On the extraction of the boundary conditions and the boundary ports in second-order field theories
Markus Schöberl Kurt Schlacher
math.OC math-ph math.DS math.MP





On the extraction of the boundary conditions and the boundary ports
in second-order field theories
    Markus Schberl and Kurt Schlacher
    Received: 25 Janauray 2018 
==================================================================================================



Institute of Automatic Control and Control Systems Technology, Johannes
Kepler University, Altenbergerstr. 69, 4040 Linz, Austria

In this paper we consider second-order field theories in a variational
setting. From the variational principle the Euler-Lagrange equations
follow in an unambiguous way, but it is well known that this is not
true for the Cartan form. This has also consequences on the derivation
of the boundary conditions when non trivial variations are allowed
on the boundary. By posing extra conditions on the set of possible
boundary terms we exploit the degree of freedom in the Cartan form
to extract physical meaningful boundary expressions. The same mathematical
machinery will be applied to derive the boundary ports in a Hamiltonian
representation of the partial differential equations which is crucial
for energy based control approaches. Our results will be visualized
for mechanical systems such as beam and plate models.




§ INTRODUCTION


Many distributed-parameter systems described by partial differential
equations (pdes) allow for a variational formulation such that they
can be analyzed based on the corresponding Lagrangian density. In
a geometric setting the Euler-Lagrange equations are constructed by
using the so-called Cartan form, which provides a coordinate free
way to perform integration by parts, see e.g. <cit.>
and references therein. It is well-known that a unique formulation
for the Euler-Lagrange equations can be found but this is not true
for the Cartan form when the variational problem involves derivative
variables of second-order or higher and more than one independent
coordinate. This is connected to the commutativity of repeated partial
differentiation. In the literature many attempts have been made to
fix the Cartan form also in the higher-order field theories, e.g.
by using connections, see e.g <cit.> or to avoid the ambiguities
by using different approaches e.g. the Skinner and Rusk formalism,
see e.g. <cit.>. Another consequence of the ambiguities of
the Cartan form (i.e. the need for repeated integration by parts)
is the fact that also the terms in the boundary integral are affected
by this non-uniqueness, when non-trivial variations on the boundary
are explicitly allowed. In the mathematical physicist literature this
effect is usually ignored and only trivial boundary conditions are
considered, but e.g. for mechanical structures such as beams, plates
or shells it is of great importance to allow for non trivial boundary
terms and consequently also for non-zero energy flow over the boundary,
especially for engineering applications, e.g. for control purposes
and especially for boundary control systems.

In this paper we consider non-trivial variations on the boundary and
we restrict ourselves to second-order field theories. The case of
second-order field theories is in some sense special, see e.g. <cit.>,
but to our best knowledge a systematic derivation of non-trivial boundary
conditions based on the Cartan form has not been reported in the literature
so far. In <cit.> it is shown by evaluating the
variational principle (using integration by parts) that the boundary
term is indeed connected with the Cartan form but the extraction of
the meaningful boundary conditions is not discussed in this setting.
In <cit.> natural boundary conditions for higher-order field
theories in a variational setting are discussed based on flag fibrations
and again using integration by parts.

We demonstrate that by posing extra conditions on the set of possible
boundary terms, one is able to derive the physically admissible boundary
conditions which will be demonstrated for mechanical systems where
the Lagrangian depends on second-order derivatives. To this end, we
use the ambiguities in constructing the Cartan form and we exploit
the special structure of the boundary manifold of the underlying variational
problem, where we make use of an adapted coordinate system as it has
been suggested e.g. in <cit.> to present our result
in a compact and simple manner. Some preliminary work has been reported
in <cit.> and <cit.>.

Furthermore, we will focus on partial differential equations (pdes)
that allow for a Hamiltonian formulation. Given pdes in a Lagrangian
setting, different approaches to construct Hamiltonian counterparts
have been discussed in the literature including symplectic, polysymplectic
and multisymplectic cases see, e.g. <cit.>.
From a control and a system theoretic point of view the class of port-Hamiltonian
systems with dissipation <cit.> is a very prominent
system class, which has been extended from the lumped-parameter case
(systems described by ordinary differential equations) to the distributed-parameter
pde case. However, different port-Hamiltonian formulations exist,
see e.g. <cit.>
for an approach based on a Stokes-Dirac structure whereas in <cit.>
an alternative strategy based on jet-bundles has been considered.
In this paper we will consider the jet-bundle approach and apply the
theory of Cartan forms to be able to extract the correct boundary
ports when the Hamiltonian depends on second-order derivative variables
in a formal and systematic way.

The main contribution of the paper is i) to provide a simple and algorithmic
procedure to derive the physical admissible boundary conditions for
variational problems by means of the Cartan form where the Lagrangian
depends on second-order derivative variables and ii) to apply this
mathematical machinery developed for the Lagrangian scenario to a
port-Hamiltonian representation in order to analyze the power balance
relation. The paper is organized as follows. In Section 2 some geometric
fundamentals are presented and in the third section we discuss first-order
theories where we consider the well-known case of the Cartan form
approach to derive the pdes and the boundary terms in a Lagrangian
setting. Then, Section 4 is devoted to the second-order case in the
Lagrangian framework, where we show how to extract the boundary conditions
by using the presented machinery. In Section 5 we discuss the Euler-Bernoulli
beam and the Kirchhoff plate in the Lagrangian framework. In the sixth
section the port-Hamiltonian perspective is analyzed within this mathematical
setting and the Kirchhoff plate example is rearranged in a Hamiltonian
formulation to make the power ports visible. 



