Geometric network modeling and control of physical systems

Geometric network modeling and control of physicalsystems

Arjan van der Schaft

Johann Bernoulli Institute for Mathematics and Computer ScienceUniversity of Groningen, the Netherlands

La Cristalera, October 4-6, 2010

Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systemsLa Cristalera, October 4-6, 2010 1 /

52

Outline of Part III

1 Port-Hamiltonian dynamics on graphs

2 Consensus algorithms as port-Hamiltonian systems

3 Port-Hamiltonian dynamics on k-complexes

4 Thermostatics

5 Thermodynamical systems


52

Port-Hamiltonian dynamics on graphs

Outline of Part III




4 Thermostatics



52


Directed graphs

Definition

A directed graph G consists of a finite set V of vertices and a finite set Eof directed edges, together with a mapping from E to the set of orderedpairs of V (with no self-loops allowed).

• Thus, to any edge e ∈ E there corresponds an ordered pair(v ,w) ∈ V × V, with v 6= w , representing the tail vertex v and thehead vertex w of this edge.

• A directed graph is specified by its incidence matrix B .

• B is an v × e matrix, with (i , j)-th element equal to 1 if the j-th edgeis an edge towards vertex i , equal to −1 if the j-th edge is an edgeoriginating from vertex i , and 0 otherwise.

• In the sequel ’graphs’ will throughout mean ’directed graphs’.


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The vertex and edge space

Definition

The vertex space Λ0 is the vector space of all functions from V to R.Hence Λ0 can be identified with R

v .The edge space Λ1 is the vector space of all functions from E to R. ThusΛ1 is identified with R

e .The dual spaces of Λ0 and Λ1 are denoted by Λ0, respectively Λ1.

The incidence matrix B induces the linear incidence operator

B : Λ1 → Λ0

whileBT : Λ0 → Λ1,

is called the co-incidence operator.


52


Open graphs

Definition

An open graph G is obtained from an ordinary graph with set of vertices Vby identifying a subset Vb ⊂ V of boundary vertices.These are the vertices that are open to interconnection (i.e., with otheropen graphs).The remaining vertices Vi := V − Vb are the internal vertices of the opengraph.

Corresponding splitting of the vertex space and its dual:

Λ0 = Λi ⊕ Λb

Λ0 = Λi ⊕ Λb

while the incidence operator B : Λ1 → Λ0 splits as

B = Bi ⊕ Bb

with Bi : Λ1 → Λi and Bb : Λ1 → Λb.Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systems

La Cristalera, October 4-6, 2010 6 /52


The vertex-edge Dirac structure

For any open graph G define the Dirac structure

Dve(G) = {(f , e, fi , ei , fb, eb) ∈

Λ1 × Λ1 × Λi × Λi × Λb × Λb |

Bi f = −fi ,Bbf = −fb, e = BTi ei + BT

b eb}

Dve(G) is called the vertex-edge Dirac structure.


52

Consensus algorithms as port-Hamiltonian systems

Outline of Part III




4 Thermostatics



52


A standard consensus algorithm

Consider a network of agents moving in linear space R, whose interactiontopology is described by an undirected graph G (symmetric interaction),with set of undirected edges E (G).

Distinguish between leader and follower agents, associated to theboundary, respectively, internal vertices.

Corresponding to each agent v there is a variable xv ∈ R describing themotion in the linear space R. In a standard consensus algorithm the vectorxv of each follower agent v satisfies the following dynamics

xv(t) = −∑

(v ,w)∈E(G)

g(v ,w)(xv (t)− xw (t))

where g(v ,w) > 0 denotes a certain positive-definite weight associated toeach edge.


52


Port-Hamiltonian formulation

Collecting all follower variables xv into one vector x ∈ Rvi , and all leader

variables xv into one vector u ∈ Rvb the dynamics can be written as

x = −BiGBTi x − BiGB

Tb u

with B the incidence matrix of the graph endowed with an arbitraryorientation, and G the diagonal matrix with elements g(v ,w) for each edge(v ,w).This can be seen to define a port-Hamiltonian system with respect to thevertex-edge Dirac structure Dve(G).


