Classical Problems in Calculus of Variations and Optimal Control

Stephen Lamb

Supervised by Lyle Noakes

The University of Western Australia

Vacation Research Scholarships are funded jointly by the Department of Education and Training and the Australian Mathematical Sciences Institute.

1 Introduction

Calculus of variations (COV) is a field of mathematics that deals with finding the extremals ofLagrangian functions defined by functionals, in an attempt to find optimal solutions. Although thesubject has a long and rich history, current research in the field is still producing new results. Op-timal control is the generalisation of the calculus of variations. My project focuses on the classicalproblem of COV, and how the tools of optimal control can be used to simplify results, and evenproduce results where COV was too blunt an object. My project started with a brief study of someclassical problems in COV and then I started generating the differential equations that we solve toanswer the problems. These include such problems as the brachistochrone and the catenary.

The next task was to learn about optimal control and its link to COV. Once able to write outour control problem, I needed a similar tool to solve for extremals as above, which in optimal con-trol is the Pontryagin Maximum Principle. The most crucial part of the project was about learningand understanding how to use the Pontryagin Maximum Principle to solve the optimal controlproblems. We then extended it from Rn to other smooth manifolds using Riemannian geometry.We used our framework of optimal control to solve two very important classical problems: geodesicson Riemannian manifolds and elastica curves. Lastly, after generating solutions to these problemsfor different spaces, we had to have a method of solving these boundary value problems numerically,for problems where no closed form solution exists.

The study of classical problems in calculus of variations and optimal control has provided me witha good foundation for further study into differential geometry, optimisation methods and functionalanalysis. It has also given good insight into the topics of topology, Riemannian geometry and anal-ysis. Although I have aquired a deeper level of understanding through this research project, I amleft with far more questions to answer.

Future work into the topic involves study of dependence of these classical problems to assumedconditions. e.g. The study of the brachistochrone under different gravitational field conditions orthe catenary under a changing viscocity of air as a function of height. Further research into geodesicsphrased using optimal control could involve looking into the geodesics on manifolds that aren’t asnice and symmetric as the ones chosen. Also research into the existence of abnormal geodesicsin sub-Riemannian manifolds would yield enlightening results. For the problems of elastic curves,future work could involve the study of elastic curves in other riemannian manifolds. Lastly furtherinvestigation into other numerical solving methods for solving these problems, aiming at reducingcomputing cost could prove a good idea.

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2 Calculus of Variations

To understand what calculus of variations is, and in turn what optimal control is, we requireunderstanding of the lagrangian function and how to determine extremals from it.

Definition 2.1. The Lagrangian (L) is an energy function defined on the Tangent bundle, thatmaps onto the space of real numbers. We can write L as L(q(t), q(t))

L : TM → R

Calculus of variations is concerned with finding the extremals of functions defined by:

J(t) =

∫ t2

t1

L(q(t), q(t))dt, (1)

where extremals are the critical points of the Lagrangian.

Theorem 2.2. If q is an extremal of the Lagrangian, then it must satisfy the Euler-Lagrangeequation:

d

dt

(∂L

∂q

)=∂L

∂q(2)

Proof. Consider the system defined above by (1), and let’s assume that q is an extremal. Withoutloss of generality, we will assume that q is a minimising function. Hence we can conclude that:

L(q(t), q(t)) < L(q(t) + εη(t), q(t)εη(t))

Therefore if we take the derivative with respect to ε, the following result is obtained:

d

dεJ(t)|ε=0 =

d

dε

∫ t2

t1

L(q(t) + εη(t), q(t) + εη(t))dt|ε=0 = 0

=⇒∫ t2

t1

(∂L

∂qη +

∂L

∂qη

)dt = 0

Using integration by parts on the second term above, we get the following:

=⇒∫ t2

t1

(∂L

∂qη − d

dt

∂L

∂qη

)dt = 0

Factoring out η we get the Euler-Lagrange equation:

d

dt

(∂L

∂q

)=∂L

∂q

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2.1 Classical Problems in COV

2.1.1 Brachistochrone

The Brachistochrone is a classical problem that deals with finding a curve between two points:(x0, y0) and (x1, y1) with x0 < x1 and y0 < y1, such that the time taken for a bead to go along thecurve between the two points on the curve is minimal.

