simulated inverted pendulum using analytical mechanics

INOM EXAMENSARBETE TEKNIK,GRUNDNIVÅ, 15 HP

, STOCKHOLM SVERIGE 2019

Simulated inverted pendulum using analytical mechanics

CARL CLAUSON

MALTE SLOMA

KTHSKOLAN FÖR TEKNIKVETENSKAP

INOM EXAMENSARBETE TEKNIK,GRUNDNIVÅ, 15 HP

, STOCKHOLM SVERIGE 2019

Simulering av inverterad pendel med analytisk mekanik

CARL CLAUSON

MALTE SLOMA

KTHSKOLAN FÖR TEKNIKVETENSKAP

Contents

1 Abstract 2

2 Sammanfattning 3

3 Introduction 43.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.3.1 Double Pendulum . . . . . . . . . . . . . . . . . . . . . . 53.3.2 Kapitza Pendulum . . . . . . . . . . . . . . . . . . . . . . 63.3.3 Kapitza Pendulum with elliptic oscillation. . . . . . . . . 7

4 Theory 84.1 Newtonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Functionals and Variational Calculus . . . . . . . . . . . . . . . . 84.3 Lagrange’s equations . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Method 115.1 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5.1.1 Maple with Sophia package . . . . . . . . . . . . . . . . . 115.1.2 Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5.2 General method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.3 Double pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.3.1 Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . 125.3.2 Solving with Euler-Lagrange equations . . . . . . . . . . . 13

5.4 Kapitza pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . 145.4.1 Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . 145.4.2 Solving with Euler-Lagrange equations . . . . . . . . . . . 14

5.5 Kapitza pendulum with elliptical oscillation . . . . . . . . . . . . 155.5.1 Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . 155.5.2 Solving with Euler-Lagrange equations . . . . . . . . . . . 15

6 Results 166.1 Double pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 Kapitza pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6.2.1 Minimum angular velocity to maintain stability . . . . . . 176.2.2 Maximum initial angle to maintain stability . . . . . . . . 20

6.3 Kapitza pendulum with elliptical oscillation . . . . . . . . . . . . 226.3.1 Maximum width of the orbit to maintain stability . . . . 226.3.2 Influence of width in macro oscillations . . . . . . . . . . 24

A Appendix code 29A.1 Double pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . 29A.2 Kapitza’s pendulum . . . . . . . . . . . . . . . . . . . . . . . . . 30A.3 Kapitza’s pendulum with elliptical oscillation . . . . . . . . . . . 31

1

1 Abstract

Analytical mechanics is an alternative to Classical (Newtonian) mechanics forcalculating the movement of particles and systems of particles. In this paper weanalyze three different mechanical systems using analytical mechanics. We thenimplement this into the maple based software ”Sophia” to further analyze howchanging some of the parameters alter the behavior of the system. One of ourmain focuses in the report is studying the stability of an inverted pendulum. In1951 the Russian physicist Pyotr Kapitza, figured out that the pendulum couldbe stabilized in the upside-down position by forcing it into a rapid vertical oscil-lation. We examine the boundaries of this stability and explore if the pendulumcan instead be stabilized using an elliptical oscillation.

2

2 Sammanfattning

Analytisk mekanik ar ett alternativ till klassisk (Newtonsk) mekanik for attberakna rorelsen av partiklar och system av partiklar. I den har rapporten anal-yserar vi tre olika mekaniska system med analytisk mekanik. Vi implementerarsedan detta i den maple-baserade programvaran ”Sophia” for att ytterligareanalysera hur andringar av vissa parametrar forandrar systemets beteende. Ettav vara huvudsyften med den har rapporten ar att studera stabiliteten hos eninverterad pendel. Ar 1951 fann den ryska fysikern Pyotr Kapitza att pendelnkunde stabiliseras i dess upp-och-ner-lage genom att tvinga den till en snabbvertikal svangning. Vi studerar granserna for denna stabilitet och undersokerom pendeln istallet kan stabiliseras genom en elliptisk svangning.

