The Geometry Underlying Port-Hamiltonian Systems · •Review on classical Hamiltonian systems •...

The Geometry Underlying Port-Hamiltonian SystemsPre-LHMNC School, UTFSM Valparaiso, April 30 - May 1, 2018

Arjan van der Schaft

Jan C. Willems Center for Systems and ControlJohann Bernoulli Institute for Mathematics and Computer Science

University of Groningen, the Netherlands

Arjan van der Schaft (Univ. of Groningen) Geometry of pH Systems 1 / 66

• Review on classical Hamiltonian systems

• Dirac structures on linear spaces

• Definition of port-Hamiltonian systems on linear spaces

• Examples from different physical domains

• Dirac structures on manifolds

(Almost) everything (and much more !) can be found in

A.J. van der Schaft, D. Jeltsema,Port-Hamiltonian Systems Theory: An Introductory Overview,

NOW Publishers, 2014

The pdf of the book is freely available from my home page

www.math.rug.nl/˜arjan


Outline

1 Review on classical Hamiltonian systems

2 From network interconnection to geometric structure

3 Port-Hamiltonian systems

4 Definition of port-Hamiltonian systems

5 Examples from different physical domains

6 Dirac structures on manifolds

7 Conclusions and Outlook


Review on classical Hamiltonian systems

Euler-Lagrange equations:

Consider a mechanical system with n degrees of freedom, and positioncoordinates

q = (q1, . . . , qn)

for the configuration manifold Q. Determine the kinetic energy

K (q, q) =1

2qTM(q)q, M(q) > 0

and the potential energy P(q), where q ∈ TqQ.Define the Lagrangian function L : TQ → R

L(q, q) := K (q, q)− P(q)

The equations of motion are

d

dt

(

∂L

∂q(q, q)

)

−∂L

∂q(q, q) = τ



Define the momenta

p :=∂L

∂q(q, q) ∈ T ∗

qQ

and the Hamiltonian H : T ∗Q → R defined as

H(q, p) :=1

2pTM−1(q)p + P(q)

(kinetic energy plus potential energy = total energy).

Then we obtain the classical Hamiltonian equations of motion

q = ∂H∂p

(q, p)

p = −∂H∂q

(q, p) + τ



Geometrically (coordinate-free) this is described by the triple

(T ∗Q, ω,H)

• Q is the configuration manifold

• ω is canonical symplectic form on the cotangent bundle T ∗Q

• the dynamics given by the Hamiltonian vector field XH satisfying

ω(XH ,−) = −dH


The symplectic form ω on the cotangent bundle T ∗Q is defined as follows.T ∗Q is endowed with a canonical one-form θ (called Liouville one-form),defined as

θ(α)(Z ) = α(π∗Z )

where α ∈ T ∗Q,Z ∈ Tα(T∗Q) and π : T ∗Q → Q natural projection.

Choosing any set of coordinates q = (q1, · · · , qn) for Q, and naturalinduced coordinates

(q, p) = (q1, · · · , qn, p1, · · · , pn)

for T ∗Q, it follows that

θ =

n∑

i=1

pidqi

Now define the symplectic form ω := dθ.In local coordinates

ω =

n∑

i=1

dpi ∧ dqi



Equivalently, let {, } denote the Poisson bracket on T ∗Q given as

{F ,G} := ω(XF ,XG ), F ,G : T ∗Q → R

In natural coordinates for T ∗Q

{F ,G} =

n∑

i=1

(∂F

∂qi

∂G

∂pi−∂F

∂pi

∂G

∂qi)

Then XH is determined by the requirement

XH(F ) = {F ,H}

for all F : T ∗Q → R.In an arbitrary set of local coordinates x the Hamiltonian dynamics takesthe form

x = J(x)∂H

∂x(x)

where J(x) = −JT (x) is the (invertible) Poisson structure matrix

Jij = {xi , xj}, i , j = 1, · · · , 2n


Hamiltonian systems obtained by symmetry reduction

On the other hand, it is well-known that many dynamical equations ofphysical interest are not precisely of this form.Typical example is formed by the Euler equations for rigid body motion

px

py

pz

=

0 −pz py

pz 0 −px

−py px 0

∂H∂px

∂H∂py

∂H∂pz

with p = (px , py , pz) the body angular momentum vector along the three

principal axes, and H(p) = 12

(

p2xIx

+p2yIy

+ p2zIz

)

the kinetic energy

(with Ix , Iy , Iz principal moments of inertia.)In general, many physical systems are of the Hamiltonian form

x = J(x)∂H

∂x(x)

with J(x) = −JT (x), but not of full rank.Arjan van der Schaft (Univ. of Groningen) Geometry of pH Systems 9 / 66

Hamiltonian systems obtained by symmetry reduction

The Euler equations can be regarded as the reduction of classicalHamiltonian equations on the cotangent bundle T ∗Q where Q = SO(3).

