 # On the information content of the difference from hamiltonian evolution

A dissipative version of hamiltonian mechanics is proposed via a principle of minimal information content of the deviation from hamiltonian evolution. We show that we can cover viscosity, plasticity, damage and unilateral contact. This article continues arXiv:1807.10480

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## 1 Introduction

In hamiltonian mechanics a physical system is described by a state vector

and a momentum vector . The evolution in time of the system is governed by a hamiltonian function , via the equations:

 ⎧⎪⎨⎪⎩˙q=∂H∂p(q,p,t)−˙p=∂H∂q(q,p,t) (1)

where , denote derivatives with respect to time. The evolution is reversible.

We propose a dissipative modification of these equations, which extends the work from , , .

###### Definition 1.1

Given a hamiltonian and a likelihood function

 H:X×Y×R→R,π:(X×Y)3→[0,1]

a physical system described by the state vector and the momentum vector evolves such that the vector defined by:

 ⎧⎪⎨⎪⎩˙q=∂H∂p(q,p,t)+ηq−˙p=∂H∂q(q,p,t)+ηp (2)

called the difference (or gap) from hamiltonian evolution driven by , maximizes the likelihood:

 π(z,˙z,η)=maxz"∈X×Yπ(z,˙z,z") (3)

with the notations , .

The likelihood function cannot be taken arbitrarily, otherwise the proposed modification would be too vague. We propose the following constraints on the likelihood function.

First we define the information content function associated to the likelihood as

 I:(X×Y)3→[0,+∞],I(z,z′,z")=−lnπ(z,z′,z") (4)

with the convention that .

###### Definition 1.2

The likelihood and the associated information content satisfy:

1. The information content function (4) is convex in each of the 2nd and 3rd variables and it has the needed degree of smoothness required (for example it is lower semi-continuous with respect to the relevant topologies on and )

2. for any , the following maxima exist

 maxz"∈X×Yπ(z,z′,z"),maxz"∈X×Yπ(z,z",z′)

and they are either or .

The equation (3) can be rephrased as: given the hamiltonian and the information content function

, the physical system evolves such that at any moment it minimizes the information content of the gap from a hamiltonian evolution. Indeed, an evolution of the system is a curve

with the property that it minimizes the information content gap functional:

 G(c)=∫T0I(c(t),˙c(t),˙q(t)−∂H∂p(c(t),t),−˙p(t)−∂H∂q(c(t),t))dt (5)

among all admissible evolution curves .

### Pure Hamiltonian evolution.

Let’s pick the information content (4) to be:

 I(z,z′,z")=χ0(z")={0 if z"=0+∞ otherwise

This corresponds to a likelihood function:

 π(z,z′,z")={1 if z"=00 otherwise

The maximization of the likelihood (3) implies that the gap vector , therefore the evolution equations (2) reduce to the pure Hamiltonian evolution equations (1).

This example is trivial, we need a method to construct more interesting ones. One such method is based on the following observation, adapted from , section 2. We use the notations explained in section 3, in particular we use the duality and we suppose that we have on a topology compatible with it, so that the information content function is lower semicontinuous (lsc).

###### Proposition 1.3

The information content function (4) satisfies the conditions from Definition 1.2 if and only if the function

 b:(X×Y)3→R∪{+∞},b(z,z′,z")= I(z,z′,z")+⟨⟨z′,z"⟩⟩ (6)

is a bipotential, i.e. it satisfies: for any ,

1. for any the functions and are convex (and lsc),

2. for any we have the equivalences

 z′∈∂b(z,z′,⋅)(z")⟺z"∈∂b(z,⋅,z")(z′)⟺I(z,z′,z")=0 (7)

where ”” denotes a subgradient, see section 3 for notations.

Bipotentials were introduced in  as a convex analysis notion which is well adapted for applications to non-associated constitutive laws. Bipotentials were used in soil mechanics, plasticity, damage or friction. For the theory of bipotentials see the review paper .

By concentrating our attention to the function , instead of the information content , we can build a host of examples. Indeed, for any lsc and convex function

 Φ:X×Y→R∪{+∞}

the associated function

 b(z,z′,z")=Φ(z′)+Φ∗(z") (8)

satisfies the conditions (a), (b) from Proposition 1.3 , where is the polar, or Fenchel conjugate of . Indeed, the mentioned condition (b) is just a reformulation of the Fenchel inequality .