3

3 Introduction

3.1 Background

Newtonian mechanics has throughout its existence been used to determine vec-tor quantities of forces, momenta and motion in a certain system. When scien-tists discovered that the Newtonian mechanics was not enough to solve certaintypes of mechanical problems, analytical mechanics was invented which can bedefined as a collection of closely related alternative formulations of Newtonianmechanics. Analytical mechanics utilizes scalar properties of motion, i.e usuallythe systems total potential energy and kinetic energy and not vector quantitiesof accelerations and forces of individual particles. Lagrangian mechanics is ananalytic method which is ideal for calculating the motion of different systemswith various types forces, for example pendulums. In the present work we willrestrict the forces acting on the system to conservative fields.

A regular pendulum is in it’s ground state stable and will all the time strive tokeep the center of mass as low down as possible. In 1951 the Russian physicistPyotr Kapitza, figured out that the pendulum could be stabilized in the upside-down position by forcing the pendulum into a quick vertical oscillation. Thispendulum is one of threee systems that will be studied in this thesis.

3.2 Purpose

The methods of analytical mechanics will be applied to model the three differentphysical systems using the dynamic tool for Maple; Sophia [2]. To compute themotion of these systems using regular Newtonian mechanics is possible but evenwith only two degrees of freedom it becomes very complicated. Therefore willthe selected systems firstly be modeled quite easily using analytical mechanicsand then use these methods to solve and analyze three different problems. Thesystems are a regular double pendulum, and then two different versions of theKapitza pendulum [3].

4

3.3 Problems

3.3.1 Double Pendulum

Problem 3.29 from Apazidis [3].

Figure 1: Double pendulum.

’Consider a double pendulum consisting of two bars OA and AB of masses m1

and m2 and lengths l1 and l2 respectively that are free to rotate in the verticalplane according to the figure 1. Choose the following numerical values of theparameters m1 = 3 kg, m2 = 1 kg, l1 = l2 = 1 m and calculate and plot thetrajectory of end B by means of the Sophia and Graphics packages. Choosethe following initial conditions q1(0) = 1 rad, q1(0) = 0 rad, q2(0) = 1.5 rad,q2(0) = 0 rad. Calculate then the trajectory of the point B for slightly differentinitial conditions and show the sensitive dependence of the motion of the systemon initial conditions by comparing the two trajectories.’

5

3.3.2 Kapitza Pendulum

Figure 2: Kapitza pendulum.

Consider an inverted pendulum consisting of one bar OA of mass m1 and lengthl1 that is free to rotate in the vertical plane according to the figure. The bar willbe forced into a harmonic vertical oscillation which will stabilize the pendulum.In order to change the behaviour of the pendulum, parameters will be changedand examined.

The stability of the pendulum will be studied by deciding the minimum angularvelocity ω and the maximum initial angle to maintain stability. The parametersof the pendulum is described in the method section.

6

3.3.3 Kapitza Pendulum with elliptic oscillation.

Figure 3: Kapitza pendulum with elliptic oscillation.

Consider an inverted pendulum consisting of one bar OA of mass m1 and lengthl1 that is free to rotate in the vertical plane according to the figure. The barwill be forced into a harmonic oscillation around an ellipse which will stabilizeit vertically. In order to change the behaviour of the pendulum, the parameterswill be changed and examined.

The stability of the pendulum will be studied by deciding the maximum re-lationship between the height and the width of the ellipse in order to maintainvertical equilibrium. The effect of an increasing width will also be studied,causing macro oscillations.