Reduced space is the orbit space of the action of a Lie group that leavesthe Hamiltonian invariant.

In fact, the cotangent bundle T ∗SO(3) can be reduced by the action ofSO(3) on T ∗SO(3) into so(3)∗,since the Hamiltonian (= kinetic energy) is invariant under this action.

This holds in many situations, both in the finite- and infinite-dimensionalcase (“Marsden-Weinstein reduction by symmetry program”).

Thus in these cases the Poisson structure is still derivable from a standardsymplectic structure on a cotangent bundle.


If we move away from mechanical systems the story becomes different.What is the Hamiltonian formulation of a general LC-circuit?

Example (LC-circuits)

Two inductors with magnetic energies H1(ϕ1),H2(ϕ2)(ϕ1 and ϕ2 magnetic flux linkages),and capacitor with electric energy H3(Q) (Q charge).

$\varphi_1$ $\varphi_2$

$L_1$ $L_2$

$C$

$Q$

Figure: LC-circuit


Same question holds for multi-physics systems (e.g., electro-mechanicalsystems).

Furthermore

• How to include energy-dissipation into a Hamiltonian formulation?

• How to interconnect systems?

• What if the interconnected system is a differential-algebraic equation(DAE) system?


Outline









A paradigm-shift in modeling is needed, originating e.g. from electricalnetwork theory; later on generalized to port-based modeling andbond-graphs.

Example (The ubiquitous mass-spring system)

Instead of starting with the position coordinate q of the mass, and itsvelocity q yielding the kinetic energy 1

2mq2, together with the potentialenergy 1

2kq2 of the spring,

we consider the system as the network interconnection of twoenergy-storing elements corresponding to them:

• Spring Hamiltonian Hs(q) =12kq

2 (potential energy)

q = −fs = velocity (= em)

es = dHs

dq(q) = kq = force

• Mass Hamiltonian Hm(p) =12mp2 (kinetic energy)

p = −fm = force (= −es)

em = dHm

dp(p) = p

m= velocity


A different starting point

Network modeling of physical systems

Prevailing trend in modeling and simulation of lumped-parameter systems(multi-body systems, electrical circuits, electro-mechanical systems,robotic systems, cell-biological systems, etc.).

Main advantages of network modeling:

• Systematic modeling procedure, starting from simple components tocomplex systems.

• Re-usability of component models. Flexible adaptation. Design andcontrol.

• Suited to multi-physics systems.

Originates from electrical circuit theory (Kirchhoff) and mechanicalengineering (Newtonian modeling).

What is the underlying geometric structure ? Not a cotangent bundle nora reduced cotangent bundle !


Port-based network modeling

Interaction between ideal system components is modeled by power-portsmodeling the energy exchange between the components.

Associated to every power-port there are conjugate pairs of (vectors of)variables, called flows f and efforts e, whose product eT f equals power.

For example,

• voltages and currents

• forces and velocities

• pressure and volume change

• chemical potentials and concentration fluxes


Example (LC-circuits)

Two inductors with magnetic energies H1(ϕ1),H2(ϕ2)(ϕ1 and ϕ2 magnetic flux linkages),and capacitor with electric energy H3(Q) (Q charge).

$\varphi_1$ $\varphi_2$

$L_1$ $L_2$

$C$

$Q$

Figure: LC-circuit

Question: How to write this LC-circuit as a Hamiltonian system ?


Example (LC-circuit continued)

Dynamical equations for the three components:

Inductor 1 : ϕ1 = f1 (voltage)

(current) e1 = ∂H1∂ϕ1

Inductor 2 : ϕ2 = f2 (voltage)

(current) e2 = ∂H2∂ϕ2

Capacitor : Q = f3 (current)

(voltage) e3 = ∂H3∂Q

If Hi are quadratic, e.g., H3(Q) = 12CQ

2, then the elements are linear.

