4 Variational Calculus
4.2 Extremal, Variational and Strong Formulation of a Problem
Based on the following “standard” exercise in calculus of variation, the three different formulations of one and the same problem can be understood. Suppose a task demands to find the two times differentiable functiony=y x of the variablex, where the scalar valued quantityJis maximal or minimal respectively extremal. It is assumed that by means of theLagrangian density functionalZthat the following formulation of the quantityJcan be used to describe appropriately the task
J = ˆb
a
Zdx= ˆb
a
Z x, y, y0
dx → extremal, (4.1)
where the boundary conditions,y x = a
= yaandy x = b
= yb, are given andy0 = dydx. In calculus of variations, the above formulation is considered to be a principle, avariational principleto be precise, which is believed that a certain class of processes follows. For example, the Fermat’s principle also known as theprinciple of least timepostulates that a ray of light traversed between two points in the least amount of time. In this case,J is the total traverse time that has to be minimal. The path between this two points is described by the functiony.
The density functionalZrepresents the obstacles such as different media that lie between the two points. The path yhas to be determined. Another example is the aforementioned Brachistochrone problem. A curveyof fastest descent of a particle has to be determined. Therefore, the total descending timeJhas to be minimal. The gravity that the particle is subjected to as well as the consideration of kinetic energy are expressed by the density functional Z. The variational principle in Eq. (4.1) is a formulation that is valid and applicable for a specific class of problems.
In dependence of a given problem the formulation in Eq. (4.1) might be insufficient. In this case, the variational principle has to be extended. Nevertheless, in general a variational principle formulates mathematically a given problem in form as a extremal postulation. Therefore, we can refer variational principles asextremal formulations of problems.
The objective is to find the curveywhere the extremal formulation of a specific problem in Eq. (4.1) is maximal or minimal. As mentioned in the brief history of calculus of variations, it was Lagrange who found a non-geometrical approach to determine the curvey. The main difficulty is, simply put, to find the derivative ofJ with respect toyso that it can be set to zero to find the optimal curvey. To overcome this issue, Lagrange varied
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around the optimal solutionywith a disturbanceηin Eq. (4.1) to obtain a non-optimalJ. It reads
J = ˆb
a
Z x, y+η, y0+η0
dx . (4.2)
Furthermore, he decomposedηinto two parts
η =εδy , (4.3)
whereδy=δy x
is any function of variablexand the small parameterεis a scalar value quantity independent of variablex. Inserting Eq. (4.3) into Eq. (4.2), it reads
J = ˆb
a
Z x, y+εδy, y0+εδy0
dx . (4.4)
The functionδy is arbitrary in the sense that this function is freely selectable. As a result of this, the arbitrary functionδyoperates as a given quantity. However, the following constraint applies to this functionδy. Since the starting and end points of the curveyare fixed, all possible disturbed curvesy+ηmust also pass through these two points. Thus, the disturbanceηhas to vanish atx=aand atx=b. To achieve this, it must hold for
δy
x=a=δy
x=b = 0. (4.5)
The small parameterεis used to regulate the perturbationη. In case thatε= 0,Jis optimal, sinceηvanishes and yis the optimal solution. Any other values forε,J becomes non-optimal. Since the optimal solutionyand the arbitrary function are both treated as given, the extremal formJ =J ε
becomes a function which depends only on the one single small parameterε. Forε= 0,Jhas the optimal value, therefore its first (directional) derivative has to vanish respectively it must hold for
dJ dε ε=0
= 0! . (4.6)
Inserting Eq. (4.4) into the above equation leads to dJ dε ε=0
= ˆb
a
dZ dε ε=0
dx . (4.7)
The total differentialdZin the above equation reads dZ = ∂Z
∂x dx+ ∂Z
∂ y+εδyd y+εδy
+ ∂Z
∂ y0+εδy0d y0+εδy0
. (4.8)
This in turn yields dZ
dε = ∂Z
∂x dx dε
|{z}
=0
+ ∂Z
∂ y+εδy
d y+εδy dε
| {z }
=δy
+ ∂Z
∂ y0+εδy0
d y0+εδy0 dε
| {z }
=δy0
= ∂Z
∂ y+εδyδy+ ∂Z
∂ y0+εδy0δy0. (4.9)
When the disturbance is set to zero, i. e.ε= 0, it reads dZ
dε ε=0
= ∂Z
∂yδy+ ∂Z
∂y0δy0. (4.10)
The second expression on the right-hand side in the preceding equation can be rewritten by means of the product rule as
d dx
Å∂Z
∂y0δy ã
= d dx
Å∂Z
∂y0 ã
δy+∂Z
∂y0δy0. (4.11)
Therefore, Eq. (4.10) can be rewritten as dZ
dε ε=0
= ∂Z
∂yδy− d dx
Å∂Z
∂y0 ã
δy+ d dx
Å∂Z
∂y0δy ã
. (4.12)
Inserting the equation above into Eq. (4.7) yields dJ
dε ε=0
= ˆb
a
∂Z
∂yδy− d dx
Å∂Z
∂y0 ã
δy
! dx+
ˆb
a
d dx
Å∂Z
∂y0δy ã
dx . (4.13)
Due to Eq. (4.5), in the preceding equation the second term on the right-hand side vanished ˆb
a
d dx
Å∂Z
∂y0δy ã
dx= ∂Z
∂y0 x=b
δy x=b
| {z }
=0
−∂Z
∂y0 x=a
δy x=a
| {z }
=0
= 0. (4.14)
Finally, the following expression dJ dε ε=0
= ˆb
a
∂Z
∂y − d dx
Å∂Z
∂y0 ã!