Time is the variable that is going to be minimised.Let the total length of the curve be L. Using Newtons equations of motion, we can derive theequation for total time taken to traverse the curve:

T (y) =

∫ 0

L

ds

v(s)ds (3)

(Where v(s) is the velocity and ds is the arclength)

Using the conservation of energy principle, we can define this expression in terms of y(x).

KE + PE = E = constant

E = mgy(x) +1

2mv(x)2 (4)

Rearranging for v(x), we get the following:

v(x) =

√2(E −mgy(x))

m(5)

Thus using (3), we get the following result:

T (y) =

∫ x1

x0

√1 + y2√

2(E−mgy(x))m

dx (6)

With y(x0) = y0, and y(x1) = y1

Simplifying this expression with the substitution: z = 12g

(2E−2gmy(x)

m

)we get the following func-

tional:

J(z) =√

2g

∫ x1

x0

√1 + z2

zdx (7)

Using the Euler-Lagrange equation, we derive the following differential equation:

z(1 + z2) = c (8)

Using the substitution z = tan(θ), then 1+z2 = sec(θ). Hence the curve that satisfies the conditionsof the brachistochrone curve is the parametric equation of a cycloid:

y(θ) = d1(1 + cos(2θ)) (9)

x(θ) = d2 − d1(2θ + sin(2θ)) (10)

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Figure 1: Brachistochrone curve plotted between fixed points (−π2 , 1) and (0, 0)

This problem can be extended and studied in many ways for various reasons. As such, some of theseinclude having a forcing/damping system, or even considering a non-constant gravitational field,dependent on the y(x). Harry H. Denman goes into other possible solutions to the brachistochroneproblem by varying fields. See [2] for more.

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2.1.2 Tautochrone

The tautochrone problem questions finding a curve such that under constant acceleration, a particleplaced anywhere on the curve will take the same amount of time to reach the bottom of the curve,no matter its start position. In a constant gravitational field, it can be easily shown that thebrachistochrone curve satisfies such a property. [2]

2.1.3 Catenery

The catenary is a sometimes refered to as the hanging chain problem. It seeks to determine theshape of a curve made between two fixed points in a constant gravitational field. In practice it isthe curve made when hanging a chain/wire between two fixed points, making a curve called thecatenary curve, who’s name originates from the problem. The problem can also be described asa curve that minimises gravitational potential energy, which we take advantage of to derive ouranswer.We start by defining the functional to minimise:

PE =

∫ L

0

m g y(s) ds (11)

(where PE is the potential energy of the whole curve, L is the length of the curve and s is thearclength of the curve)

Using the fact that ds2 = dx2 + dy2 =⇒ ds =

√1 +

(dydx

)2dx, we can rewrite the above equation

as:

PE =

∫ L

0

m g y(x)

√1 +

(dy

dx

)2

dx (12)

Now we can employ the Euler-Lagrange equation above. Hence any extremal of the above equationmust satisfy the following:

d

dx

(∂PE

∂y

)−(∂PE

∂y

)= 0

Simplifying the above equation gives the following differential equation

yy y√1 + y2

− y√

1 + y2 = Constant

y2

1 + y2= D2

1 (13)

We ignore the trivial case and only consider D 6= 0 to get:

y =

√y2

D21

− 1 (14)

Separating variables and integrating we get the follwoing equation:

x = D1 ln

(y +

√y2 −D2

1

D1

)+D2 (15)

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Rearanging the above equation, we get the solution of the catenary curve in terms of y as:

y = D1 cosh

(x−D2

D1

)(16)

which is graphed below. (D1 and D2 are constants)

Figure 2: Plot of equation (16) - Catenary curve/Hanging chain

This curve looks similar to the parabolic curve, however the two are very different. Catenary curvesfind themselves in nature and also lend themselves to building the perfect arches for buildings.