§ NOTATION AND PRELIMINARIES


In this paper we will apply differential geometric methods, in particular
we use jet-bundles and induced geometric objects, see <cit.>
for an extensive treatise. We use tensor notation and especially Einsteins
convention on sums where we will not indicate the range of the indices
when they are clear from the context. Given a manifold ℳ
with local coordinates (x^α) where α=1,…,dim(ℳ)
we denote with 𝒯(ℳ)→ℳ and
𝒯^*(ℳ)→ℳ the tangent
bundle and the cotangent bundle, respectively. Local coordinates for
𝒯(ℳ) and 𝒯^*(ℳ) are
(x^α,ẋ^α) as well as (x^α,ẋ_α)
with respect to the holonomic fibre basis {∂_α}
and {dx^α}. Furthermore, we use ∧ for
the exterior product (wedge product), d is the exterior
derivative and ⌋ denotes the natural contraction between
tensor fields. With L_v(ω) we denote the Lie-derivative
of the geometric object ω (e.g. a differential form) with
respect to the vector field v. Furthermore, C^∞(·)
denotes the set of smooth functions on the corresponding manifold.

Let 𝒳π→𝒟 be a bundle
with dim(𝒳)=m+r where 𝒟 is an orientable
r-dimensional manifold. Local coordinates for 𝒳 adapted
to the fibration are (X^i,x^α) with x^α , α=1,…,m
where X^i , i=1,…,r is a coordinate chart for 𝒟.
We fix the volume form Ω on the manifold 𝒟 as
Ω=dX^1∧…∧dX^r and we
set Ω_i=∂_i⌋. Let
σ be a section of the bundle 𝒳→𝒟,
i.e. a map σ:𝒟→𝒳 with coordinate
representation σ(X^i)=(X^i,σ^α(X^i)). The
set of sections of the bundle 𝒳π→𝒟
will be denoted as .
Then the k-th order partial derivatives of σ^α are

    ∂^kσ^α/(∂ X^1)^j_1⋯(∂ X^r)^j_r=∂_Jσ^α

where J is an ordered multi-index with ∑ j_i=k=#J. The
components of the multi-index J are J(i) and 1_k is the
special multi index defined as J(i)=δ_k^i such that all
components of 1_k are zero except the k-th entry which is
one. Two multi-indices J and I can be added and subtracted component
wise, where the result of a subtraction may not be a multi-index.

Given two sections, such that at a point p∈𝒟 the sections
and their partial derivatives up to order k coincide, these sections
are called k-equivalent. Given now a smooth local section σ∈
at point p∈𝒟, the equivalence class of n-equivalent
sections that contains σ is denoted j_p^n(σ) (called
the n-jet at p) and the set of all these n-jets is the n-th
order jet-manifold 𝒥^n(𝒳) with adapted coordinates
(X^i,x_J^α) with 0≤#J≤ n. Furthermore, we
have π_k,l:𝒥^k(𝒳)→𝒥^l(𝒳)
for k>l as well as π_k,0:𝒥^k(𝒳)→𝒳
and π_k:𝒥^k(𝒳)→𝒟.
The tangent and the cotangent bundles of 𝒥^k(𝒳)
can be easily constructed and the holonomic fibre bases for 𝒯(𝒥^k(𝒳))
and 𝒯^*(𝒥^k(𝒳)) read as {∂_i, ∂_α^J}
and {dX^i, dx_J^α}, where 0≤#J≤ k,
respectively, with ∂_i=∂/∂ X^i and ∂_α^J=∂/∂ x_J^α.

The total derivative d_1_i with respect to the independent
variable X^i reads as

    d_1_i=∂_i+x_J+1_i^α∂_α^J

and (d_1_if)∘ j^n+1(σ)=∂_i(f∘ j^n(σ))
is met by construction for all f∈ C^∞(𝒥^n(𝒳))
and σ∈Γ(π). The k-th order total derivative meets
d_1_i∈π_k+1,k^*(𝒯(𝒥^k(𝒳))).
The dual elements to the total derivatives are the so-called contact
forms. In local coordinates we have

    ω_J^α=dx_J^α-x_J+1_i^αdX^i ,    #J≥0,

where a k-th order contact form meets ω_J^α∈π_k+1,k^*(𝒯^*(𝒥^k(𝒳)))
with #J=k and j^k(σ)^*(ω_J^α)=0 is met
for σ∈Γ(π). Furthermore, based on the structure of
the underlying fibre bundle the operator of horizontalisation hor,
see e.g. <cit.>, can be introduced as 

    hor(dX^i)=dX^i, hor(dx_J^α)=x_J+1_i^αdX^i

which has the property that hor(ω_J^α)=0. 