E.g., voltage over capacitor = ∂H3∂Q

= QC, and similarly for the inductors.


Example (LC-circuit continued)

Kirchhoff’s voltage and current laws are

−f1−f2−f3

=

0 0 10 0 −1−1 1 0

e1e2e3

Substitution of eqns. of components yields port-Hamiltonian system

ϕ1

ϕ2

Q

=

0 0 −10 0 11 −1 0

∂H∂ϕ1

∂H∂ϕ2

∂H∂Q

with H(ϕ1, ϕ2,Q) := H1(ϕ1) + H2(ϕ2) + H3(Q) total energy.Can be directly extended to RLC-circuits.


Preliminary conclusions

• The structure matrix J is completely determined by theinterconnection structure of the circuit (in this case, Kirchhoff’scurrent and voltage laws).

• Skew-symmetry of J corresponds to the interconnection beingpower-conserving (Tellegen’s theorem derived from Kirchhoff’s laws).

• There is no clear underlying cotangent bundle nor symplectic manifold!

• Building blocks are open dynamical systems (systems point of view).

Not yet general enough: how do we formulate the LC-circuit with thesame topology, but with capacitors and inductors swapped (correspondingto an algebraic constraint)?


Outline









The basic picture

Dstorage dissipation

eS

fS

eR

fR

eP fP

Figure: Port-Hamiltonian system


The basic elements

Port-based modeling is based on viewing the physical system as theinterconnection of ideal energy processing elements, all expressed in(vector) pairs of flow variables f ∈ F , and effort variables e ∈ E , where Fand E are linear spaces of equal dimension.Furthermore, there is a pairing between F and E defining the power

< e | f >

Canonical choice: E = F∗ with < e | f >= eT f , but more involved in e.g.multi-body systems.

• Energy-storing elements:

x = −f

e = ∂H∂x

(x)

• Purely energy-dissipating elements:

R(f , e) = 0, eT f ≤ 0


The basic elements

• Energy-routing elements: generalized transformers, gyrators:

f1 = Mf2, e2 = −MT e1, f = Je, J = −JT

They are power-conserving:

eT f = 0

• Ideal interconnection constraints

0-junction :e1 = e2 = · · · = ek , f1 + f2 + · · ·+ fk = 01-junction :f1 = f2 = · · · = fk , e1 + e2 + · · ·+ ek = 0Ideal flow or effort constraints :f = 0, or e = 0

Also power-conserving:

e1f1 + e2f2 + · · ·+ ek fk = 0


From power-conserving elements to Dirac structures

All power-conserving elements/interconnection constraints have thefollowing properties in common.They are described by linear equations in f , e ∈ R

k satisfying

eT f = e1f1 + e2f2 + · · · + ek fk = 0,

while furthermore the number of independent equations is equal to k .

All power-conserving elements/interconnection constraints will be groupedinto one geometric object: the Dirac structure.

(Note: the linearity of the relations between efforts and flows breaks downin thermodynamic systems.)


Dirac structures on linear spaces

Definition

A (constant) Dirac structure (on a linear space) is a subspace

D ⊂ F × E

such that

(i) eT f = 0 for all (f , e) ∈ D,

(ii) dimD = dimF .

For any skew-symmetric map J : E → F its graph given as{(f , e) ∈ F × E | f = Je} is a Dirac structure. Similarly, the graph of anyskew-symmetric map ω : F → E is a Dirac structure.

But not all Dirac structures are of this type ! E.g, for any subspaceV ⊂ F , the product V × V⊥ ⊂ F × E is also a Dirac structure.


Alternative definition of Dirac structure

Symmetrized form of power

< e | f >= eT f , (f , e) ∈ F × E .

Symmetrization leads to the indefinite bilinear form ≪,≫ on F × E :

≪(f a, ea), (f b, eb) ≫ := < ea | f b > + < eb | f a >,

(f a, ea), (f b, eb) ∈ F × E .


Alternative definition of Dirac structure

Definition

A (constant) Dirac structure is a subspace

D ⊂ F × E

such thatD = D⊥⊥,

where ⊥⊥ denotes orthogonal companion with respect to the bilinear form≪,≫.

Due to Weinstein & Courant, Dorfman; with a connection to the Diracbracket of constrained Hamiltonian systems.