7

4 Theory

4.1 Newtonian Mechanics

From Newton’s laws of motions, the concept of energy can be derived. Whendescribing a conservative mechanical system, energy becomes a key factor tosolve the often non-linear differential equations for the system. In classicalmechanics the energy of the system can be divided into two parts. The kineticenergy is defined as

T =m(v · v)

2(4.1)

for a particle with all of it’s mass distributed in one single point. When talkingabout larger and more complicated structures than particles, e.g a bar, thekinetic energy will consist of two parts, translational and rotational:

Ttot =m(vG · vG)

2+

ωT Iω

2(4.2)

Where the inertia I and the angular velocity ω per definition Apazidis [3].Potential energy, is defined per the Cambridge Dictionary as the ”energy storedby something because of position (as when an object is raised), because of itscondition (as when something is pushed or pulled out of shape), or in chemicalform (as in fuel or an electric battery)” [5].

4.2 Functionals and Variational Calculus

A functional is a mathematical operator that uses functions as its input argu-ments and returns a scalar. One fundamental problem of variational calculusis to find a real function y(x) of a real variable x so that the functional of I[y]becomes an extreme value.

I[y] =

∫ x2

x1

f(y(x), y′(x), x

)dx (4.3)

I[y] is the functional of y with fixed endpoints x1 and x2. The purpose of thefunctional is to determine those functions y(x), which uses the given valuesy1 = y(x1) and y2 = y(x2) as endpoints and makes the functional I[y] anextremum. In other words, for which functions y(x) the functional I[y] assumesa maximum, minimum or a saddle point. Now assume

I[β] =

∫ x2

x1

f(y(x, β), y′(x, β), x

)dx (4.4)

8

where y(x, β) = y(x) + βη(x) with η(x1) = η(x2) = 0. This implies that y(x)is different from y(x, β) but have the same boundary conditions, i.e. multiplepaths are possible between the two endpoints.

Figure 4: Example of functions with same boundary conditions.

The variation of I is given by

δI =dI

dβdβ =

∫ x2

x1

dx

{∂f

∂y

dy

dβ+∂f

∂y′dy′

dβ

}dβ. (4.5)

When the second term in the integral is integrated by parts∫ x2

x1

dx∂f

∂y′d

dx

(dy

dβ

)= −

∫ x2

x1

dxdy

dβ

d

dx

(∂f

∂y′

)+∂f

∂y′dy

dβ

∣∣∣∣∣x2

x1

(4.6)

the conclusion is that the boundary terms do not contribute because dy/dβ =η(x). Therefore

δI =

∫ x2

x1

dx

{∂f

∂y− d

dx

∂f

∂y′

}dy

dβdβ. (4.7)

The expression

∂f

∂y− d

dx

∂f

∂y′=δf

δy(4.8)

9

is called the variational derivative of f by y. δy can be seen as a small variationof the curve y(x) since (dy/dβ)dβ = δy. When δI = 0 i.e. I(β) assumes aextreme value, the integrand in (4.7) can be negligible. This results in

∂f

∂y− d

dx

∂f

∂y′= 0 (4.9)

which is Euler’s differential equation of variational calculus.

4.3 Lagrange’s equations

The Lagrange function is defined as the difference of the systems kinetic andpotential energy.

L = T − V (4.10)

As implemented, the Lagrange function is a function of generalized coordinatesqk, their time derivatives qk and the time t

L = L(qk, qk, t) (4.11)

where the index k varies depending on the degree of freedom. The action integralS is defined as an integral of the Lagrange function (4.11) along a path in theconfiguration space between two fixed times t1 and t2.

S =

∫ t2

t1

L(qk, qk, t) (4.12)

The action integral (4.12) can be used in Hamilton’s Variational Principle whichis a central concept for Langranian mechanics. The principle states that the pathof the system between the times t1 and t2 has a extremum along the actual path.This implies that the action intregal calculated using the actual path is lessthan for any another possible path, if the extremum is a minimum. Applied toall possible paths, the Hamilton’s Variational Principle can be rewritten usingvariation calculus from Section 2

δS = δ

∫ t2

t1

L(qk, qk, t) = 0 (4.13)

and with this derive the Euler-Lagrange Equation which is the equation used inSophia and throughout this thesis.