2.1.4 Isoperimetric Problem

The isoperimetric problems set to solve constrained variational problems by first transformingthem into unconstrained problems using Lagrange multipliers. These problems allow solutionsof problems constrained to have extremals exist on curves or other constraints. These are veryinteresting problems, however further research is needed to fully understand the importance ofLagrange multipliers.

2.1.5 Minimal Surfaces

Minimal surfaces are objects on spaces that have zero mean curvature (See [1] for definition). Theyare also more commonly defined as surfaces that minimise surface area given a set of boundaryconditions. These problems are usually posed in R3, which coincides with the common definition of

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how we view surfaces. In mathematical terms, if the surface is parameterized as x=(u, v, f(u, v)),then it is a minimal surface if it satisfies the following condition:

(1 + f2v )fuu + 2fufvfuv + (1 + f2u)fvv = 0 (17)

The simplest minimal surface is just the space R3. Another example of a minimal surface is thecatenoid, which is just a surface of revolution of a catenary curve. This is pictured below:

Figure 3: Catenoid figure

Minimal surfaces are wonderful mathematical objects that are very fascinating to study. Theidea of minimal surfaces can be extended to minimal submanifolds. See [12] for more

3 Optimal Control

Optimal control is an important extension of COV to a more general framework, allowing a greaterrange of problems to be solved as a result. All COV problems can be posed as optimal controlproblems. This is a result of how optimal control problems are set out, and the similarities it shareswith COV. Rather than operate with a Lagrangian, optimal control problems use a different energyfunction known as the Hamiltonian.

Definition 3.1. The Hamiltonian is an energy function on the cotangent bundle of manifold,making the problem in the phase space rather than the state space.

Definition 3.2. A Legendre Transformation is a type of transformation that provides a connectionbetween the Euler-Lagrange equation and the Hamiltonian system of equations. The transforma-tion can be phrased as such:

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Let y : [t1, t2]→ R, and let u = y(t).

Now suppose there is a Lagrangian defined as L(y, y), then letting λ = ∂L∂y . From this, we can

construct our Hamiltonian:H(y, λ0, λ, u) = λ u− λ0L(y, u) (18)

Using a Legendre transformation, we can generate our problem in phase space and solve 2nfirst-order equations instead of n second-order ones. However, do they generate the same extremalsfor the problem.

Theorem 3.3. The extremals of the Lagrangian function are the exact same as those of theHamiltonian function under the Legendre Transformation. [10]

Now that we know that we can solve the same types of problems in optimal control as COV,we’ll need to know how to solve them.

3.1 Pontryagin Maximum Principle

The Pontryagin Maximum Principle (PMP) is a very elegant tool used in optimal control to solvethese control problems. In some sense, it does the equivalent to a Hamiltonian as the Euler-Lagrangeequation does to the Lagrangian. However because it lends itself to a more general framework ofproblems, it can be used for so much more. Phrasing the PMP goes like this:Imagine that we had a control problem, solving for the extremals of the following Hamiltonian:

H(y, λ, λ0, u) = λ0L(y, u, t) + λ(u) (19)

and also assume that we know the optimal solutions of the trajectory is (y*,u*), then the followingconditions must be satisfied.

1. Non-triviality: (λ0, λ) 6= 0

2. Costate equations: λ = −∂H∂y

3. Maximum condition: H(y∗, λ, λ0, u∗) = max

u∈VH(y, λ, λ0, u)

The u’s in these problems are part of a larger set U called the set of admissible functions. This isthe functional space that fits our conditions to be a solution. For the purpose of this project, wehave only considered piecewise continuous functions with a finite number of discontinuities, howeverthere exist much larger functional spaces that are required to solve some optimal control problemsas the answer isn’t contained in the smaller set we consider. In his book, Pontryagin considers sucha larger set to solve problems.