52


Port-Hamiltonian formulation continued

Define the Hamiltonian function as

H(x) =1

2‖ x ‖2

leading to the energy storage equations

x = −fi , ei =∂H

∂x(x) = x

Furthermore, let the variables f , e satisfy the relations for energydissipation given by

f = −Ge

Then the consensus dynamics is given as the port-Hamiltonian system

x = −BiGBTi

∂H∂x

(x)− BiGBTb u

y = BbGBTi

∂H∂x

(x) + BbGBTb u

with u = eb ∈ Λb, y = fb ∈ Λb, and∂H∂x

(x) = x .


52


Note the additional output vector y , satisfying

d

dtH = −

[

xT uT]

BGBT

[

xu

]

+ yTu ≤ yTu,

and thus is a passive output for the consensus dynamics.

By interpreting H to be the energy stored in the consensus dynamics, theterm yTu corresponds to the power supplied to the system.

Remark

A physical analogue is the dynamics of a number of unit masses(corresponding to each internal vertex), with linear dampers associated tothe edges, and externally prescribed boundary velocities u = ebcorresponding to the boundary vertices, with outputs y = fb being theboundary forces.


52


Coordination control

A different port-Hamiltonian dynamics with regard to the same Diracstructure occurs in coordination control.

In this case there is energy-storage associated to the edges (instead ofdamping as in the case of consensus algorithms.

Remark

In a typical formation control context, pv is a momentum variable (orvector), while qe is a configuration variable. Furthermore, eb is anexternal (reference) velocity vector, and fb the corresponding generalizedforce vector. Thus the total system is a mass-spring system with masses

corresponding to the vertices and springs corresponding to the edges.


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Port-Hamiltonian dynamics on k-complexes

Outline of Part III




4 Thermostatics



52


Lumped port-Hamiltonian modeling of distributed

phenomena

• Discretize the spatial domain, and lump the system variables.

• Important: lump the variables in such a manner that their ’nature’ ispreserved.(Thus provide the direct discrete analogues of the 0-forms, 1-forms,2-forms, etc., employed in the description of distributed-parameterport-Hamiltonian systems).

• The resulting finite-dimensional system is port-Hamiltonian.

• Provides an alternative to discretizing the port-Hamiltonian pdemodel in a structure-preserving manner (using mixed finite elements).

Secondary goal: Study of systems on networks of higher complexity thangraphs.


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Discretization of the spatial domain: k-complexes

A k-complex Λ is a sequence of linear spaces Λ0,Λ1, · · · ,Λk , withincidence operators

Λk∂k→ Λk−1

∂k−1→ · · ·Λ1

∂1→ Λ0∂0→ R

with the property that

∂j−1 ◦ ∂j = 0, j = 1, · · · , k

The vector spaces Λj , j = 0, 1 · · · , k , are called the spaces of j-chains.


52


A typical example of a 2-complex is the triangularization of a 2-dimensionalmanifold (surface), with the j-cells, j = 0, 1, 2, being the sets of vertices,edges, faces, and ∂1, ∂2 capturing the incidence structure.

Example

Consider a triangularization of a 2-sphere with 4 faces, 6 edges, and 4vertices. The incidence matrix D2 (corresponding to ∂2) is

< v1v2v3 > < v1v3v4 > < v1v4v2 > < v2v4v3 >< v1v2 > 1 0 −1 0< v1v3 > −1 1 0 0< v1v4 > 0 −1 1 0< v2v3 > 1 0 0 −1< v2v4 > 0 0 −1 1< v3v4 > 0 1 0 −1


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Example (Example continued)v3v4 v4

v2

v4

v1

The incidence matrix D1 (corresponding to ∂1) is given as

< v1v2 > < v1v3 > < v1v4 > < v2v3 > < v2v4 > < v3v4 >< v1 > −1 −1 −1 0 0 0< v2 > 1 0 0 −1 −1 0< v3 > 0 1 0 1 0 −1< v4 > 0 0 1 0 1 1


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k-complexes continued

Denoting the dual linear spaces by Λj , j = 0, 1 · · · , k , we have thefollowing dual sequence

Λ0 d1→ Λ1 d2→ Λ2 · · ·Λk−1 dk→ Λk

having the analogous property

dj ◦ dj−1 = 0, j = 2, · · · , k

The elements of Λj are called j-cochains.


