]

So then, a holonomic constraint is a function of position and time, while a non-holonomic constraint is a function of position and momentum?

More simply a non-holonomic costraint is any constraint that cannot be reduced (e.g. by integration) to an holonomic one; in general it will be a function of coordinates, momenta and time.