The first condition of the PMP ensures the non-existence of trivial solutions when the Hamil-tonian is equal to 0. The second follows from the existence of the Hamiltonian, satisfying theHamiltonian system of equations.

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The last condition is by far the most important. Note that the set V is the set of all u, suchthat any u ∈ V satisfies the Hamiltonian system of equations. So the optimal solution is onlydetermined by the control, and the u is such that the value of the Hamiltonian is maximised. Thisis sometimes refered to as the strong condition as it lends itself to problems that are bounded. Thisis one reasong that the optimal control framework of problems is more efficient, and may generateresults where COV might fail.

The weaker and more familiar condition requires that our optimal solution satisfy the followingcondition:

∂H

∂u= 0 (20)

which is used for unbounded problems, where the minima/maxima coincide with the local min-ima/maxima. This condition will be used throughout the rest of the report as most of the problemsconsidered are unbounded.

3.2 Riemannian Geometry and Differential Geometry

Optimal control can be used to solve problems in Riemannian geometry. With use of RiemannianGeometry we can put a give a smooth manifold a natural structure called a Riemannian metric(defined below), which allows us to measure lengths in the space. With use of this, we can definea metric on the smooth manifold, allowing extension of the PMP to spaces other than Rn.

Definition 3.4. A Riemannian Metric on smooth Manifold M is a smooth function that assignsan inner product on every pair of vectors V,W ∈ TpM for each p ∈ M . The inner product issymmetric, positive definite and smoothly varies as x varies. [4]

The metric is defined as:

G = gikdxidxk (In local coordinates) (21)

We can therefore define our vectors above as the following:

V = V i∂

∂xiW = W i ∂

∂xi(22)

So given two vectors V, W ∈ TpM , the inner product can be seen as follows:

< V,W >G |p = (V 1, ..., V n)

g11(x) ... g1n(x)... ...

gn1(x) ... gnn(x)

W 1

...Wn

(23)

where the following is defined: gik = < ∂∂xi ,

∂∂xk >

4 Geodesics on Riemannian Manifolds

As seen earlier, geodesics are curves that we can characterise as energy minimizing curves on themanifold, that satisfy the Euler-Lagrange equation above. Taking an optimal control approach, wecan derive equivalent conditions for geodesics on smooth manifolds defined in phase space.

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Using a Legendre transform on the Lagrangian geodesic function, the following Hamiltonian isgenerated:

H(y, u, λ, λ0) = λ0uTGu+ λ(u) (24)

Taking λ0 = −1, we generate our Hamiltonian system of equations:

y(t) = u =∂H

∂λ(25)

λ(t) = −∂H∂yi

= uT∂gij∂yi

u (26)

We are not considering bounded cases, hence the only extremals to consider is those critical pointsof ∂H

∂u . We also assume that we are only looking at normal geodesics; i.e. λ0 6= 0; so we normaliseit to λ0 = 1 without loss of generality. Hence our extremals are y(t) such that:

∂H

∂u= λ − 2gu =⇒ λ = 2gu (27)

We take the derivative of (27) with respect to y and get the following:

λ = ˙(2gu) = 2yi∂gij∂yk

yk + 2gij yi = yi∂gij∂yk

yk (28)

Rearranging the above and relabeling indices, we get the following equation know as the geodesicequation:

d2yk

dt2+ Γkij

yi

dt

yj

dt= 0 (29)

With the symbol Γkij defined below as a Christoffel Symbol:

Γmij =1

2gmk

(∂gik∂xj

+∂gkj∂xi

− ∂gij∂xk

)(30)

N:B. The Einsteinian summation convention was used in the derivation of this equation, so we mustsum over all the indices when we use it.