52


Directed graphs as 1-complexes

Consider a directed graph G with incidence matrix B (from its edges tovertices).

This defines a 1-complex. Indeed, the sequence

Λ1B→ Λ0

11→ R

satisfies 11 ◦ B = 0.


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Open k-complexes

An open k-complex is obtained by identifying a subset N(k−1)b of the setof all (k − 1)-cells, called the boundary (k − 1)-cells.

Define the linear space of functions from the boundary (k − 1)-cells to R

as Λb ⊂ Λk−1 (the space of ’boundary currents’).


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Hamiltonian dynamics on a k-complex

Similar to the vertex-edge Dirac structure we define the Dirac structurecorresponding to the following relations:

fx = −dk

[

eieb

]

, fx ∈ Λk ,

[

eieb

]

∈ Λk−1, eb ∈ Λb

[

fifb

]

= ∂kex , ex ∈ Λk ,

[

fifb

]

∈ Λk−1, fb ∈ Λb

with fb, eb corresponding to the boundary (k − 1)-cells, and fi , eicorresponding to the internal (k − 1)-cells.


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Associate to every k-cell an energy storage, leading to a total energystorage H(x), where x ∈ Λk denotes the vector of energy variables, with

x = −fx , ex =∂H

∂x(x)

Furthermore, associate to the internal (k − 1)-cell the resistive relation

ei = −Rfi , R = RT ≥ 0

This yields the (relaxation) Hamiltonian dynamics

x = −d rk R ∂r

k∂H∂x

(x) + dbk eb

fb = ∂bk∂H∂x

(x)

where we have split dk as dk =[

d rk db

k

]

and ∂k =

[

∂rk

∂bk

]

.


52


This defines a port-Hamiltonian system with inputs eb and outputs fb.

It is the discrete analog of Stefan-Maxwell diffusion (in vector calculusnotation)

∂∂tx(z , t) = div(R(z)grad∂H

∂x(x(z , t))

y(z , t) = ∂H∂x

(x(z , t))

with z ∈ Z (spatial domain), plus the boundary conditions

fb(z , t) = −R(z)grad∂H∂x

(x(z , t)) · n

eb(z , t) = ∂H∂x

(x(z , t))

on ∂Z .


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These equations immediately imply the following passivity property

dH

dt= −

∫

Z

ex fx = −

∫

Z

f TR(z)fdz +

∫

∂Z

ebfb ≤

∫

∂Z

ebfb

where H is the total energy

H(x(·, t)) =

∫

Z

H(x(z , t))


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Heat transfer on a 2-complex

We will write the heat transfer as a conservation law in terms of theconservation of internal energy:The internal energy u of the 2-complex is a 2-cochain, u ∈ Λ2 (with everycomponent of u denoting the energy of the corresponding 2-cell).The thermodynamic properties are defined by Gibbs’ relation, andgenerated by the entropy function s = s(u) as thermodynamic potential.Since we consider transformations with constant volume and without masstransfer, Gibbs’ relation reduces to the definition of the vector of intensivevariables eu, conjugated to the extensive variables u by

eu =∂s

∂u(u)

The components of the vector eu ∈ Λ2 are equal to the reciprocal of thetemperature in each 2-cell.