The geometric version of the calculus of variations rests on the use
of so-called variational vector fields η, whose flow is used
to perform variations of sections σ∈Γ(π), where we
consider the special case of vertical variations. Hence, these vector
fields η are tangent to the fibration of 𝒳π→𝒟
and they are referred to as vertical vector fields, i.e. ∈𝒱(𝒳)
with the coordinate representation η=η^α(X,x)∂_α,
i.e. meeting Ẋ=0. Furthermore, we need the n-th jet-prolongation
of a vertical vector field η∈𝒱(𝒳) which
is given as

    j^n(η)=η^α∂_α+d_J(η^α)∂_α^J ,   d_J=(d_1_1)^j_1…(d_1_r)^j_r

with 1≤#J≤ n, see e.g. <cit.>. It is well-known
that these jet-prolongations of vertical vector fields will play a
prominent role when formulating a variation problem as recapitulated
next, see again <cit.>.

Let us consider a variational problem on the bundle 𝒳π→𝒟
where the Lagrangian is of the form 𝔏=ℒΩ
with ℒ∈ C^∞(𝒥^k(𝒳)) and
induces a functional L(σ)=∫_𝒮j^k(σ)^*𝔏
for sections σ∈Γ(π) where 𝒮 is a submanifold
of 𝒟. In the calculus of variations, the task is to find
so-called critical sections σ that extremize the functional
L(σ). To find these critical sections we will consider vertical
variations ϕ_ϵ (fiber preserving maps described by
a flow). A critical section σ meets

    .d_ϵ(∫_𝒮(j^k(ϕ_ϵ∘σ)^*𝔏)|_ϵ=0=0,

which is equivalent to

    ∫_𝒮j^k(σ)^*(L_j^k(η)(𝔏))=0,

see e.g. <cit.>, where η is the infinitesimal generator
of the flow ϕ_ϵ. 



§ FIRST-ORDER CASE


In this section we will recapitulate the well-known facts of a variational
problem for first-order Lagrangians, i.e. ℒ∈ C^∞(𝒥^1(𝒳)).
However, contrary to most of the differential geometric approaches
in the literature we allow for non-trivial boundary conditions and
we will demonstrate how the Cartan form is connected to these boundary
terms. 

These non-trivial boundary terms play a crucial role in many applications,
e.g. in engineering, where systems can be interconnected via their
boundaries, see e.g. <cit.>
for this port based approach, or where actuators directly influence
the boundary variables - boundary control systems.

This introductory section serves mainly as a reminder but it should
clarify some of the ideas that are then used in second-order case
k=2 in the forthcoming section. Let us consider the equation (<ref>)
for k=1 then using 

    L_j^1(η)(𝔏)=j^1(η)⌋d𝔏+d(j^1(η)⌋ 𝔏)

there is a natural decomposition into a domain and a boundary term,
as by Stokes' theorem d(j^1(η)⌋ 𝔏)
contributes to the boundary. Unfortunately, we cannot pick the domain
conditions directly as j^1(η)⌋d𝔏
may depend on derivatives of the variational field η that induces
further terms on the boundary. However, if we are able to reformulate
the problem such that no prolongation of η is necessary we are
done. The well-known solution of this problem is based on the use
of contact forms (<ref>) and the Cartan form, see
e.g. <cit.>, with 

    ℭ=ℒΩ+ρ_α^j ω^α∧Ω_j ,   ω^α=dx^α-x_1_i^αdX^i

implying ∫_Sj^1(σ)^*𝔏=∫_𝒮j^1(σ)^*ℭ
as hor(ℭ)=𝔏 is met, where the derivation
of the functions ρ_α^j will be described next. Instead
of (<ref>) we consider 

    ∫_𝒮j^1(σ)^*(L_j^1(η)(ℭ))=0

and j^1(η)⌋dℭ will deliver the domain
conditions (pdes) and from j^1(η)⌋ ℭ the
boundary expressions can be extracted, if we choose the functions
ρ_α^i such that 

    hor(j^1(η)⌋dℭ)=hor(η⌋dℭ)

is met. Consequently, 

    hor(j^1(η)⌋dℭ)=hor(j^1(η)⌋(dℒ∧Ω+dρ_α^j∧ω^α∧Ω_j-ρ_α^jdx_1_j^αΩ))

follows, which can be evaluated as

    hor(j^1(η)⌋dℭ)=(∂_αℒη^α+∂_α^1_iℒη_1_i^α-ρ_α^iv_1_i^α)Ω-hor(dρ_α^i)∧η^αΩ_i

where we have used the fact that hor(ω^α)=0
and d_1_iη^α=η_1_i^α. This expression
(<ref>) is independent of the jet-prolongation of the variational
vector field η_1_i^α iff 

    ρ_α^i=∂_α^1_iℒ

such that 

    hor(j^1(η)⌋dℭ)=η^α(∂_αℒ-d_1_iρ_α^i)Ω=η^αδ_αℒΩ

holds, where we have used 

    hor(dρ_α^i)∧Ω_i=d_1_i(ρ_α^i)Ω=d_1_i(∂_α^1_iℒ)Ω.

From (<ref>) we see that we have recovered the Euler Lagrange
operator that delivers the domain condition on 𝒮, where
δ_α=∂_α-d_1_i∂_α^1_i
denotes the variational derivative. 



 §.§ Boundary Conditions


The boundary conditions on ∂𝒮 follow from

    j^1(η)⌋ ℭ=η^αρ_α^iΩ_i=η^α(∂_α^1_iℒ)Ω_i.