4.1 Geodesics on a Sphere

As the sphere is a smooth manifold, we can take coordinate charts and find the geodesics by thismethod. Using the usual stereographic projections, we get two charts S2/N and S2/S (where Nand S are the north and south poles respectively).Let x(t) ∈ S2 and y(t) ∈ R2, then we can write x(t) in terms of y(t) as follows:

x(t) =1

‖y(t)‖2 + 1(2y(t), ‖y(t)‖2 − 1) (31)

As the stereographic projection is diffeomorphic to R2, the metrics must be the same. Hence thewfollowing condition must be satisfied:

‖y(t)‖2G = ‖x(t)‖2E (where E stands for the Euclidean norm) (32)

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From the above equations, G is founds to be:

G =4δij

(1 + ‖y(t)‖2)2(where δij is the kronecker delta) (33)

Now finding the geodesics reduces to finding the extremals of the following equation:

H(y, λ, λ0, u) = λ0‖y(t)‖2G + λ(u) (34)

Solving the system of equations, we see that the geodesics on a sphere are arcs of great circles. Weshould also note that geodesics between two fixed points are not unique as another geodesic wouldbe the arc joining the two fixed points along the larger arc. Hence the geodesic pictured below canbe referred to as the minimal geodesic of the sphere given these two fixed points.

Figure 4: Geodesic on S2 - Great circle arcs (code adapted from [11])

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4.2 Geodesics on a cylinders

For the geodesics, we employ a quicker and more elegant solution, using the theorem below.

Theorem 4.1. If two manifolds M and N are isometric, then the geodesics of M are preservedunder the isometric mapping to N. [6]

This means instead of solving the geodesic equation, (which may not always give closed formsolutions), we can instead prove that two surfaces are isometric, and derive the desired geodesicsfrom there. So our goal is to show that the plane and the cylinder are isometric to each other.

We construct our isomerism as follow. We take the open interval (0, 2π), which is clearly isometricto the unit circle (minus a point), under the mapping:

π : θ → (cos(θ), sin(θ)) θ ∈ (0, 2π) (35)

We take the cartesian product of both curves with the plane, which is also isometric. Hence ourinfinite strip in R2 is isometric to the cylinder locally. So locally our geodesics are the same. Belowis a plot of one of the three possible geodesics on the cylinder. This geodesic is isometric to adiagonal line on the plane. The other two geodesics are just parameterised vertical and horizontallines on the plane.

Figure 5: Circular helix geodesic on cylinder (code adapted from [11])

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4.3 Abnormal Geodesics

Definition 4.2. An abnormal geodesic is a geodesic that satisfies the condition that λ0 = 0 whilenot violating the nontriviality condition of the Pontryagin maximum principle.

Hence we have that our co-state variable must be non-zero. Applying this condition to ourgeodesic equation on a Riemannian manifold, we get the following hamiltonian:

H(u, λ) = λ(u) (36)

From our definition, as geodesics are locally minimising curves, it is a necessary condition that anyabnormal geodesic on a riemannian manifold satisfy:

∂H

∂u= 0 =⇒ λ = 0 (37)

This contradicts our non-triviality condition of the PMP,Hence there exists no abnormal geodesics on riemannian manifolds.

I was not able to investigate the existence of abnormal geodesics in Sub-Riemannian manifolds,but their existence has been proven by numerous sources, and are an interesting topic of research.See [5] for more on this.

5 Elastica

The problem of elastica goes back to the time of Euler. The problem sets out minimise the bendingenergy of a thin wire. For all the our problems considered, the wire is of negligible thickness, withconstant length. Mathematically, we are looking for functions that are minimsers of the followingfunction: ∫ t1

t0

k(t)2dt (38)

(where k is the curvature of the curve y.) The other condition palced on these curves is that theymust have unit speed. i.e.