52


The heat conduction is given by the heat flux

e ∈ Λ1

describing the heat flux through every 1-cell (edge). This flux arises fromthermal non-equilibrium, defined by the fact that the temperature isvarying over the 2-cells.Its conjugate vector of variables is the thermodynamical driving forcevector

f ∈ Λ1

given as the vector of the differences of the reciprocals of the temperaturesof the 2-cells with common edges (1-cells)

f = ∂2eu


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By Fourier’s law the heat flux due to thermal non-equilibrium is expressedas

e = R(eu) f ,

with R(eu) = RT (eu) ≥ 0 depending on the heat conduction coefficients.(Note the sign-difference !). Finally, we have the conservation law

du

dt= d2e

Hence the resulting system is a port-Hamiltonian system (of relaxationtype) defined on the 2-complex, with vector of state variables x given bythe internal energy vector u, and Hamiltonian s(u).By the different sign the entropy s(u) satisfies

ds

dt= (∂2

∂s

∂u(u))TR(eu)∂2

∂s

∂u(u) ≥ 0

expressing the fact that the entropy s(u) is monotonously increasing.


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Exchange of heat with the environment

The exchange of heat through the boundary of the system can beincorporated by splitting the edges (1-cells) into internal edges with theresistive relation and external (boundary) edges. This would lead to

ds

dt= (∂2

∂s

∂u(u))T R(eu) ∂2

∂s

∂u(u) + ebfb

with eb, fb denoting the heat flux, respectively, thermodynamical drivingforce, through the boundary 1-cells.


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Another type of Hamiltonian dynamics is obtained by considering therelations

fx1 = dkex2 , fx1 ∈ Λk , ex2 ∈ Λk−1

fx2 = −∂kex1 , ex1 ∈ Λk , fx2 ∈ Λk−1

defining again a Dirac structure, together with two energy-storagerelations

x1 = −fx1, ex1 =∂H1

∂x1(x1)

x2 = −fx2, ex2 =∂H2

∂x2(x2)

leading to the oscillatory dynamics

x1 = −dk∂H∂x2

(x1, x2)

x2 = ∂k∂H∂x1

(x1, x2)

Example 1: (k = 2) Discretized Maxwell’s equations on a 2-dimensionaldomain.


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Thermostatics

Outline of Part III




4 Thermostatics



52

Thermostatics

Thermostatics and contact geometry

(cf. Hermann, Mrugala)

The thermodynamic phase space, on which the thermodynamic propertiesof a system are defined, is a contact manifold endowed with a contact

structure.

Typical example of a contact manifold:

Let N be an n-dimensional manifold. Define the associatedthermodynamic phase space T as R× T ∗N whose elements are denoted(x0, x , p). It has a canonical contact structure defined by the contact form

θ = dx0 −n

∑

k=1

pkdxk ,

where d denotes the exterior derivative.


52

Thermostatics

Definition

A 1-form θ on a 2n + 1-dimensional manifold T is a contact form ifθ ∧ (dθ)n is a volume form. Then the pair (T , θ) is called a contactmanifold.

Theorem

Given a contact form θ on a 2n + 1-dimensional manifold T . Then thereexists local coordinates (x0, x , p) for T such that

θ = dx0 −n

∑

k=1

pkdxk ,

In thermodynamic systems x0 is associated with a thermodynamicpotential, such as the energy U, the enthalpy H, etc., and (x i , pi ) denotesthe pairs of conjugated extensive and intensives variables, such as the pairsof volume V and pressure P , entropy S and temperature T , number ofmoles Ni and chemical potentials µi .


52

Thermostatics

The Gibbs’ relation

dU = TdS − PdV + µidNi

corresponds to the definition of a canonical submanifold of a contactmanifold, called Legendre submanifold.

Definition

A Legendre submanifold of a contact manifold (T , θ) is an n-dimensionalsubmanifold of T that is an integral manifold of θ.


52

Thermostatics

Legendre submanifolds are locally generated by a generating function.

Theorem

For a given set of canonical coordinates and any partition I ∪ J of the setof indices {1, . . . , n} and for any differentiable function F (x I , pJ) of nvariables, i ∈ I , j ∈ J, the formulas

x0 = F − pJ∂F

∂pJ, xJ = −

∂F

∂pJ, pI =

∂F

∂x I(1)

define a Legendre submanifold of R2n+1. Conversely, every Legendresubmanifold of R2n+1 can be defined in a neighborhood of every point bythese formulas, for at least one of the 2n possible choices of the subset I .