∂L

∂qk− d

dt

(∂L

∂qk

)= 0 (4.14)

10

5 Method

5.1 Software

To solve the non-linear differential equations described, different kinds of soft-ware were used to solve and present the given problems. The softwares usedwill be described briefly and the areas of application.

5.1.1 Maple with Sophia package

Maple is a computer software system, that provides symbolic solutions of math-ematical problems and technical calculations [2]. Sophia is a package custommade for Maple, which both simplifies the construction of a mechanical systemand the method of understanding the system.

The main perks of the Sophia package is that time dependent variablescan be declared. If the variables are declared in different coordinate systems,a relationship between the coordinate systems can be established. Sophia alsoprovides a function which makes it easy to evaluate different vectors in whicheveralready declared coordinate system. These functions are crucial in order toconstruct the description of the mechanical systems for this thesis (see AppendixA for each system).

5.1.2 Matlab

Matlab is a computer software mainly used for visualization of functions throughgraphs and numerical computing. To present the data from the different mecha-nial systems, graphs and animations were created by importing data from Mapleto Matlab.

5.2 General method

The three different mechanical systems in this thesis were analyzed using thesame procedure. This model describes a general solution for each system. Thespecific equations for each system is described later in this section.

I. By examining the mechanical system, the generalized coordinates, con-stants and relations between coordinate systems are defined.

II. Assemble the Lagrangian L = T − V , where T is the kinetic energy andV the potential energy of the system.

III. Determine and calculate the Euler-Lagrange equations.

IV. Animate and plot the generalized coordinates and their respective deriva-tives numerically using Maple’s ode-solver.

11

5.3 Double pendulum

The two bars OA and AB have masses m1 and m2 and lengths l1 and l2 re-spectively. Since there is no friction in the joins O and A the pendulum canmove freely in the xy-plane. The system can be considered conservative dueto that the only acting force on the pendulum is gravitation. The generalizedcoordinates q1 and q2 are defined as the angle between the two bars OA andAB respectively.

Figure 5: Double pendulum.

5.3.1 Lagrangian mechanics

The Lagrangian L is defined as the difference between the kinetic and the po-tential energy of the system, from equation (4.10). The kinetic energy of thebar OA is defined as

TOA =m1L

21ω

2OA

3(5.1)

where ωOA is the angular velocity. Since the bar OA is fixed in O, the contri-bution to the kinetic energy of the system will only depend on the rotationalmotion. With the parallel axis theorem, equation (5.1) can be derived. How-ever, the bar AB will experience both translative and rotational motion. Thekinetic energy for the bar AB is

12

TAB =m2L

22

(ωOA + ωAB

)26

+m2(u2 · u2)

2(5.2)

where u2 is the speed of the bar AB. The total kinetic energy is then calculated

TTOT = TOA +TAB =m1L

21ω

2OA

3+m2L

22

(ωOA + ωAB

)26

+m2(u2 · u2)

2. (5.3)

The reference level of the system is the x-axis of the N -frame and the potentialenergy of the system is therefore calculated by the vertical displacement of thecentre of masses with regard to this level. The potential energy of the bar OAis

VOA = −m1gL1 cos(q1)

2(5.4)

and for the bar AB

VAB = gm2

((−L2 cos(q2) − 2L1) cos(q1)

2+L2 sin(q1) sin(q2)

2

). (5.5)

The total potential energy is then calculated

VTOT = −m1gL1 cos(q1)

2+gm2

((−L2 cos(q2) − 2L1) cos(q1)

2+L2 sin(q1) sin(q2)

2

).

(5.6)

With equation (4.10) the Lagrangian L can be calculated using the definedenergies in this section.

5.3.2 Solving with Euler-Lagrange equations

Using the theory from the previous section, the Euler-Lagrange equations can bederived. The ode-function of Maple is then used to numerically solve the systemfor given initial conditions. Afterwards, plots can be designed and presented.