< y, y > = 1 (39)

5.1 Elastica in E2

We now consider the elastic curves in Euclidean-2 space. We seek to find a curve y defined as such:

y : [t0, t1]→ E2 (40)

with the y having unit speed as mentioned above. Taking derivatives of both sides, we get:

< y, y > = 0 (41)

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Hence we have that the speed and acceleration of the curve are perpendicular to each other. AsE2 ∼= C, we can write this as:

y = iky (42)

where k is curvature as before. Working in the complex plane, from (39) we get expressions for yand y:

y = eiθ (43)

y = i θ eiθ (44)

From these equations, we can formulate our optimal control problem. Letting u(t) = θ, this impliesfrom (42) that

k(t) = u(t) (45)

From the above equations, we formulate the following Hamiltonian:

H(y, θ, λ, ω, u, λ0) = λ0 u2 + λ(eiθ) + ω(u) (46)

(Where λ and ω are our co-state variables.) As we have an unbounded space, we can use theweaker condition of the PMP to generate our curves. Using the PMP, we get the following systemof equations:

λ = −∂H∂y

= 0 (47)

ω = −∂H∂θ

= −λ(ieiθ) (48)

∂H

∂u= 2λ0 u+ ω = 0 (49)

λ0 = 0 =⇒ ω = 0

=⇒ ω = 0 =⇒ λ(ieiθ) = 0

Hence we either have that λ = 0 or θ = constant. The only acceptable solution is the latter, as theformer violates the non-triviality condition of the PMP. Thus our first solution to elastica is thestraight line.

Now assuming that λ0 6= 0 we proceed. Equation (47) implies that λ is constant. We can normalise(49) by setting λ0 = 1 without loss of generality. Hence our new equation for the extremal is:

∂H

∂u= 2 u+ ω = 0 (50)

Taking the derivative of (50) and using (48), we get the following differential equation

2θ = R Cos (θ − α) (51)

(with R ≥ 0) Solving (51), we get the following equation:

k2(t) = κ20

(1− p2sn2

(κ02s, p

)(52)

This is a special case of the general equation for elastica generated by Singer [8], when w = 1

k2(t) = κ20

(1− p2

w2sn2(κ02w

s, p

)(53)

Here sn is the elliptical sine function, which is defined in [8].

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5.2 Catalogue of elastica curves

Depending on the set of parameters, we can generate different curves satisfying our differentialequation. These curves below were generated by David Singer in [8]

(a) Borderline Elastica [p = w = 1] (b) Orbit Elastica [w = 1]

Figure 6: Elastica figures [8]

(a) Wavelike Elastica (b) Wavelike Elastica

Figure 7: Wavelike Elastica [w = p] Function will oscilated between the +k0 and −k0 [8]

This list however is not exhaustive. Helices and the figure-eight elastica also satisfy the aboveequations. The last trivial elastica is that of a straight line which is obtained by setting k0 = 0,hence has a squared curvature of 0.

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6 Numerical Methods of solving

For solving differential equations, symmetry is a key feature required for closed form solutions. Forsome problems in optimal control, the spaces are too complicated to give closed form solutions,hence other methods are used to produce results. We use numerical methods and techniques tosolve them. Another problem faced is that most optimal control and COV problems are boundaryvalue problems (BVP), hence there is no assurance of a unique solution or even the existence ofone, but we persevere. There are many numerical methods one can use to solve ODE’s and PDE’s,such as these:

6.1 Shooting methods

Studied extensively by Keller [3], shooting is a numerical approach that aims to transform our BVPinto and initial value problem (IVP). It’s simplicity works well for computational time and with aidof some other numerical methods, has a fast convergence.

Shooting works as follows. Imagine we have a second order equation:

y(x) = f(y, y, x) (54)

With y(a)=A and y(b)=B, (a < b)We take a guess at y(a) [usually guess b−a

t = g1, where t is the time wanting to reach other bound-ary condition; however the initial guess is not of high importance.]