52

Thermostatics

Legendre submanifold continued

Consider the particular case of a generating function F which is adifferentiable function on N . The Legendre submanifold is then the set

LF :=

{

x0 = F (x), x , p =∂F

∂x(x)

}

For thermodynamic systems, the generating functions are potentials suchas U, G , etc., while the associated Legendre submanifold defines thethermodynamic properties of the system, an ideal gas.


52

Thermostatics

Example (Ideal gas)

One way to express its thermodynamic properties is to use the contactform

θG := dG + SdT − VdP − µdN

with generating function being the Gibbs free energy G (where R is theideal gas constant):

G (T ,P ,N) =5

2R(1− ln(

T

T0))− NT (S0 − R ln(

P

P0)

The generated Legendre submanifold is

x0 = G (T ,P ,N)

p1 = −S(T ,P ,N) = ∂G∂T

= NS0 + 5/2NR ln( TT0)− NR ln( P

P0) = S

p2 = V (T ,P ,N) = ∂G∂P

= NR TP

p3 = µ(T ,P ,N) = ∂G∂N

= 5/2RT − T SN


52

Thermostatics

Example (Ideal gas continued)

However, one may also take as coordinates for the Legendre submanifoldthe extensive variables S ,V ,N. In this case the generating function is theinternal energy U obtained by the Legendre transform of G as

U(S ,V ,N) = G − P∂G

∂P− T

∂G

∂T=

3

2NRT0 exp[γ(S ,V ,N)]

where

γ(S ,V ,N) = (S − NS0 + NRln(NRT0)− NR ln(VP0))/(3

2NR)

with the contact form given as

θ = dU − TdS + PdV − µdN


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Thermodynamical systems

Outline of Part III




4 Thermostatics



52


Thermodynamics

Definition

A vector field X on (T , θ) is a contact vector field if and only if thereexists a differentiable function ρ such that

LX θ = ρ θ,

where LX denotes the Lie derivative with respect to the vector field X .

The set of contact vector fields forms a Lie subalgebra of the Lie algebraof vector fields.


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Contact vector fields continued

There exists a mapping between contact vector fields and differentiablefunctions. To every contact vector field X one associates the function

f = θ(X )

This function f is called the contact Hamiltonian. Conversely, to everyfunction f there corresponds the contact vector field Xf given as

Xf =(

f −∑n

k=1 pk∂f∂pk

)

∂∂x0

+ ∂f∂x0

(

∑nk=1 pk

∂∂pk

)

+∑n

k=1

(

∂f∂xk

∂∂pk

− ∂f∂pk

∂∂xk

)

.


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Contact vector fields continued

The mapping f → Xf defines an isomorphism from the Lie algebrastructure of the contact vector fields to the Lie algebra structure on thespace of contact Hamiltonians defined by the following bracket (’Jacobibracket’):

{f , g} = θ([Xf ,Xg ])

where [·, ·] is the usual Lie bracket.(This replaces the usual isomorphism between the Lie algebra ofHamiltonian vector fields and the Lie algebra of functions with respect tothe Poisson bracket (modulo R), on any symplectic manifold (e.g. acotangent bundle T ∗)).


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Theorem (Mrugala)

Let (T , θ) be a contact manifold and L a Legendre submanifold. Then XK

is tangent to L if and only if K is identically zero on L.

Example

Let the generating function of L be U(x i , pj), i ∈ I , j /∈ I , I ⊂ {1, · · · , n}.Consider the contact Hamiltonian

K = x0 − U + pj∂U

∂pj

This generates a reversible transformation. In the example of the ideal gas

K = PV − NRT

’Real’ dynamics corresponds to the case where the contact Hamiltonian Krepresents the (irreversible) dynamics in the system due to non-equilibriumconditions (e.g., due to heat conduction or chemical reaction kinetics).