It should be noted that the boundary conditions (<ref>)
on ∂𝒮 can be easily met, if one does not consider
variations on the boundary, i.e. if η vanishes. Conversely,
arbitrary variations on ∂𝒮 lead to conditions
for the first-order partial derivatives of ℒ. Furthermore,
in the case of boundary control systems it also possible to extend
the variational problem to include external boundary inputs that equate
with ∂_α^1_iℒ such that for arbitrary
variations the boundary conditions are met.

The functions ρ_α^j in (<ref>)
have been derived in such a way that in the variational principle
the domain integral becomes independent of derivatives of the variational
field. Furthermore, ρ_α^j enter the boundary integral
and determine the boundary conditions. As the choice for ρ_α^j
is unique for first-order field theories, so are the expressions in
the boundary integral.

To summarize, (<ref>) for k=1 is equivalent
to 

    ∫_𝒮j^2(σ)^*(η^αδ_αℒ)Ω+∫_∂𝒮j^1(σ)^*(η^α∂_α^1_iℒ)Ω_i=0.


Alternatively, (<ref>) can be derived using the
integration by parts method in a straightforward and unique manner,
as 

    L_j^1(η)(𝔏)     =     (η^α∂_αℒ+(d_1_iη^α)∂_α^1_iℒ)Ω
        =    η^α(∂_αℒ-d_1_i(∂_α^1_iℒ))Ω+d_1_i(η^α ∂_α^1_iℒ Ω)

gives the same result as in (<ref>) using Stokes'
theorem. Thus, the Cartan form approach indeed qualifies for a coordinate
free version of the integration by parts technique.




 §.§ Example


Let us consider the simple model of a Timoshenko beam, see e.g. <cit.>,
where the dependent variables x^1 and x^2 correspond to
the deflection of a cross section of the beam and the angle of rotation
due to bending. We have that m=2 and r=2, such that 𝒳
possesses the coordinates (X^1,X^2,x^1,x^2) and the independent
coordinates (X^1,X^2) correspond to the time X^1=t and
a spatial coordinate X^2=X. 

The Lagrangian density corresponding to the difference of the kinetic
and potential energy densities is given as ℒ with
and 

    ℒ=1/2(ρ(x_10^1)^2+J(x_10^2)^2-EI(x_01^2)^2-kAG(x_01^1-x^2)^2)

where for simplicity we assume constant beam parameters ρ, J, E, k, A, G.
Using (<ref>) the pdes on the domain immediately
follow as 

    [                  ρ x_20^1                         =        kAG(x_02^1-x_01^2);                   Jx_20^2                         = EIx_02^2+kAG(x_01^1-x^2), ]

as the variational field η is arbitrary on 𝒮. The
boundary expression follows from the boundary integral in (<ref>)
and leads to a condition of the form 

    -η^1kAG(x_01^1-x^2)-η^2EIx_01^2

which has to vanish at X=0 and X=L (the spatial boundary) as
in mechanics no variation on the time boundary takes place. For arbitrary
η^1 and η^2 we derive the classical conditions that
the shear force kAG(x_01^1-x^2) and the bending moment EIx_01^2
have to vanish on the spatial boundary.

As already pointed out in remark <ref> it is possible to
extend the variational problem, such that for external boundary inputs
Q(t) and M(t) at X=0 and/or X=L the boundary conditions
read as kAG(x_01^1-x^2)=Q and EIx_01^2=M. This can
be achieved e.g. by considering an additive boundary integral in the
variational principle such that external boundary inputs are incorporated.




§ SECOND-ORDER CASE


Let us consider the case ℒ=𝒞^∞(𝒥^2(𝒳))
and consequently the equation (<ref>) for k=2.
Again, using 

    L_j^2(η)(𝔏)=j^2(η)⌋d𝔏+d(j^2(η)⌋ 𝔏)

we aim to decompose the variational problem into an expression on
the domain and one on the boundary.



 §.§ The domain conditions


Similar as in the first-order case we modify the Lagrangian density
by extending it with contact forms of the type 

    ω_I^α=dx_I^α-x_I+1_k^αdX^k ,   0≤#I≤1.

Thus, we make the ansatz 


    ℭ    =    ℒΩ+ρ_α^j(dx^α-x_1_i^αdX^i)∧Ω_j+ρ_α^1_k,j(dx_1_k^α-x_1_k+1_i^αdX^i)∧Ω_j

which reads as

    ℭ=ℒΩ+ρ_α^j(dx^α∧Ω_j-x_1_j^αΩ)+ρ_α^1_k,j(dx_1_k^α∧Ω_j-x_1_k+1_j^αΩ).

