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# Energy Balance Invariance for Interacting Particle Systems

This paper studies the invariance of balance of

energy for a system of interacting particles under groups of

transformations. Balance of energy and its invariance is first

examined in Euclidean space. Unlike the case of continuous media,

it is shown that conservation and balance laws do not follow

from the assumption of invariance of balance of energy under

time-dependent isometries of the ambient space. However, the

postulate of invariance of balance of energy under arbitrary

diffeomorphisms of the ambient (Euclidean) space, does yield

the conservation laws. These ideas are then extended to the case

when the ambient space is a Riemannian manifold. Pairwise

interactions in the case of geodesically complete Riemannian

ambient manifolds are defined by assuming that the interaction

potential explicitly depends on the pairwise distances of

particles. Postulating balance of energy and its invariance under

arbitrary time-dependent spatial diffeomorphisms yields balance of

linear momentum. It is seen that pairwise forces are directed

along tangents to geodesics at their end points. One also obtains

a discrete version of the Doyle-Ericksen formula, which relates

the magnitude of internal forces to the rate of change of the

interatomic energy with respect to a discrete metric that is

related to the background metric.

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## Comments

## Question

Arash,

After Eq. 3.19 you say that it is not always true that the interatomic forces are self-equilibrated. For the type of pairwise interactions you are looking at I am not sure that I understand this statement. Specifically, aren't Eq. 3.4 and Eq. 3.19 in agreement? If so, then how is the statement after 3.19 true? If not, can you explain why?

Thanks,

Chad

P.S. I am still working my way through the rest.

## I see it, sorry.

Arash,

I see where my confusion was. For some reason I read 3.19 as a double sum, instead of the single sum that it is.

Chad

## Riemannian part

Dear Chad:

Thanks for your interest. Please contact me if there is anything unclear about the other parts.

Regards,

Arash

## But what does it mean, philosophically?

So, what does it mean philosophically? Should we expect energy conservation plus arbitrary diffeomorphic transformation to give balance of linear moment? On first thought, it would see like it shouldn't. However, on second thought, maybe it's obvious since each particle is a system itself and should have its own invariance requirements. This would imply that the converse of Prop. 3.1 would not be true as the diffeomorphism would only have to be discrete.

How does this relate to the interpretation of a system of particles as a single particle in a higher dimensional space. Would rigid coordinate translations in this space along with energy conservation give balance of linear momentum.

Also, since it's not hard to go from Newtonian mechanics to Lagrangian or Hamiltonian, what does this imply about these?

Eric Mockensturm

## invariance

Dear Eric:

Yes, energy conservation of the whole system plus invariance under arbitrary time-dependent diffeomorphisms (actually constant speed would suffice) would give conservation of linear momentum. I'm not sure if I understand what you mean by invariance requirement for each particle. To me invariance means that some functional, function, action, etc. remains unchanged under the action of a group (here the diffeimoroisms). In a way (if I understand you correctly?) having arbitrary diffeomorphisms means that energy balance is invariant under a diffeomorphism that acts nontrivially only on one particle at a time. I don't see how this would imply that the converse of 3.1 would not hold. If you assume balance of linear momentum and look at the difference between energy balance in two different frames invariance of energy balance under arbitrary diffeomorphsms will tell you that this difference is zero. Please also note that the invariance group is conitnuous; there are a finite number of particles but the group (diffeomorphisms) acts continuously on them. Again maybe I'm not ffollowing what you mean by "discontinuous"?

The main conclusion in this part is the following. In classical elasticity invariance of energy balance under arbitrary time-dependent isometries (rigid translations and rotations) will give all the balance laws (the converse is also true) but in particle systems this is not true; one needs to enlarge the group. This is also similar to what one sees in nonlocal theories of elasticity.

I don't understand what you mean exactly by interpreting a collection of particles as a single particle in a higher dimensional system. Maybe by that you mean a single particle with some microstructure, which is exactly what we have here? I don't think isometries would give you the balance laws you're looking for. But perhaps you can give more details here.