13

5.4 Kapitza pendulum

The bar OA have mass m1 length l1. Since there is no friction in the joint O,the pendulum can move freely in the xy-plane. The system can be consideredconservative due to that the only acting force on the pendulum is gravitation.The generalized coordinate q1 is defined as the angle between the y-axis and thependulum in an upright position.

Figure 6: Kapitza pendulum. The bar OA oscillates vertically.


The Lagrangian is defined as the difference between the kinetic and the potentialenergy of the system, from equation (2.8). The kinetic energy of the bar isdefined as

T =m(u1 · u1)

2+m1L

21ω

2

3(5.7)

The reference level of the system is the x-axis of the N-frame and the potentialenergy of the system is therefore calculated by the vertical displacement of thecentre of mass with regard to this level. The potential energy of the bar

V = m1g

(A sin(ωt) +

cos(q1)L1

2

)(5.8)



14

5.5 Kapitza pendulum with elliptical oscillation

The bar OA have mass m1 length l1. Since there is no friction in the joint O,the pendulum can move freely in the xy-plane. The system can be consideredconservative due to that the only acting force on the pendulum is gravitation.The generalized coordinate q1 is defined as the angle between the y-axis and thependulum in an upright position.

Figure 7: Kapitza pendulum with elliptic oscillation. The bar OA oscillateselliptically.


The Lagrangian is defined as the difference between the kinetic and the potentialenergy of the system, from equation (2.8). The kinetic energy of the bar isdefined as

T =m(u1 · u1)

2+m1L

21ω

2

3(5.9)

The reference level of the system is the x-axis of the N-frame and the potentialenergy of the system is therefore calculated by the vertical displacement of thecentre of mass with regard to this level. The potential energy of the bar

V = m1g

(A sin(ωt) +

cos(q1)L1

2

)(5.10)



15

6 Results

6.1 Double pendulum

(a) Initial condition q1 = 1,0 q2 = 1,4. (b) Initial condition q1 = 1,1 q2 = 1,5.

Figure 8: Trajectory plots for the pendulum from given initial conditions forthe first 10 seconds of the simulation.

(a) Initial condition q1 = 1,0 q2 = 1,4. (b) Initial condition q1 = 1,1 q2 = 1,5.

Figure 9: Phase portraits for the pendulum from given initial conditions for thefirst 100 seconds of the simulation.

16

6.2 Kapitza pendulum

6.2.1 Minimum angular velocity to maintain stability

The first study of this pendulum is to find the smallest angular velocity of thependulum ω to maintain stability. The initial angle is set to q1 = 10−5 rad sincesmaller initial angles have a smaller effect on the pendulums trajectory. Thegraphs in this section describes the angle q1 (y-axis) over time (x-axis).

Figure 10: ω = 36 rad/s. Not stable.

The pendulum is close to upright to begin with but turns unstable quickly. Sincethe initial angle is very small, one can see that small changes could have bigimpact for larger t.

17

Figure 11: ω = 36, 3 rad/s. Not stable but falls down slower.

When we increase the frequency to 36, 3 rad/s, the pendulum falls over muchslower compared to the frequency of 36 rad/s.

18

Figure 12: ω = 36, 324 rad/s. Stable pendulum.

The pendulum has now reached the required oscillation frequency required forstability. This means that the critical ω is less than 37 rad/s and bigger than36 rad/s. Our closest approximation of 36, 324 rad/s can be seen in figure 12.

19

Figure 13: ω = 37 rad/s. Stable and increased frequency of maximum. Thependulum is stable for all ω > 36, 324 rad/s.

Discussion:In this study we set out to find the minimum angular velocity required to keepthe pendulum stable. To mimic reality where a small disturbance would causethe pendulum to swing slightly from its state of equilibrium we initiated thesimulation with the pendulum at a very slight angle. Specifically q1 = 10−5. Bygradually increasing the angular velocity of the oscillation we can see in Figure(10), (11) and (12) the pendulum is unstable all the way up to a frequency ofω = 36, 324, after which it is stable.