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Control contact systems

Definition

Let (T , θ) be a contact manifold and L a Legendre submanifold. A controlcontact system on T is determined by an input space U = R

m and inputsuj , j = 1, ..,m, together with m + 1 contact Hamiltonian functions : K0

the internal contact Hamiltonian and Kj , j = 1, . . . ,m the interactioncontact Hamiltonians, all satisfying the invariance condition :Kl |L ≡ 0, l = 0, . . . ,m. The dynamics of the control contact system isgiven by

XK0+

m∑

j=1

uj XKj.


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• The Legendre submanifold L represents the thermodynamicproperties of the system.

• The internal contact Hamiltonian K0 represents the law giving thedynamics due to non-equilibrium conditions in the system (forinstance due to heat conduction or chemical reaction kinetics).

• The interaction contact Hamiltonians Kj provide the flows due to thenon-equilibrium of the system with its environment.

Example (Ideal gas in a cylinder with moving piston)

Define the interaction contact Hamiltonian

K (x , p) = (P − p2)

This leads to dVdt

= u.


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How to merge thermodynamical systems with port-Hamiltonian systems ?

Lifting of conservative systems on N to R× T ∗N

Consider a closed dynamical system on N :

x = f (x), x ∈ N

This can be lifted to the contact vector field XKfon R× T ∗N with

contact HamiltonianKf (x

0, x , p) = −pT f (x)

Lemma

Consider a Legendre submanifold LH with generating function H(x). ThenKf is zero restricted to LH if and only if

H = Lf H = 0

Thus: XKfdefines a dynamics on LH if and only if x = f (x) is

conservative with respect to H.Arjan van der Schaft (Univ. of Groningen) Geometric network modeling and control of physical systems

La Cristalera, October 4-6, 2010 46 /52


Remark

In this case the dynamics XKfon LH can be described in terms of the

extensive variables x asx = f (x),

and equally well in terms of the intensive variables p, provided that themap

p =∂H

∂x(x)

is injective. The dynamics in extensive variables p is then given by

p = −∂K

∂x= −(Df )T (x)p

with Df the Jacobian of f . (For LC-circuits this yields theBrayton-Moser equations.)


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In particular, the Hamiltonian dynamics

x = J(x)∂H

∂x(x)

can be lifted to a contact vector field with contact Hamiltonian

K (x , p) = −pTJ(x)∂H

∂x(x)

which is clearly zero on the Legendre submanifold

LH :=

{

x0 = H(x), x , p =∂H

∂x(x)

}

.

Note that the contact Hamiltonian has the dimension of power (and notof energy).


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A Hamiltonian dynamics with energy dissipation

x = [J(x)− R(x)]∂H

∂x(x)

may be also lifted to a contact vector field by adding an extra extensivevariable S (’entropy’), and defining the contact Hamiltonian

K (x ,S , p,T ) = −pT [J(x)− R(x)]∂H

∂x(x)−

T

T0

∂H

∂x(x)R(x)

∂H

∂x(x)

which obviously is zero on the Legendre submanifold

{

x0 = H(x), x ,S , p =∂H

∂x(x),T = T0

}

.


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Finally, port-Hamiltonian systems with inputs

x = [J(x) − R(x)]∂H

∂x(x) +

∑

gj(x)uj

are lifted to control contact systems by defining the interaction contactHamiltonians

Kj (x , p) = gTj (x)[

∂H

∂x(x) − p]


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Conclusions for thermodynamical systems

• Definition of the thermodynamic phase space and Legendresubmanifold corresponding to thermostatics.

• Thermodynamics defined by contact vector fields with contactHamiltonian zero on the Legendre submanifold.

• Lifting of conservative, in particular Hamiltonian, systems to theLegendre submanifold.

• Mixed ’thermodynamical’ and ’non-thermodynamical’ systems.


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Slides of the talks will be available at my homepage:

www.math.rug.nl/˜arjan

together with references.


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Geometric network modeling and control of physical systems

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