There is a connection between energy balance invariance and Lagrangian mechanics. Of course, the classical result is that if Lagrangian density is invariant under time shifts, then energy has to be conserved (Noether's theorem). The exact connection between energy balance (a global quantity) invariance and Euler-Lagrange equations, Noether's theorem, etc. is not completely clear in the literature as far as I know (I have a paper in preparation with Jerry Marsden and a couple of other mathematicians on this and would be happy to send you a draft when it's in a more complete form). In the context of particle systems, you could easily write an action and then extremize it. In doing so, each particle will have its own independent variation (this is a finite dimensional field theory, each particle position being a field). This is similar to having arbitrary diffeomorphisms (and not rigid motions only).

Regards,

Arash

## A small query

Hi Dr. Yavari

I have gone through the paper on energy balance under group of transformations. It is very interesting. Specially for Molecular dynamic simulation the model seems to be quite useful. I have a small question. If the particles have Stochastic Movement; say Brownian Movement (Having Jumps) can we apply the model? I am little confused about this. Do we need any interim operator for this. Please kindly let me know.

Thanks

Rezwan

## I am also confused: can you tell me in 3 lines the application?

Dear Yavari

can you explain me in 1 or 2 slides what the application is? I don't see any figure, I am lost in the maths. It may be that "figurare e descrivere" by Leonardo is a good suggestions (Leonardo had at least 4 drawings each page). Here I see text and maths... Please help for the poor practical engineer like me!!

Regards

mike

michele ciavarella

www.micheleciavarella.it

## Figures

Dear Mike:

Actually, there is one figure in the paper! I agree that figures can always help but in the math literature it's not common to have too many fugues. If you believe in figures, every equation might need one figure; it is implicitly assumed that the reader will draw some figure(s) in his/her mind.

Now the short description. In every theory there are balance laws, conserved quantities, inequalities, etc. One can always ask whether a given balance law is an "assumption" simplifying the theory or there is more into it, e.g. balance of linear momentum. The other thing to hope for is understanding the connection between different formulations of the same physical phenomenon and how to go from one formulation to another. "Invariance" or lack of invariance is a key idea in any physical theory. Here, we look at invariance of "energy balance", something that can be written for many systems unambiguously. Invariance is always defined with respect to some group of transformations and in this sense it would be meaningless to discuss invariance without specifying the group of transformations. Here we discuss invariance of energy balance for particle systems under different groups of transformations and consequences of postulating invariance.

On potential application (I haven't been able to work out the details yet) is to simplify a given (multibody) interatomic potential by going to an imaginary space in which particles interact in a simpler way (very much like going to Fourier space or Laplace space and simplify a linear differential equation).

Regarding Leonardo's approach, I have the highest respects for him but at the same time don't think we have to or even should follow his writing style in every paper.

Regards,

Arash

## Arash thanks for your reply and your effort: I remain confused!

I am not implying you should change style from mathematical one, but then you should NOT expect engineers to read you! This is life

michele ciavarella

www.micheleciavarella.it

## good point

Dear Mike:

You have a very good point here. I agree that it is important to write clearly and in a way that people can follow the ideas, derivations, etc. I personally do not like unnecessary abstraction and complicating things that are intrinsically simple. I think good ideas are simple and should be explainable in simple words. Having said all that, one should just hope that people follow what s/he writes. In the case of this paper, I've done the best I could.

Again in simple words, there are different approaches in deriving the governing equations of a given theory. It is, in principle, useful and important to understand the connection between these different formulations. For example, you can start from F=ma or from Principle of Virtual Work. These are "equivalent". But it seems some formulation(s) is preferable in a specific application. A good example is the weak formulation of finite element method.

Regards,

Arash

## Stochastic Movement

Dear Rezwan:

I don't know much about stochastic processes. I would say if you know what the modified energy balance is (if there are any modifications), then the invariance arguments should not be that different. If you send me some more details on what you have in mind (arash.yavari@ce.gatech.edu) I might be able to have some suggestions/ideas.