6.2.2 Maximum initial angle to maintain stability

The second study of this pendulum is to calculate the largest initial angle pos-sible in order to maintain stability.

20

Figure 14: q1 = 0, 5 rad. Stable pendulum.

Figure 15: q1 = 0, 521 rad. Stable pendulum.

21

Figure 16: q1 = 0, 522 rad. Not stable pendulum.

Discussion:In this study we wanted to find what the maximum initial angle while maintain-ing stability is. The frequency used for this was very high at ω = 1000 rad/s.This was to make sure that any instability was not caused by a lack of angularvelocity, By gradually increasing the initial angle of the oscillation we can seein Figure (14), (15) and (16) the pendulum is stable all the way up to an initialangle of 0, 521 rad, which is approximately 29, 9°, after which it is unstable.1,55 rad, approximately 88.8°, is the maximum angle of the pendulum whichcan be seen in figure (15).

6.3 Kapitza pendulum with elliptical oscillation

6.3.1 Maximum width of the orbit to maintain stability

The first study of this pendulum is to decide the maximum width of the orbit,with a given fixed height A = 0, 1.

22

Figure 17: B = 0, 098 m. Stable pendulum.

Figure 18: B = 0, 09854 m. Stable pendulum.

23

Figure 19: B = 0, 09855 m. Not stable pendulum.

From these graphs we can see that when using a very high frequency ω =1000 rad/s and very small initial angle q1 = 10−5 m the pendulum is stable foroscillation widths of 0, 09854 m and lower.

Discussion:When using an elliptical oscillation pattern instead of a vertical one we noticedthat the pendulum is unstable for big enough widths of the ellipse. To find thebreaking point between stability and instability we increased the width gradu-ally until the system was no longer stable. A high frequency (ω = 1000 rad/s)and small initial angle (q1 = 10−5) was used to minimize their affect on themaximum possible width. We found that the pendulum is stable for oscillationwidths of 0, 09854 m and lower.

6.3.2 Influence of width in macro oscillations

Here we are considering a Kapitza pendulum with the frequency of ω = 50 rad/sand an initial angle of 0, 2 rad.

24

Figure 20: B = 0 m.

25

Figure 21: B = 0, 03 m.

26

Figure 22: B = 0, 05 m.

Discussion:When studying the graphs of different Kaptiza pendulums it is clear that whilethe base is oscillating quickly with a low amplitude, the pendulum is oscillatingslowly and with a larger amplitude around it’s equilibrium point. In this studywe looked at the impact on these macro oscillations of changing the width ofthe base oscillation, specifically with the frequency of ω = 50 rad/s and aninitial angle of 0, 2 rad. From Figure 20-22 we can see that by making the baseoscillation more elliptical the frequency of the macro oscillations decreases.

27

References

[1] Kojevnikov, A. (2019). Pyotr Leonidovich Kapitsa. [online] Encyclopæ-dia Britannica. Available at: https://www.britannica.com/biography/Pyotr-Kapitsa [Accessed 30 Apr. 2019]

[2] Maple. (2019). Ontario, Canada. University of Waterloo.

[3] Apazidis, N. (2012). Mekanik 2, Partikelsystem, stel kropp och analytiskmekanik. Lund: Studentlitteratur.

[4] Apazidis, N. (2012). Mekanik 1, Statik och partikeldynamik. Lund: Stu-dentlitteratur.

[5] Cambridge Dictionary. Potential energy. [online] Available at:https://dictionary.cambridge.org/dictionary/english/potential-energy [Ac-cessed 02 May 2019].

[6] Scheck F. Mechanics : From Newton’s Laws to Deterministic Chaos. Berlin,Heidelberg : Springer Berlin Heidelberg; 2010.