Outline









The basic model

Consider a Dirac structure D ⊂ F × E , where

f = (fS , fR , fP), e = (eS , eR , eP)


eS

fS

eR

fR

eP fP

Figure: Port-Hamiltonian system


The port-Hamiltonian system is defined by closing the energy-storing andenergy-dissipating ports of the Dirac structure D by their constitutiverelations

−x = fS ,∂H

∂x(x) = eS

respectivelyR(fR , eR) = 0

This leads to, in principle, a mixture of differential and algebraic equations(DAE systems)

(−x(t), fR(t), fP(t),∂H∂x

(x(t)), eR (t), eP(t)) ∈ Dt ∈ R

R(fR(t), eR(t)) = 0


Energy-balance

Power-conservationeTS fS + eTR fR + eTP fP = 0

implies the energy-balance

dHdt(x(t)) = ∂TH

∂x(x(t))x(t) =

eTR (t)fR(t) + eTP (t)fP(t) ≤

eTP (t)fP t)

(showing passivity if H is bounded from below).


DAE example

Consider the same LC-circuit as before, but now the inductors andcapacitors are swapped.replacements

ϕ

Q1 Q2+

−

C1 C2

L

Figure: LC circuit.


The capacitors (assumed to be linear) are described by

Qi = −Ii ,

Vi =Qi

Ci,

for i = 1, 2. Here Ii and Vi are the currents through, respectively voltagesacross, the two capacitors. Ci are their capacitances. Qi are the charges ofthe capacitors.Similarly, the linear inductor is described by

ϕ = −VL,

IL =ϕ

L,

where IL is the current through the inductor, and VL is the voltage acrossthe inductor. ϕ is the flux-linkage of the inductor, and L its inductance.


Parallel interconnection of these three subsystems by Kirchhoff’s lawsamounts to the same interconnection equations

V1 = V2 = VL, I1 + I2 + IL = 0,

and thus to the same Dirac structure as before, where however someefforts are replaced by flows and conversely.The equation V1 = V2 gives rise to the algebraic constraint

Q1

C1=

Q2

C2,

relating the two state variables Q1,Q2.

The resulting system is a port-Hamiltonian DAE system F (x , ∂H∂x

(x)) = 0.


Outline









Example (The mass-spring system revisited)

Two storage elements:

• Spring Hamiltonian Hs(q) =12kq

2 (potential energy)

q = −fs = velocity

es = dHs

dq(q) = kq = force

• Mass Hamiltonian Hm(p) =12mp2 (kinetic energy)

p = −fm = force

em = dHm

dp(p) = p

m= velocity


Example

Dirac structure linking fs , es , fm, em, fP , eP as

fs = −em = −fP , fm = es − eP

(power-conserving since fses + fmem + uy = 0) yields the port-Hamiltoniansystem

[

q

p

]

=

[

0 1

−1 0

][

∂H∂q

(q, p)

∂H∂p

(q, p)

]

+

[

0

1

]

eP

fP =[

0 1]

[

∂H∂q

(q, p)

∂H∂p

(q, p)

]

withH(q, p) = Hs(q) + Hm(p)


Example (Electro-mechanical systems)

q

p

ϕ

=

0 1 0−1 0 00 0 −R

∂H∂q

(q, p, φ)∂H∂p

(q, p, φ)∂H∂ϕ

(q, p, φ)

+

001

V , I =∂H

∂ϕ(q, p, φ)

Coupling electrical/mechanical domain via Hamiltonian

H(q, p, ϕ) = mgq +p2

2m+

ϕ2

2L(q)


DC motor

_

V

I

J

b

R L

K

ω

τ

+

Figure: DC motor.

6 interconnected subsystems:◦ 2 energy-storing elements: inductor L with state ϕ (flux), and rotationalinertia J with state p (angular momentum);◦ 2 energy-dissipating elements: resistor R and friction b;◦ gyrator K ;◦ voltage source V .


The energy-storing elements (here assumed to be linear) are given by

Inductor:

ϕ = −VL

I =d

dϕ

(

1

2Lϕ2

)

=ϕ

L,

Inertia:

p = −τJ

ω =d

dp

(

1

2Jp2)

=p

J.