δydx= 0. (4.15)
is obtained. Since the above expression ˆb
a
∂Z
∂y − d dx
Å∂Z
∂y0 ã!
δydx= 0 (4.16)
is derived from the idea of Lagrange’s variation, it is referred to as thevariational formulation. This form plays an important role in the numerical computation for the finite element method as well as for the least squares ad-justment as it shall be seen later. Since this formulation is often used to determine various approximate solutions, equation Eq. (4.16) is also known as theweakformulation.
In a nonchalant way, Lagrange concluded that from Eq. (4.16)
∂Z
∂y − d dx
Å∂Z
∂y0 ã
= 0 (4.17)
must hold. This conclusion can be made due to the so-calledfundamental lemma of calculation of variationsthat was not known to him at that time. According to this lemma, the arbitrary functionδycan be anything due to its arbitrariness and the only way to ensure that the variational formulation in Eq. (4.16) is fulfilled for any arbitrary functionδyis to force the expression in the parenthesis in Eq. (4.16) to be always zero. This leads to the equation in Eq. (4.17) that is known as theEuler-Lagrange equation. It can be solved analytically to determine the optimal curve y. Since an approximate solution is unable to satisfy Eq. (4.17) respectively only the exact one can fulfil, equation Eq. (4.17) is also referred to as thestrong formulation.
What insights can be learnt from this exercise in calculus of variation? A given problem can be formulated in three different ways: extremal, variational and strong form. Moreover, these three formulations can be converted among each other. In this particular exercise a specified problem is restated from the extremal form in Eq. (4.1) to the variational form in Eq. (4.16) to the strong form in Eq. (4.17). It is also possible start at any form and converting to another one. Then, each formulation can be served as an entry point to various methods in order to solve the problem. Some main aspects of the three different formulations are presented as follows.
extremal form A problem is stated as a certain valueJthat has to be minimal or maximal. This valueJ may be scaled and shifted by a constant factor and a fixed offset without changing the problem statement. Therefore, the extremal formulation is not unique in that sense thatJmay take different extremal forms, but they may all describe the same problem. Some examples of extremal formulated problems are: The aforementioned Brachistochrone problem where the total descending time has to be minimal, maximising profits or minimising losses in economics,
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in optics the principle of least time states that a ray of light prefers a path to travel from one point to another at least amount of time, in least squares adjustment the sum of weighted squared residuals has to be minimised and many more.
variational form This formulation is seemingly an intermediate state in the exercise of calculus of variation. But by far, this form serves as the most interesting entry point for many different numerical methods. If one looks closely, a simplified depiction of the variational form in Eq. (4.16) has this outline´
f g= 0. In a vector interpretation, this expression can be seen as a dot product betweenfandg. Therefore, it demands thatfis orthogonal togrespectively it requests the perpendicular distance offtogto be minimal. Iff can be seen as some sort of measure on how well some solutions can fulfilf(also known asresidual), the variational formulation can be used to minimizefin order to determine some optimal solution. Sincefcan be non-zero in the variational formulation, this implies that fcan accept an approximate solution. Due to this reason, the variational formulation is considered less restrictive and this non-strictness property is often described as “weak”. Therefore, the variational formulation is also known as theweak formulation. In this particular exercise, this form is derived from the extremal formulation, but it is also possible to reach the variational form from the strong formulation in Eq. (4.17) by multiplying it by an arbitrary functionδyand integrating it over the domain of interest. This approach is for example often used in finite element method. And one may notice that the functionδyis none other than the so-calledtest functionin finite element method. It is important to note that for numerical analysis the functionδyhas to be specified and at the same time this function has to fulfil certain compatibility aspects in order to obtain computable results. Furthermore, one needs to state a certaintrial functionrespectivelyansatz functionfory.
strong form In the exercise this formulation is extracted from variational form. If the simplified depiction of the variation formulation´
f g = 0is re-examined, one notices that the strong form isf = 0. This implies that in the strong formulation the solution has to satisfyfcompletely. Only the exact solution can fulfil this requirement.
This strictness is described as “strong”. One may realize that in the exercise in calculus of variation the strong formu-lation is a differential equation. By thinking ahead, the strong formuformu-lation can be any kind of equation. As long as it can be assumed that a system of (differential) equations is being able to describe a given problem properly enough, an exact solution of the problem can be obtained by solving the equations. In case an exact solution is unobtain-able, one can still try to approximate the solution by reformulating the strong formulation into its variational form.
Furthermore, in comparison to other formulations, the strong formulation of a given problem has an exclusive characteristic when this formulation is represented independently of a particular system of coordinates. Aforemen-tioned, an extremal formulation can be scaled and shifted without changing the problem statement. Therefore, it is possible to have different in fact infinite amount of extremal formulations that are describing the same problem. In contrast, a modification of the strong formulation leads to a different problem description and vice versa. In other words, a given problem is clearly assigned to one single representation in the strong form. This implies that there is only one theory of anything. Some examples of strong formulated problems are: equations of elastodynamics in mechanics in Eq. (2.161), the Lamé-Navier equations in Eq. (2.162), the Euler-Bernoulli beam equation in Eq. (2.182), heat equation in Eq. (2.206) in thermodynamics, Maxwell’s equations in electrodynamics and even overdetermined system of equations in least squares adjustment.
The questions “Where to begin?” or “Which order one must follows to obtain a solution?” is unnecessary. All three formulations exist side-by-side. An expedient procedure results rather from an abstract comprehension of a given problem and one’s intention.
In conclusion, a problem can be formulated in three different ways: As anextremal formulationwhere one has to look for an extremal valueJto determine an optimal solution, as avariational, respectively,weak formulation where a numerical method is applied to compute an approximate solution and as astrong formulationwhere one has to find an analytical solution.