###### Corollary 1.4

For an information content function of the form

 I(z,z′,z")=Φ(z′)+Φ∗(z")−⟨⟨z′,z"⟩⟩ (9)

the equation(3) is equivalent with the symplectic Brezis-Ekeland-Nayroles principle  definition 4.1,  definition 1.1:

 η∈∂Φ(˙z) (10)

## 2 Examples

### Viscosity, Rayleigh dissipation.

In particular, let’s pick

 Φ(z′)=Φ(q′,p′)=ϕ(q′)

where is a convex, lsc function. A straightforward computation of gives:

 Φ∗(z")=Φ∗(q",p")=sup{⟨⟨(q′,p′),(q",p")⟩⟩−ϕ(q′)∣q′∈X,p′∈Y}=
 =sup{⟨q′,p"⟩+⟨q",p′⟩−ϕ(q′)∣q′∈X,p′∈Y}=
 =χ0(q")+ϕ∗(p")

therefore the information content has the expression:

 I(z,z′,z")=ϕ(q′)+ϕ∗(p")+χ0(q")−⟨q′,p"⟩

By the corollary 1.4 we obtain the equations:

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩˙q=∂H∂p(q,p,t)−˙p=∂H∂q(q,p,t)+ηpηp∈∂ϕ(˙q) (11)

This shows that is a Rayleigh dissipation potential.

### Elasto-plasticity.

Take a hamiltonian system with space state and momentum space and supplement the state and the momentum with a new pair of state and momentum spaces:

 (q,qI)∈X×XI,(p,pI)∈Y×YI

Suppose further that the pair of spaces are in duality, so that we can define

 ⟨(q,qI),(p,pI)⟩=⟨q,p⟩+⟨qI,pI⟩

which leads us to a duality product of with itself:

 ⟨⟨(q′,q′I,p′,p′I),(q",qI",p",pI")⟩⟩=⟨q′,p"⟩+⟨q",p′⟩+⟨q′I,pI"⟩+⟨qI",p′I⟩

In this setting, we take and

 Φ(q′,q′I,p′,p′I)=ϕ(p′I)

where is a convex lsc function.

There are two differences with respect to the viscosity example: there is a cartesian decomposition of the state space and momentum space, and the dissipation potential depends on (a component of) the momentum variable, while previously the dependence was on the state variable.

By a computation analoguous with the one from the previous example we obtain:

 Φ∗(q",qI",p",pI")=ϕ∗(qI")+χ0(p")+χ0(pI")+χ0(q")

Via the corollary 1.4 we obtain the equations:

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩˙q=∂H∂p(q,p,qI,pI,t)˙qI=∂H∂pI(q,p,qI,pI,t)+ηq,I−˙p=∂H∂q(q,p,qI,pI,t)−˙pI=∂H∂qI(q,p,qI,pI,t)ηq,I∈∂ϕ(˙pI) (12)

In particular, let’s take , , Hilbert spaces, and a hamiltonian of the form:

 H(q,p,qI,pI,t)=K(p)+E(q−qI)−⟨q,f(t)⟩

where is the kinetic energy and is the elastic energy. We denote the elastic force by

 σ=∂E∂q(q−qI)

The 2nd and the 4th equations of the system (12) give:

 ηq,I=˙qI,˙pI=σ

The system (12) reduces to the familiar equations:

 ⎧⎪ ⎪⎨⎪ ⎪⎩˙q=∂K∂p(p)˙p=f(t)−σ˙qI∈∂ϕ(σ) (13)

In the next example we encounter a small extension of (9).

### Damage.

We consider here a standard model of damage. In  we introduced a formulation of hamiltonian evolution with dissipation which is the ancestor of the one proposed in this article. Then we applied that formulation to a more involved damage model, where we used an Ambrosio-Tortorelli functional  in the hamiltonian of the model.

As in the elasto-plasticity example, we supplement the spaces , with a new pair . Here is a damage variable and is a conjugated variable to the damage one.

Thefore we have a duality product over :

 ⟨⟨(q′,d′,p′,r′),(q",d",p",r")⟩⟩=⟨q′,p"⟩+⟨q",p′⟩+d′r"+d",r′⟩

We choose a hamiltonian

 H(q,p,d,r,t)=K(p)+k(r)+(1−d)E(q)−⟨q,f(t)⟩

where is the kinetic energy, is a term akin to the kinetic energy, but for the variable and is the elastic energy.

The difference from the previous examples is that the damage variable and the sign of are constrained (i.e. damage cannot decrease):

 d∈[0,1],˙d≥0

We have to include these constraints in the expression of the information content:

 I(q,d,p,r,˙q,˙d,˙p,˙r,ηq,ηd,ηp,ηr)=χ[0,1](d)+
 +χ[0,+∞)(˙d)+Y˙d+χ(−∞,Y](ηr)+
 +χ0(ηp)+χ0(ηq)+χ0(ηd)−⟨⟨(˙q,˙d,˙p,˙r),(ηq,ηd,ηp,ηr)⟩⟩

The expression of the information content does not have the form (9), but instead it has the form

 I(z,˙z,η)= { \it constraints }+Φ(˙z)+Φ∗(η)−⟨⟨˙z,η⟩⟩

with, in this case

 Φ(˙z)=ϕ(˙d)=χ[0,+∞)(˙d)+Y˙d

We obtain the equations:

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩˙q=∂K∂p(p)˙d=∂k∂r(r)−˙p=(1−d)∂E∂q(q)−f(t)−˙r=−E(q)+ηr (14)

and the inequations:

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩d∈[0,1]˙d≥0ηr∈∂ϕ(˙d) (15)

In the next example we shall need a smoothness assumption. The alternative would be to renounce at the condition 1.2 (a) and to keep only the condition (b).