Now comes the more important part of shooting, and that is evaluating the guess, then correctingit to reach the solution of the original problem. We solve our IVP, evaluate our solution at thesecond boundary point and then correct the previous guess. Let φ be defined as the difference ofsolutions at x = b

φ(g1) = A(g1, b)− y(b) (55)

(where A(g1, b) is the solution to the IVP for y(a) = g1). If φ(g1) = 0, then we are done, as bothsolutions give the same result. However let’s suppose that we have undershot our guess, and thatin reality φ(g1 + dg1) = 0 where dg1 is just a small variation of g1. Then we can use Taylor seriesexpansion to give:

φ(g1 + dg1) = φ(g1) + dg1dφ

dg1(g1) +O2 = 0 (56)

=⇒ dg1 = −φ(g1)

φ(g1)(57)

This gives Newtons method for finding roots, of which we get closer with every iteration. [9] Hencethe corrected guess is:

gn+1 = gn −φ(gn)

φ(gn)(58)

This formulating works well, and converges fast (quadratically actually if sufficiently close to truezero), however how do we determine φ(gn). To obtain an expression for this, we must first solve

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another IVP, this time a second order one. This is constructed as follows, we take the derivative ofequation (54) with respect to time, we get:

∂A(x, t)

∂t=∂A

∂y

∂y

∂t+∂A

∂y

∂y

∂t(59)

Letting z = ∂A∂t , we get the following IVP:

z =∂A

∂yz +

∂A

∂yz (60)

with z(a) = 0 and z(a) = 1We solve the IVP’s: (54) and (60) and then re-evaluate our expression (58) using the fact thatφ(gn) = z(x, t).Newtons method is not the only one we can use when shooting. We can also use the secant method,which has less steps to compute, however has a slower rate of convergence than the Newton method.It will be used for problem where solving (60) would prove to be too expensive for every iteration.

Other numerical methods can be used to solve these problems, however they may be more ex-pensive. Such a method is the finite difference method that involves discretising our continuoussystem and solving on these intervals. Future research into this would be needed to verify whichproblems this method would be most effective on.

Acknowledgments

I would like to thank Prof. Lyle Noakes for his guidance and effort spent supervising this project,and also AMSI for giving me the opportunity to complete this Summer Research Scholarship.

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References

[1] Abbena, E., Salamon, S. and Gray, A., 2006. Modern differential geometry of curves andsurfaces with Mathematica. CRC press.

[2] Denman, H.H., 1985. Remarks on brachistochronetautochrone problems. American Journal ofPhysics, 53(3), pp.224-227.

[3] Keller, H.B. , 1976, Numerical solution of two point boundary value problems, 1st Edition,Blaisdell Publishing Company, .

[4] Lebanon, G., 2002, August. Learning riemannian metrics. In Proceedings of the Nineteenth con-ference on Uncertainty in Artificial Intelligence, (pp. 362-369). Morgan Kaufmann PublishersInc..

[5] Montgomery, R., 2006. A tour of subriemannian geometries, their geodesics and applications(No. 91). American Mathematical Soc..

[6] Petersen, P., 2006. Riemannian geometry (Vol. 171, pp. xvi+-401). New York: Springer.

[7] Pontryagin, L.S., Boltyanski, V.G., Gamkrelidze, R.V. and Mischenko, E.F. (translated byK.N. Trirogoff), 1962, The Mathematical Theory of Optimal Processes, 1st Edition, Inter-science, John Wiley

[8] Singer, D.A., 2008, April. Lectures on elastic curves and rods. In O.J. Garay, E. GarcaRo andR. VzquezLorenzo eds.,, AIP Conference Proceedings (Vol. 1002, No. 1, pp. 3-32). AIP.

[9] Sung N. Ha, 2001, ’A nonlinear shooting method for two-point boundary value problems’,Computers & Mathematics with Applications, Volume 42, Issue 10, pp. 1411-1420

[10] Van Brunt, B, 2004 The Calculus of Variations, 1st Edition, Springer-Verlag, New York

[11] Wang,L , 2004 , geodesic, Available from: http://au.mathworks.com/matlabcentral/fileexchange/6522-geodesic [10 January 2017]

[12] Xin, Y.L., 2003. Minimal submanifolds and related topics (Vol. 8). World Scientific.

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