Regards,

Arash

## even if you have just 5 minutes, it is good to read Leonardo

michele ciavarella

www.micheleciavarella.it

## You should try to have this as a Liquid paper

Dear Arash

your paper will be read by few, and you say you are not willing nor able to make it simpler and more accesible. But in the future, this will change. This is because your paper is "solid", and there are too many papers around.

See my recent posts

LiquidPub Project: Scientific Publications meet the Web, a project from University of Trento

My letter of resignation from the board of Int J Solids and Structures / ELSEVIER

“Leading Opinion: Peer review as professional responsibility: A quality control system only as good as the participants.

## Concept of Manifold

Hi

I am trying to give some basic introduction on the works related to Topologial/Differentiable Manifolds. It is very common that a illustrative explanation makes any topic more lively. Unfortunately sometimes the abstract mathematical concepts are very difficult to be illustrated. However, the concept of manifold can be visualized as a geometric entity; more clearly surfaces. Simply any plane, surface of sphere and ellipsoid etc are manifolds in a sense. But the basic concept extentends to more complexity. In eucliden space we can use Pythagorian theorem but in a curvilinear space its not true in glabal sense. If we take the limiting condition it is then possible to think as euclidean space (Suppose a point on the sphere, a very small region surrounding that point may be dealt as euclidean space but if we keep going far from the point on the sphere we can't conserve this idea). So in order to deal these kind of problems the concept of manifold came. So if we draw a line on a sphere that is no longer a line. Hence we need to use Differential Geometry. Using the theories of differential geometry we can find the length, tangent sets on the line (Actually path). If we have more than one lines we can fine the shortest line or path which are called Geodesics. The Riemannian manifold comes when we deal with these kind of concepts. As mentioned earlier, a path on the sphere is not necessarily a straight line, so the pythagorian theorem must be modified little bit. Hence Riemannain metric comes into the scenario. We can say sphere as smooth surface. So it is Riemannian manifold. Because we can have a continous tangent space on the sphere. The concept of Riemannian manifold made the analysis easier.If we use topological operation such as a sphere is deformed to be an elliosoid, we can say a smooth manifold becomes another smooth maniflod. So it is a homeomorphism which preserves the differentiability condition on every point. Now if a question comes in mind that when a sphere became an ellipsoid, will they become totally independently seperate entities? In order to find this question we have to look for invariance property. The invariance features will say that both of the manifolds are the same at some extent. I tried to give this above explanation in order to give a physical idea of manifolds. In theoretical physics maniy fields such as relativity (Obvious;y general one), magnetohydrodynamics, plasmadynamics, quantum machanics, string theory etc they use these beautiful mathematical tools in order to explain plysical phenomena. Now a days in many theories such as fracture mechanics, damage mechanics, particale dynamics (Dr. Yavari's paper) etc are being tried to be explained by the concept of classical differential geometry. Please kindly have alook on the following link if some one is more interested on the theory of manifold.

"http://www.math.washington.edu/~lee/Books/Manifolds/c1.pdf" Instead of going through the text books this above document is very helpful to have a basic idea at a glance.

Thanks

Rezwan

## John M. Lee

Dear Rezwan:

John M. Lee is an excellent writer indeed.

Regards,

Arash

## Concept of Manifold (Books)

Hi Dr. Yavari

I would like to add some more books on topology and theory of manifolds. These books are the very nice in discussing about these bastract mathematical topics, as far as I have seen:

Topogy : James Munkers.Riemannian Manifolds, An introduction to Curvature: John M. LeeIntroduction to Topological Manifolds: John M. LeeThese books are very lively in explaining the physical concept of basic general topology, Homotopy theory,differntial geometry and theory manifolds.

Thanks

Rezwan

## Sorry about the mistake

Hi

I had little mistake. The name of the book is:

Topology : James MunkersI am sorry for this.

Thanks

Rezwan