To obtain the domain conditions in an analogous fashion as in the
first-order case we compute hor(j^2(η)⌋dℭ)
with


    dℭ    =    dℒ∧Ω+dρ_α^j∧(dx^α∧Ω_j-x_1_j^αΩ)-ρ_α^jdx_1_j^α∧Ω
           +dρ_α^1_k,j∧(dx_1_k^α∧Ω_j-x_1_k+1_j^αΩ)-ρ_α^1_k,jdx_1_k+1_j^α∧Ω

and hor(j^2(η)⌋dℭ)
can be stated as

    (∂_α^Iℒη_I^α-ρ_α^jη_1_j^α-ρ_α^1_k,jη_1_k+1_j^α)Ω-hor(η^αdρ_α^j+η_1_k^αdρ_α^1_k,j)∧Ω_j

with 0≤#I≤2 which simplifies to (<ref>) in the
first-order case. Hence, to cancel out the expressions that involve
a jet-prolongation of the variational vector field η_I^α,
we have to choose for the case #I=2 


    [   ρ_α^1_k,j+ρ_α^1_j,k                     = ∂_α^1_k+1_jℒ ,   k≠ j;             ρ_α^1_k,k                     =          ∂_α^1_k+1_kℒ ]

and to guarantee that η_I vanishes for the case #I=1 we
assign 

    ρ_α^l    =    ∂_α^1_lℒ-d_1_j(ρ_α^1_l,j).

Finally, 

    hor(j^2(η)⌋dℭ)=(∂_αℒ-d_1_j(ρ_α^j))η^αΩ

where we have used the fact that all expressions η_I for #I≠0
are canceled out. Using (<ref>) and (<ref>)
we obtain 


    ∂_αℒ-d_1_l(ρ_α^l)=∂_αℒ-d_1_l(∂_α^1_lℒ-d_1_j(ρ_α^1_l,j))

and consequently 

    hor(j^2(η)⌋dℭ)=δ_αℒη^αΩ

with

    δ_αℒ=∑_#I=0^2(-1)^#Id_I∂_α^Iℒ.

Hence, the domain conditions involve again the variational derivatives
(the components of the Euler-Lagrange operator) (<ref>) which
is of second-order in this case. The preceding calculation verifies
that despite of the fact that the choice of contact forms in (<ref>)
is not unique, it has no influence on the pdes, which is a well-known
fact. But this non uniqueness will have an impact on the boundary
conditions as demonstrated in the following.



 §.§ The boundary conditions


From (<ref>) it is obvious that there is no unique
choice for the functions ρ_α^1_k,j when k≠ j
which has the consequence that in the boundary expression, reading
as 

    j^2(η)⌋ ℭ=j^1(η)⌋ ℭ=(ρ_α^iη^α+ρ_α^1_k,iη_1_k^α)Ω_i,

a different choice for ρ_α^1_k,i leads to different
boundary conditions. To overcome this problem we pose an extra condition
on the set of possible boundary terms in order to extract the physical
meaningful boundary terms. To this end, we assume that the boundary
of the submanifold 𝒮 which is denoted as ∂𝒮
is parameterized as X^r=const. such that the coordinates for
𝒟 are (X_∂^i, X^r) with X_∂^i=(X^1,…,X^r-1)
which is possible at least locally by a change of the independent
coordinates, such that in these coordinates adapted to the boundary
the boundary volume form Ω_∂=∂_r⌋Ω
can be introduced. It is now remarkable that for a section σ_∂
on the boundary, i.e. x^α=σ_∂^α(X_∂^i),
we can compute all the first-order jet-variables apart from the X^r
direction, which represents an additional degree of freedom on the
boundary. 

Let us consider the inclusion mapping ι:∂𝒟→𝒟
in adapted coordinates. Then ι^*(𝒳) and ι^*(𝒥^1(𝒳))
are equipped with induced coordinates (X_∂^i,x^α)
and (X_∂^i,x^α,x_1_l^α) , l=1,…,r,
respectively. It is worth noting that ι^*(𝒥^1(𝒳))
does not qualify as a jet-manifold as we have too few independent
coordinates, but the coordinate x_1_r^α will play an
essential role in deriving the boundary conditions. This construction
has been used also in <cit.> to introduce the so-called extended
Cartan form on the boundary. We will pursue a different target, namely
we will exploit this boundary structure to give conditions for the
choice of the under determined functions ρ_α^1_k,i.

The boundary conditions are derived by means of 

    j^1(η)⌋ ℭ=(ρ_α^rη^α+ρ_α^1_k,rη_1_k^α)Ω_∂=(ρ_α^rη^α+ρ_α^1_1,rη_1_1^α+…+ρ_α^1_r,rη_1_r^α)Ω_∂

where the summation is carried out over k as r is fixed in the
adapted coordinates. On the boundary, independent variations with
respect to x^α and x_1_r^α can be carried
out, as discussed above, such that we are led to set 

    ρ_α^1_k,r=0 ,    k=1,…,r-1

such that

    j^1(η)⌋ ℭ=(ρ_α^rη^α+ρ_α^1_r,rη_1_r^α)Ω_∂

is met. 

From a variational point of view ρ_α^1_k,r=0, k≠ r
guarantees that only independent variations arise in the boundary
integral, which can be interpreted as an additional postulate which
selects the appropriate Cartan form out of a set of possible Cartan
forms that all deliver the same domain conditions. It should be noted
that this choice always leads to the minimal possible boundary conditions.
The coordinate representation is well justified in adapted coordinates
only, i.e. when the boundary is parameterized as X^r=const.

This means that in the relations (<ref>) we have to
choose ρ_α^1_r,k=∂_α^1_r+1_kℒ.
From relation (<ref>) we observe that then the coefficient
of η^α namely ρ_α^r which is 

    ρ_α^r=∂_α^1_rℒ-d_1_k(ρ_α^1_r,k).

involves exactly the terms ρ_α^1_r,k. Hence we have
exploited the degree of freedom regarding the functions ρ_α^1_k,i
in (<ref>) to generate the admissible boundary expressions. 