### Unilateral contact.

We take , to be dual, finite dimensional Banach spaces and to be a closed, nonempty set of admissible states . With the notations about Fréchet normals from  1.4, we shall need that is smooth enough, in the sense that the tangent cone at any is the polar of the normal cone at x, denoted by .

There is no special form of the hamiltonian required. The information content has the form:

 I(z,˙z,η)=χM(q)+χN(q∣M)(ηp)+(χN(q∣M))∗(˙q)−⟨⟨˙z,η⟩⟩

The smoothness assumption on is that

 (χN(q∣M))∗=χT(q∣M)

i.e. that the tangent cone is the polar of the normal cone. In general it is true that the normal cone is the polar of the tangent cone, but not the other way around. An alternative would be to not make any smoothness assumptions on and instead to take an information content of the form

 I(z,˙z,η)=χM(q)+χN(q∣M)(ηp)+χT(q∣M)(˙q)−⟨⟨˙z,η⟩⟩

Again, in general the tangent cone is not convex, therefore this form of information content contradicts the convexity condition 1.2 (a).

We obtain the equations:

 ⎧⎪⎨⎪⎩˙q=∂H∂p(q,p,t)−˙p=∂H∂q(q,p,t)+ηp (16)

and the unilateral contact conditions:

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩q∈M˙q∈T(q∣M)ηp∈N(q∣M)⟨˙q,ηp⟩=0 (17)

## 3 Notations and useful definitions

### General notations.

The space of states and the space of momenta are real topological vector spaces in duality:

 (q,p)∈X×Y↦⟨q,p⟩∈R

We suppose the usual: the duality is bilinear, continuous and for any linear and continuous functions , there exist , such that and .

The space is in duality with itself by:

 ⟨⟨(q1,p1),(q2,p2)⟩⟩=⟨q1,p2⟩+⟨q2,p1⟩

The space is also symplectic, with the symplectic form defined by: for any ,

 ω(z1,z2)=⟨⟨Jz1,z2⟩⟩=⟨q1,p2⟩−⟨q2,p1⟩

For any differentiable function the gradient of at a point is the element with the property that

 limε→01ε(H(z+εz′)−H(z))=⟨⟨DH(z),z′⟩⟩

and the symplectic gradient of is is defined in a similar way by the equality

 limε→01ε(H(z+εz′)−H(z))=ω(XH(z),z′)

If we use the partial derivatives notation

 DH(q,p)=(∂∂pH(q,p),∂∂qH(q,p))

then

 XH(q,p)=(∂∂pH(q,p),−∂∂qH(q,p))

With the introduction of the linear conjugation

 ¯⋅:N→N,¯¯¯¯¯¯¯¯¯¯¯¯(q,p)=,(q,−p)

we have .

### Convex analysis notations.

These are the classical ones from Moreau . We add the field of reals . The addition operation is extended with for any . The multiplication with positive numbers is extended with: if then .

For any function , it’s domain is .

The set of lower semicontinuous (lsc), convex functions defined on , with non-empty domain is . The indicator function of a convex and closed set is

 χA(x)={0 if x∈A+∞ otherwise

For any natural number , a non empty set is -monotone monotone if for any collection we have the inequality:

 ⟨xn−x0,yn⟩+n∑1⟨xk−1−xk,yk−1⟩≥0.

The set is maximally -monotone if it is -monotone and maximal with respect to the inclusion of sets.

The set is cyclically monotone if it is monotone for any natural number . It is cyclically maximal monotone if it cyclically monotone and maximal with respect to the inclusion of sets.

The subdifferential of a function at a point is the set:

 ∂ϕ(x)={u∈Y∣∀z∈X ⟨z−x,u⟩≤ϕ(z)−ϕ(x)} .

The polar of a function is

 ϕ∗(y)=sup{⟨x,y⟩−ϕ(x)∣x∈X}

The polar is always convex and lsc.

Polars and subgradients are related by the Fenchel inequality. For any function which is convex, lsc, we define

 c(x,y)=ϕ(x)+ϕ∗(y)−⟨x,y⟩

for any , . The Fenchel inequality has two parts:

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