Proving Conservation Laws for Galilean Boosts - Norm

[Norman]

Anyone know what quantity is conserved if the Hamiltonian (classical) is invariant under a Galilean boost? Also how would I prove that it is this quantity that is conserved?
Cheers,
Norm

 

[HallsofIvy]

A Galilean "boost"? If you mean a Galilean transformation, since the Hamiltonian of a system is its total energy, invariant Hamiltonian means unchanging energy- that's conservation of energy.

 

[Norman]

I don't think that you are quite understanding my question. If a Hamiltonian is unchange under rotation, this implies that angular momentum is conserved. If the Hamiltonian is unchanged under translation, this implies that linear momentum is conserved. So if the classical Hamiltonian is unchanged under a Galilean boost (x-> x+vt and p->p+mv) what is the physical quantity conserved? Any obvious ways to see this? I think that it may actually be energy, but am unsure of it.

 

[selfAdjoint]

Umm, that's the Lagrangian that has that property, not the Hamiltonian. Noether's theorem. If the Lagrangian is invariant under some symmetry, then the equations of motion will contaain a conserved quantity corresponding to that symmetry. And vice versa, if you find a conserved quantity you can look for a corresponding symmetry of the Lagrangian.

 

[Norman]

Actually it is the Hamiltonian. See for example, Shankar p99.
Cheers,
Norm

 

[Norman]

for those that care,

since no one answered this question, I assume none of you know, or didn't feel the need to enlighten me, so I will enlighten you. For the classical case, a galilean boost (or transformation if you prefer) conserves the center of mass of a system of particles. That is if your Hamiltonian (or Lagrangian for those that prefer that method) is invariant under this boost, the center of mass is conserved as the Hamiltonian is time evolved. It is not obvious at all that when it is worked out for a single body, that the quantity that conserved corresponds to this, but only through considering many bodies was I able to understand it in this physical way. It is interesting to note that for the relativistic case (lorentz boost), that the quantity conserved is the center of mass per unit energy. This is even more difficult to see in my eyes. Hope this helps anyone who was wondering like I was.
Cheers,
Norm

 

[Alan19]

Hamiltonian under Galilean boost

Norman correct only partially and only in part when he refers to the center of mass. In fact, if we consider the Hamiltonian for a system of free particles, then invariance of such a Hamiltonian under the Galilean boost would mean that the center of mass of the system moves with a constant velocity equal to 1/2 of the boost parameter ( velocity). The conservation of the center of mass would mean zero velocity of the c.m., which for the above system is possible only if its total momentum is 0. This does not tell us anything about the Hamiltonian.

 

[kinnee2000]

electric field-electric field due to acontinuous charge distribution

my aim is to find related questions on this topic

A Galilean "boost"? If you mean a Galilean transformation, since the Hamiltonian of a system is its total energy, invariant Hamiltonian means unchanging energy- that's conservation of energy.

 

[kinnee2000]

would you give me responce or an answer on questions of electric field

 

[Alan19]

You wrote
"A Galilean "boost"? If you mean a Galilean transformation, since the Hamiltonian of a system is its total energy, invariant Hamiltonian means unchanging energy- that's conservation of energy."

1) In theoretical physics sometimes the G.transformation is called the "Galilean boost".

2)Conservation of energy in A GIVEN FRAME OF REFERENCE. When there is the G.transform, this means that the energy of the system is not the same as it was in the original frame, since the new system MOVES(!) with respect to the first one.
This means that there are additional energy and momentum of the system,when measured in the new frame.
Thus invariance of the Hamiltonian under the GT does not mean conservation of energy.

3) Conservation of energy is related to the absence of the EXPLICIT dependence of the Hamiltonian on time in a given system. A change to another system moving with respect to the first one would result in the explicit appearance of such a dependence in the Hamiltonian. Even GT gives you
x=x'+Ut, t=t'

So within any of these systems ( and they must be closed) the energy is conserved, but not when the energy of the one is measured from the other.

 

[prophetlmn]

this problem also confused me for a long time，but after I saw this
http://physics.uoregon.edu/~soper/QuantumMechanics/boosts.pdf
the problem is solved.
in short,in general the hamiltonian or lagrangian is not invariant under a boost translation，you can see this by noether's theorem or simlply trans the hamiltionian,
you will get a term about central mass,to get a conservation law,you should consider only systems have no centeal mass velocity,i.e the internal energy.

 

[prophetlmn]

further more,you should define quantities in a reference that H=internal energy,you could get the internal momentum,angular momentum and so on,which are invariant under boost trans.these techniques are a little complicated and boring,and useless in paractical problems,I guess this is the reason that most authors do not like to talk about boost.

 

[prophetlmn]

and at last,you will get something seems trivial here,the boost is related to the phase of state... so it makes no physical importance.

 

[Sina]

If you know differential geometry:

1- Find the vector field X that generates this transformation (by definition of flow of a vector field you can do this directly by computation). check that it is a symmetry by getting XH =0 and check that X is a symplectomorphism.

2- Find a function such that X is the hamiltonian vector field of this function (by direct computation using hamilton's equation).

3- That function is a conserved quantity.

If you would like more details I can elaborate the technique.