It should be noted that in <cit.> a different strategy is
suggested which requires an additional integration by parts or in
terms of <cit.> an extended Cartan form which contains an
additive total differential which corrects the wrong boundary terms.
E.g. an expression involving η_1_k^α for 1_k≠1_r
can be again integrated by parts to produce an admissible η^α
term. Both strategies give the same boundary conditions as the additional
integration by parts exactly delivers the second term in the right
hand side of (<ref>).




§ EXAMPLES FOR SECOND-ORDER MECHANICAL SYSTEMS


In this section we demonstrate our approach by means of two examples
- the Euler-Bernoulli beam and the Kirchhoff plate. 



 §.§ Euler-Bernoulli Beam


We have two independent variables X^1=t and X^2=X, hence
dim(𝒟)=2, where t is the time coordinate
and X the spatial variable (we set 𝒮=𝒟 for
simplicity). The dependent variable is the deflection x^1, such
that 𝒳 is equipped with the coordinates (X^1,X^2,x^1)
and Ω=dX^1∧dX^2=dt∧dX
together with the Lagrangian (kinetic minus potential energy)

    ℒ=1/2ρ A(x_10^1)^2-1/2EI(x_02^1)^2

where again for simplicity we consider constant physical parameters
ρ, A, E, I. The pdes follow from the Euler-Lagrange operator
(<ref>), which in this case simplifies to δ_1=-d_10∂_1^10+d_02∂_1^02
such that we obtain the well-known pde

    ρ Ax_20^1+EIx_04^1=0.




  §.§.§ Boundary Conditions


To obtain the boundary conditions, we consider the relation (<ref>)
with Ω_∂=Ω_r=∂_2⌋Ω=-dt
since we are interested in the spatial boundary (no variation on the
time boundary) and derive

    (ρ_1^2η^1+ρ_1^10,2η_10^1+ρ_1^01,2η_01^1)Ω_∂=0.

As no mixed partial derivatives appear in ℒ, ρ_1^10,2=0
is met automatically and we have 

    ρ_1^01,2    =    ∂_1^02ℒ=-EIx_02^1
    ρ_1^2    =     -d_01ρ_1^01,2=EIx_03^1

such that we obtain the conditions on the spatial boundary as 

    η^1EIx_03^1=0 ,   η_01^1EIx_02^1=0.

Here, EIx_03^1 corresponds to the shear force and EIx_02^1
to the bending moment and η^1 and η_01^1 take their
values according to the support on the boundary.

Let us consider a cantilever beam that is built-in
at X=0 (clamped end) and free at X=L (free end). Then at X=0
the imposed geometric boundary conditions imply η^1=η_01^1=0
whereas at X=L we have the dynamic boundary conditions EIx_02^1=EIx_03^1=0.
Furthermore, it is again possible to apply external forces and torques
at X=L which can be included in the variational principle. 

As no mixed partial derivatives appear in this example, the Cartan
form approach is unambiguous although the example belongs to second-order
theory. This will be different for the following example.



 §.§ Kirchhoff Plate


Let us consider the case of a (rectangular) Kirchhoff plate with three
independent variables X^1=t, X^2=X and X^3=Y, such that
we have two spatial variables and the time. Hence, 𝒳
is equipped with the coordinates (X^1,X^2,X^3,x^1) where
x^1 denotes the deflection. The Lagrangian is given as ℒ=𝒯-𝒱
with 2𝒯=(x_100^1)^2 and 

    2𝒱=(x_020^1)^2+(x_002^1)^2+2ν x_020^1x_002^1+2(1-ν)(x_011^1)^2

where we have set the physical parameters to one apart from the Poisson's
ratio ν. Again the pdes follow from the Euler-Lagrange operator
(<ref>), with 

    δ_1=-d_100∂_1^100+d_020∂_1^020+d_002∂_1^002+d_011∂_1^011

such that we obtain

    x_200^1+x_040^1+x_004^1+2x_022^1=0.




  §.§.§ The boundary conditions


To obtain the boundary conditions on the boundary X^3=Y=const.
we consider

    (ρ_1^3η^1+ρ_1^010,3η_010^1+ρ_1^001,3η_001^α)Ω_∂

where we already have used ρ_1^100,3=0 since no mixed time
and spatial derivatives appear. The non-zero components for ρ_1
of regarding the jet-order #I=2 in ℒ are

    ρ_1^001,3=∂_1^002ℒ ,   ρ_1^010,2=∂_1^020ℒ

as well as

    ρ_1^010,3+ρ_1^001,2=∂_1^011ℒ.

