Can a time dependent field to be conservative?

[BarbaraDav]

(Sorry for my poor English, Please, forgive mistakes, if any.)

Dear Friends

Not doubts about what is to be meant for "conservative vector field" as far as time independent fields are concerned.

But what about non stationary fields? I thought it was a meaningless concept when field is changing in time: the line integral seems to be not well defined. All in all, why should I add up together values measured in different moments?

Nevertheless I suspect that in lagrangian and hamiltonian formalism someone consider "conservative" a non stationary field if a time varying potential function exists such as, in each instant, (at "frozen time", as Italians call it), the field is its gradient.

What do you think about that? Am I wrong?

Warmest regards.

Barabara Da Vinci
(Italy)

 

[Dale]

Hi Barbara,

If the Lagrangian is symmetric under any differentiable operation then by Noether's theorem there is a conserved quantity associated with that symmetry.

So, if the action is static then it is symmetric under time translations and the conserved quantity associated with that symmetry is called energy. On the other hand, if the action is not static then it is not symmetric under time translations and energy is not conserved.

 

[BarbaraDav]

(Sorry for my poor English, Please, forgive mistakes, if any.)

Dear Friends

Thanks for your reply!

I agree. May be I am wrong, but Noether's theorem states a sufficient condition for a conservation law to exist, not a necessary one.

As an example, as far as I know, the Laplace-Runge-Lenz vector, in a inverse square law keplerian problem, is conserved but no lagrangian's symmetry group is associated with it (possibly, one exists when the system is embedded in higher dimensional space, but I am not sure about that).

So I wonder if a line integral along a closed curve, in a time dependent field, still helds some significance despite energy being not conserved. Just to mention a non trivial conserved quantity (I evoke it here only as an example of a quantity hypothetically related with the line integral), let's think of the Jacobi's integral:

$$\sum_{i=1}^{n}\dfrac{\partial L}{\partial \dot{q}^{i}}\dot{q}^{i}-L$$

Warmest regards

Barabara Da Vinci
(Italy)

 

[Prologue]

Consider the lagrangian

$$L=\dot{q}\frac{t^{2}}{2}+qt$$

When you pop that into the euler-lagrange equation, it works. What kind of field does it correspond to? I dunno, but it may be a clue.

 

[Dale]

So I wonder if a line integral along a closed curve, in a time dependent field, still helds some significance despite energy being not conserved.

That quantity may have some significance, I don't really know. A time-varying action may exhibit many other symmetries or conserved quantities, but energy is specifically the conserved quantity associated with time-translation symmetry. So whatever other conserved quantities may exist they would not be called energy.

 

[BarbaraDav]

I see. Thanks!