Hence, the corresponding total Hamiltonian reads H(p, φ) = 12Lφ

2 + 12J p

2.The energy-dissipating relations (also assumed to be linear) are

VR = −RI , τb = −bω,

with R , b > 0, where τb is a damping torque. The equations of the gyrator(converting magnetic power into mechanical, and conversely) are

VK = −Kω, τK = KI .

with K the gyrator constant.Arjan van der Schaft (Univ. of Groningen) Geometry of pH Systems 41 / 66

The subsystems are interconnected by the equations

VL + VR + VK + V = 0, τJ + τb + τK + τ = 0.

The Dirac structure is defined by the interconnection equation, togetherwith the equations for the gyrator.

This results in the port-Hamiltonian model

[

ϕ

p

]

=

[

−R −K

K −b

]

ϕ

Lp

J

+

[

1 00 1

] [

V

τ

]

,

[

I

ω

]

=

[

1 00 1

]

ϕ

Lp

J

.

of the form x = [J − R ]∂H∂x

(x) + Gu, y = GT ∂H∂x

(x).


Synchronous machine

Classical 8-dimensional model of the Synchronous Generator (SG) can beput into port-Hamiltonian form

SG

θ ψr

p ψs

power ≈ 0excitation system

ωτ

Vs

Is

Vf Ifmechanical

powerelectricalpower

Same model for synchronous motors.


ψs

ψr

p

θ

=

−Rs 03 031 031

03 −Rr 031 031

013 013 −d −1

013 013 1 0

∂H∂ψs

∂H∂ψr

∂H∂p

∂H∂θ

+

I3 031 031

03

100

031

013 0 1013 0 0

Vs

Vf

τ

IsIfω

=

I3 03 031 031013

[

1 0 0]

0 0013 013 1 0

∂H∂ψs

∂H∂ψr

∂H∂p

∂H∂θ

where Rs =

rs 0 00 rs 00 0 rs

, Rr =

rf 0 00 rkd 00 0 rkq

, d

are the stator resistances, rotor resistances, mechanical friction constants.Arjan van der Schaft (Univ. of Groningen) Geometry of pH Systems 44 / 66

• ψs ∈ R3 are stator fluxes

• ψr ∈ R3 are rotor fluxes: field winding and two damper windings

• p is angular momentum of rotor

• θ is the angle of the rotor

• Vs ∈ R3, Is ∈ R

3 are the three-phase stator terminal voltages andcurrents

• Vf , If are the rotor field winding voltage and current

• τ, ω are the mechanical torque and angular velocity

The Hamiltonian (total stored energy) is

H(ψs , ψr , p, θ) = 12

[

ψTs ψT

r

]

L−1(θ)

[

ψs

ψr

]

+ 12J p

2

= magnetic energy + kinetic energy

where L(θ) is the 6× 6 inductance matrix.


In the round rotor case

L(θ) =

[

Lss Lsr (θ)LTsr (θ) Lrr

]

where

Lss =

Laa −Lab −Lab−Lab Laa −Lab−Lab −Lab Laa

, Lrr =

Lffd Lakd 0Lakd Lkkd 00 0 Lkkq

while

Lsr (θ) =

cos θ cos θ − sin θcos(θ − 2π

3 ) cos(θ − 2π3 ) − sin(θ − 2π

3 )cos(θ + 2π

3 ) cos(θ + 2π3 ) − sin(θ + 2π

3 )

Lafd 0 00 Lakd 00 0 Lakq


Outline









Recall of the basic picture of port-Hamiltonian systems

Consider a Dirac structure D, linking the flow and effort variables

f = (fS , fR , fP), e = (eS , eR , eP)


eS

fS

eR

fR

eP fP

leading to the port-Hamiltonian system


(x(t)), eR (t), eP(t)) ∈ Dt ∈ R

R(fR(t), eR(t)) = 0


For many systems, especially those with 3-D mechanical components, theinterconnection structure will be modulated by the energy or geometricvariables.

This leads to the notion of (non-constant) Dirac structures on manifolds.

Definition

Consider a smooth manifold X . A Dirac structure on X is a vectorsubbundle D ⊂ TX ⊕ T ∗X such that for every x ∈ X the vector space

D(x) ⊂ TxX × T ∗

x X

is a Dirac structure as before.


Special cases

(a) Let J be a Poisson structure on X , defining a skew-symmetricmapping J : T ∗X → TX . Thengraph J ⊂ T ∗X ⊕ TX is a Dirac structure.