It is remarkable that for the mixed spatial derivative we get two
possible contact forms. For #I=1 we obtain from ρ_α^l=∂_α^1_lℒ-d_1_j(ρ_α^1_l,j)
the coordinate expressions

    ρ_1^3    =    ∂_1^001ℒ-d_001ρ_1^001,3-d_010ρ_1^001,2
    ρ_1^2    =    ∂_1^010ℒ-d_010ρ_1^010,2-d_001ρ_1^010,3

and ρ_1^1=∂_1^100ℒ. Since η_010^1
is not admissible on the boundary of interest we can choose ρ_1^010,3=0
and consequently ρ_1^001,2=∂_1^011ℒ
follows. Hence,

    ρ_1^3=-d_001ρ_1^001,3-d_010ρ_1^001,2=(x_003^1+(2-ν)x_021^1)

and therefore on the spatial boundary X^3=Y=const. we obtain
(no variation on the time boundary)

    η^1(x_003^1+(2-ν)x_021^1)     =     0
    
    -η_001^1(x_002^1+ν x_020^1)     =     0

where (<ref>) is a condition for the shear force and (<ref>)
for the bending moment. As a consequence, we obtain two boundary conditions,
whereas a wrong choice for ρ_1 (corresponding to a wrong integration
by parts) would deliver three (compare with (<ref>)).
To construct the boundary conditions for the boundary X^2=X=const.
one can easily adopt the procedure or re-parameterize the independent
coordinates. Furthermore, it should be noted that in an analogous
fashion as in remark <ref> the introduction of imposed
geometric and dynamic boundary conditions can be accomplished.



§ BOUNDARY PORTS 


To motivate for the idea of boundary ports, let et us consider a Hamiltonian
of first-order based on an underlying bundle structure 𝒳→𝒟.
Hence, we have ℌ=ℋΩ with ℋ∈ C^∞(𝒥^1(𝒳)).
Depending on a chosen section s:𝒟→𝒳
we are led to compute H(s)=∫_𝒟j^1(s)^*(ℌ).
Let us consider now an evolutionary vector-field v=v^α∂_α,
corresponding to the pdes

    ẋ^α=v^α ,    v^α∈ C^∞(𝒥^2(𝒳))

and we are interested in the change of H along the section s,
where s is a formal solution of the pdes parameterized in t,
i.e.

    ∂_ts(X)=v(X,∂_Js(X)) ,   #J≤2

together with appropriate boundary conditions, where we have suppressed
the explicit dependence of s with respect to the evolution parameter
and well-posedness of the problem is assumed.

The idea of ports in a Hamiltonian setting can be visualized by computing
a power balance relation by means of the Hamiltonian density which
can be interpreted as an energy density in many applications. Thus
we evaluate 

    Ḣ=∂_tH(s)=∫_𝒮j^2(s)^*(L_j^1(v)ℌ)

and regarding the evaluation of the expression L_j^1(v)ℌ
appearing in (<ref>) the same reasoning as
in section <ref> can be applied and 

    ∂_tH(s)=∫_𝒮j^2(s)^*(v^α(δ_αℋ)Ω)+∫_∂𝒮j^1(s)^*(v^α(∂_α^1_iℋ)Ω_i)

follows which is the counterpart to (<ref>).

It should be noted that instead of a variational vector field η
that performs the variation, now an evolutionary vector field that
corresponds to a dynamical systems is applied. But form a mathematical
point of view the calculations remain valid when ℒ is
replaced by ℋ and η is replaced by v. 

Thus, we have obtained a power balance relation when ℌ=ℋΩ
corresponds to an energy density. Obviously, in this power balance
relation we have a contribution in the domain and one on/over the
boundary. We can restate (<ref>) to obtain 

    Ḣ=∫_𝒮j^2(s)^*(v⌋δℌ)+∫_∂𝒮j^1(s)^*(v⌋δ^∂ℌ)

with δℌ=(δ_αℋ)dx^α∧Ω
and δ^∂ℌ=(∂_α^1_iℋ)dx^α∧Ω_i.
Hence, choosing

    v=(𝒥-ℛ)(δℌ)+u⌋𝒢

where 𝒥 is a skew- symmetric map and ℛ is
positive semi-definite, we can express the impact on the domain as

    ∫_𝒮j^2(s)^*(-ℛ(δℌ)⌋δℌ+y⌋ u)=∫_𝒮j^2(s)^*(-(δ_αℋ)ℛ^αβ(δ_βℋ)+y_ξu^ξ)Ω

with y=𝒢^*(δℌ) and the input map 𝒢
together with its adjoint 𝒢^*. The boundary impact
(depending on the boundary conditions of the pdes) is determined by
the expression v⌋δ^∂ℌ.

Given evolutionary pdes of the form (<ref>) with the generalized
vector field v as in (<ref>), the formal change of the
first-order Hamiltonian H along solutions of (<ref>)
can be decomposed in a dissipative term in the domain -ℛ(δℌ)⌋δℌ,
in a power port in the domain y⌋ u and in a power port on
the boundary v⌋δ^∂ℌ.

The representation (<ref>) together with the collocated
output y=𝒢^*(δℌ) can be seen as a generalization
of the classical port-Hamiltonian formulation of ODEs, see e.g. <cit.>,
to the infinite-dimensional case based on the underlying jet-bundle
structure and has been proposed and discussed in <cit.>.
The power balance relation (<ref>) is valid when
the maps 𝒥, ℛ and 𝒢 do not involve
differential operators. However, also this differential operator case
can be discussed in the proposed setting, see e.g. <cit.>,
but then additional boundary expressions have to be taken into account
due to the differential operators. The first-order case can be also
handled by using the classical integration by parts technique, as
it has been done in the cited references. For second-order theories
similar problems as in the Lagrangian setting (calculus of variations)
appear and we will show that the proposed Cartan form approach can
also be exploited in the port-Hamiltonian setting for the determination
of the boundary ports in a similar way as it has been used in the
Lagrangian scenario to determine the boundary conditions. 



 §.§ Domain and Boundary Impact for the second-order case


Let us consider a Hamiltonian density ℌ=ℋΩ
with ℋ∈ C^∞(𝒥^2(𝒳)) and
an evolutionary vector field v. Then, to evaluate 

    Ḣ=∂_tH(s)=∫_𝒮j^4(s)^*(L_j^2(v)ℌ)

we follow the same consideration as in section <ref>
by augmenting ℋΩ with appropriate contact forms.
This immediately gives us the desired decomposition of Ḣ
into a domain expression Ḣ_𝒮 and a boundary expression
Ḣ_∂𝒮 with 


    Ḣ_𝒮=∫_𝒮j^4(s)^*(v⌋δℌ)

where δ involves the Euler-Lagrange operator of second-order,
i.e. 

    δ_αℋ=∑_#I=0^2(-1)^#Id_I∂_α^Iℋ.

Thus, again the consideration of (<ref>), where δℌ
involves (<ref>) in this case, gives an analogous
power balance relation. The crucial point is again the boundary impact
Ḣ_∂𝒮 and the key to solve the
problem is to consider adapted coordinates with respect to the boundary
such that 

    Ḣ_∂𝒮=∫_∂𝒮j^3(s)^*(ρ_α^rv^α+ρ_α^1_r,rv_1_r^α)Ω_∂

can be stated (which is the counterpart to (<ref>)),
where the determination of the function ρ_α^r and ρ_α^1_r,r
follows from (<ref>) and (<ref>)
by substituting ℒ with ℋ.



 §.§ Example


The port-Hamiltonian representation of the Kirchhoff plate can be
accomplished by choosing a manifold 𝒳 equipped with the
coordinates (X^1,X^2,x^1,p_1) where x^1 again denotes
the deflection and x^2=p_1 the linear momentum It should be
noted that the number of independent variables is two, in contrast
to section <ref>, as the time is serving as
an evolution parameter. The Hamiltonian is given as ℋ=𝒯+𝒱
with 2𝒯=(p_1)^2 and 

    2𝒱=(x_20^1)^2+(x_02^1)^2+2ν x_20^1x_02^1+2(1-ν)(x_11^1)^2.

The pdes read as

    [[ ẋ^1; ṗ_1 ]]=[[  0  1; -1  0 ]][[ δ_1ℋ; δ^1ℋ ]] ,    𝒥=[[  0  1; -1  0 ]]

with 

    δ^1=∂/∂ p_1 ,   δ_1=∂_1-d_1_i∂_1^1_i+d_J∂_1^J ,    #J=2

which is

    ẋ^1=p_1 ,    ṗ_1=-(x_40^1+x_04^1+2x_22^1).

The Hamiltonian includes second-order jet variables, and the application
of the variational derivative implies then forth-order pdes in our
evolutionary setting.

The pdes correspond to (<ref>) where the second time
derivative, i.e. x_200, is written as a first-order derivative
with respect to the momentum variable p_1 to obtain an evolutionary
representation in the Hamiltonian setting. Furthermore, it is easily
checked that (<ref>) is of the form v=(𝒥-ℛ)(δℌ)+u⌋𝒢
with ℛ=0 and 𝒢=0 and with the second-order
variational derivatives δ^1 and δ_1 acting on
ℋ, where δ^1 degenerates to a partial derivative
as no jet-variables with respect to p_1 are involved.

Next we wish to analyze the power balance relation in order to obtain
the corresponding ports. As we use a skew-symmetric map 𝒥
we can conclude that Ḣ_𝒟 in Ḣ=Ḣ_𝒟+Ḣ_∂𝒟
is equal to zero and that only a boundary impact has to be considered.



  §.§.§ The boundary impact


From (<ref>) we obtain on the boundary X^2=const.
the expression


    Ḣ_∂𝒟=-∫(v^1ρ_1^2+v_01^1ρ_1^01,2)dX^1=∫(ẋ^1F_S+ẋ_01^1M_B)dX^1

with the shear force F_S=(x_03^1+(2-ν)x_21^1) appearing
in the power port ẋ^1F_S and the bending moment M_B=-(x_02^1+ν x_20^1)
in ẋ_01^1M_B. The collocated quantities are ẋ^1=v^1
(velocity) and ẋ_01^1=v_01^1 (angular velocity).
Hence, to obtain the physically correct ports, it is vitally important
to perform the integration by parts properly, which in our language
amounts for a correct selection of the coefficients of the Cartan-form
in accordance with the parameterization of the boundary.



§ CONCLUSION


In this contribution we have provided a strategy to compute the boundary
conditions for second-order field theories in a variational formulation
in an algorithmic fashion suitable for the computer algebra implementation.
Furthermore, for the class of port-Hamiltonian systems formulated
on jet-bundles, the same reasoning regarding a minimal number of boundary
ports can be carried out, as the mathematical problem of computing
the variation of the Lagrangian density is equivalent to the evaluation
of the change of the Hamiltonian (the total energy) along solutions
of an evolutionary set of pdes in a port-Hamiltonian setting.

plain