(b) Let ω be a (pre-)symplectic structure on X , defining askew-symmetric mapping ω : TX → T ∗X . Thengraph ω ⊂ TX ⊕ T ∗X is a Dirac structure.

(c) Let K be a constant-dimensional distribution on X , and let annK beits annihilating co-distribution. ThenK × annK ⊂ TX ⊕ T ∗X is a Dirac structure.


Mechanical systems with kinematic constraints

Consider a mechanical system with n degrees of freedom.Kinematic constraints are constraints on the n-dimensional vector ofgeneralized velocities q:

AT (q)q = 0

with A(q) some n × k matrix (k the number of kinematic constraints).

This leads to constrained Hamiltonian equations

q = ∂H∂p

(q, p)

p = −∂H∂q

(q, p) + A(q)λ

0 = AT (q)∂H∂p

(q, p)

with H(q, p) total energy, and A(q)λ the constraint forces.


The resulting Dirac structure D on X = T ∗Q is defined by the canonicalPoisson structure on T ∗Q together with the constraints AT (q)q = 0:

D(q, p) = {(fS , eS ) ∈ T(q,p)X × T ∗

(q,p)X | ∃λ ∈ Rk s.t.

fS =

[

0 −In

In 0

]

eS −

[

0

A(q)

]

λ,[

0 AT (q)]

eS = 0}

Or in more geometric form

D(q, p) = {(fS , eS ) ∈ T(q,p)X × T ∗

(q,p)X | ∃λ ∈ Rk s.t.

−fS = J(q, p)eS +

[

0

A(q)

]

λ,[

0 AT (q)]

eS = 0}

with J the structure matrix of the canonical Poisson structure on T ∗Q.


Example (Rolling coin)

ϕϕ

θ

x

y

(x , y)(x , y)

Figure: The geometry of the rolling peso


Example

Let x , y be the Cartesian coordinates of the point of contact of the coinwith the plane. Furthermore, ϕ denotes the heading angle, and θ the angleof the coin. The rolling constraints (rolling without slipping) are(set all parameters equal to 1)

x = θ cosϕ, y = θ sinϕ

The total energy is

H =1

2p2x +

1

2p2y +

1

2p2θ +

1

2p2ϕ

and the constraints thus can be rewritten in the form AT (q)∂H∂p

(q, p) = 0as

px − pθ cosϕ = 0, py − pθ sinϕ = 0.


Now consider a Dirac structure on

X × FR ×FP ,

with X the state space manifold, and FR ,FP linear spaces, which isindependent of the position in FR ×FP .(Geometric definition can be given using symmetries of Dirac structures.)

This leads to the definition of a port-Hamiltonian system with state spacemanifold X as


(x(t)), eR (t), eP (t)) ∈ D(x(t))t ∈ R

R(fR(t), eR(t)) = 0


Special case: Consider Dirac structure D on X × FR ×FP given as graphof skew-symmetric map modulated by x ∈ X

fSfRfP

=

−J(x) −GR(x) −G (x)GTR (x) 0 0

GT (x) 0 0

eSeReP

,

together with a linear resistive relation

eR = −RfR , R = RT ≥ 0

Writing out

x = J(x)∂H

∂x(x) + GR(x)fR + G (x)eP , fR = −GT

R (x)∂H

∂x(x)

and denoting R(x) = GR(x)RGTR (x) ≥ 0, this leads to pH systems

x = [J(x) − R(x)]∂H∂x

(x) + G (x)u

y = GT (x)∂H∂x

(x)

with inputs u = eP and outputs y = fP .Arjan van der Schaft (Univ. of Groningen) Geometry of pH Systems 56 / 66

Integrability of Dirac structures

There is an important notion of integrability of a Dirac structure on amanifold.

A Dirac structure is integrable if there exist coordinates in which the Diracstructure is a constant Dirac structure.

Theorem (Dorfman, Courant)

A Dirac structure D on a manifold X is called integrable

< LX1α2 | X3 > + < LX2

α3 | X1 > + < LX3α1 | X2 >= 0

for all (X1, α1), (X2, α2), (X3, α3) ∈ D.

Indeed, for constant Dirac structures the integrability condition isautomatically satisfied.


Excursion to generalized geometry

This is strictly related to the Courant bracket on TX ⊕ T ∗X given as

[[(X1, α1), (X2, α2)]] = ([X1,X2], LX1α2 − LX2

α1 +1

2d(α1(X2)− α2(X1))

In fact, the Dirac structure is integrable if and only if the Courant bracketof any two elements in D is again in D.


Integrability of the Dirac structure is equivalent to the existence ofcanonical coordinates:

If the Dirac structure D on X is integrable then there exist coordinates(q, p, r , s) for X such that

D(x) = {(fq , fp, fr , fs , eq , ep , er , es) ∈ TxX × T ∗

x X}

fq = −ep, fp = eq

fr = 0, 0 = es

Hence the Hamiltonian system corresponding to D and H : X → R is

q = ∂H∂p

(q, p, r , s)

p = −∂H∂q

(q, p, r , s)

r = 0

0 = ∂H∂s

(q, p, r , s)


Special cases of integrability

First case

Let the Dirac structure D be given for every x ∈ X as the graph of askew-symmetric mapping J(x) from the co-tangent space T ∗

xX to thetangent space TxX .

Integrability in this case means that J(x) satisfies the conditions

n∑

l=1

[

Jlj(x)∂Jik

∂xl(x) + Jli (x)

∂Jkj

∂xl(x) + Jlk(x)

∂Jji

∂xl(x)

]

= 0, i , j , k = 1, . . . , n


In this case we may find, by Darboux’s theorem around any point x0 wherethe rank of the matrix J(x) is constant, local canonical coordinatesx = (q, p, r) in which the matrix J(x) becomes the constantskew-symmetric matrix

0 −Ik 0Ik 0 00 0 0

Then J(x) defines a Poisson bracket on X , given for every F ,G : X → R

as

{F ,G} =∂TF

∂xJ(x)

∂G

∂x

Indeed, by the integrability condition the Jacobi-identity holds:

{F , {G ,K}} + {G , {K ,F}} + {K , {F ,G}} = 0

for all functions F ,G ,K .


Second case

A similar story holds for a Dirac structure given as the graph of askew-symmetric mapping ω(x) from the tangent space TxX to theco-tangent space T ∗

x X . In this case the integrability conditions take theform

∂ωij

∂xk(x) +

∂ωki

∂xj(x) +

∂ωjk

∂xi(x) = 0, i , j , k = 1, . . . , n

The skew-symmetric matrix ω(x) can be regarded as the coordinaterepresentation of a differential two-form ω on X , that isω =

∑ni=1,j=1 dxi ∧ dxj , and the integrability condition corresponds to the

closedness of this two-form (dω = 0).


The differential two-form ω is called a pre-symplectic structure, and asymplectic structure if the rank of ω(x) is equal to the dimension of X . Bya version of Darboux’s theorem we may find, around any point x0 wherethe rank of the matrix ω(x) is constant, local coordinates x = (q, p, s) inwhich the matrix ω(x) becomes the constant skew-symmetric matrix

0 Ik 0−Ik 0 00 0 0

Third case: Let K be a constant-dimensional distribution on X , and letannK be its annihilating co-distribution. Then the Dirac structureK × annK ⊂ TX ⊕ T ∗X is integrable if and only if the distribution K isinvolutive.


The Dirac structure corresponding to mechanical systems with kinematicconstraints

D(q, p) = {(fS , eS ) ∈ T(q,p)X × T ∗

(q,p)X | ∃λ ∈ Rk s.t.

−fS = J(q, p)eS +

[

0

A(q)

]

λ,[

0 AT (q)]

eS = 0}

is integrable if and only if the kinematic constraints

AT (q)q = 0

are holonomic, which means that it is possible to find configurationcoordinates q = (q1, . . . , qn) such that the constraints are equivalentlyexpressed as

qn−k+1 = qn−k+2 = · · · = qn = 0 ,

In this case one may eliminate the configuration variables qn−k+1, . . . , qn,since the kinematic constraints are equivalent to the geometric constraints

qn−k+1 = cn−k+1, . . . , qn = cn ,

for certain constants cn−k+1, . . . , cn determined by the initial conditions.Arjan van der Schaft (Univ. of Groningen) Geometry of pH Systems 64 / 66

Outline









• Paradigm shift from cotangent bundle to network structure

• Dirac structure determined by network structure

• Port-Hamiltonian systems

• Dirac structures on manifolds and integrability


The Geometry Underlying Port-Hamiltonian Systems · •Review on classical Hamiltonian systems •...

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