28

A Appendix code

A.1 Double pendulum

> restart;

> with(plots): with(plottools):

> read ‘C:/Users/calle/OneDrive/Skrivbord/Sophia/SophiaV6N.txt‘;

> read ‘C:/Users/calle/OneDrive/Skrivbord/Sophia/Graphics.txt‘;

> dependsTime(q1, u1, q2, u2);

> &rot [N, A, 3, q1];

> &rot [A, B, 3, q2];

> r1 := A &ev [0, -L1/2, 0];

> r11 := (2 &** r1);

> r2 := r11 &++ (B &ev [0, -L2/2, 0]);

> r22 := r11 &++ (2 &**(B &ev [0, -L2/2, 0]));

> v1 := N &fdt r1;

> v2 := N &fdt r2;

> v1 := simplify(v1);

> v2 := simplify(v2);

> T := 1/3*m1*L1^2*(q1t^2) + 1/2*(1/12*m2*L2^2+m2*L2^2/4)*(q1t+q2t)^2+1/2*m2*(v2 &o v2);

> simplify(T);

> V := m1*g*(N &to r1)[1][2] + m2*g*(N &to r2)[1][2];

> L := T - V;

> L := subs(q1t=u1, q2t=u2, L);

> L_q1 := diff(L, q1);

> L_u1 := diff(L, u1);

> L_u1_t := &dt L_u1;

> eq1 := L_u1_t - L_q1 = 0;

> eq1 := subs(q1t=u1, q2t=u2, eq1);

> L_q2 := diff(L, q2);

> L_u2 := diff(L, u2);

> L_u2_t := &dt L_u2;

> eq2 := L_u2_t - L_q2 = 0;


> kde := {q1t=u1, q2t=u2};

> eq1 := {u1t = solve(eq1,u1t)};


> eqs := eq1 union eq2 union kde;

> InitCond := {q1(0)=1.1, u1(0)=0, q2(0)=1.5, u2(0)=0};

> eqst := subs(toTimeFunction, eqs);

> param := {m1=3 ,m2=1, L1=1, L2=1, g=9.8}: eqst := subs(param, eqst);

> ff := dsolve(eqst union InitCond, {q1(t), q2(t), u1(t), u2(t)}, type=numeric, maxfun=500000);

29

> odeplot(ff,[q2(t),u2(t),color=blue],0..100,thickness=1,numpoints=1000);

> #animatepath_2d(ff,r22,param,0,10,100,100,red,2,N);

> plotpath_2d(ff,r22,param,0,10,10000,blue,2,N)

A.2 Kapitza’s pendulum

> restart;




> dependsTime(q, u);

> &rot [N, A, 3, q];

> r1 := N &ev [0, b*sin(omega*t), 0];

> r2 := r1 &++ (A &ev [0, L1/2, 0]);

> v1 := simplify(N &fdt r1);


> T := 1/2*m*(v2 &o v2) + 1/24*m*L1^2*qt^2;

> V := m*g*(N &to r2)[1][2];

> L := T - V;

> L := subs(qt=u, L);

> L_q := diff(L, q);

> L_u := diff(L, u);

> L_u_t := &dt L_u;

> eq := L_u_t - L_q = 0;

> eq := subs(qt=u, qt=u, eq);

> kde := {qt=u};

> eq := {ut = solve(eq,ut)};

> eqs := eq union kde;

> InitCond := { q(0)=0.2, u(0)=0};


> param := {m=1, L1=1, g=9.8, b=0.1, omega = 50}: eqst := subs(param, eqst);

> ff := dsolve(eqst union InitCond, {q(t), u(t)}, type=numeric, maxfun=500000);

> odeplot(ff,[[t,q(t)]],0..5,thickness=0,numpoints=1000);


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A.3 Kapitza’s pendulum with elliptical oscillation

> restart;

> restart;




> dependsTime(q2, u2);

> &rot [N, A, 3, q2];

> r1 := N &ev [a*cos(omega*t), b*sin(omega*t), 0];

> r2 := r1 &++ (A &ev [0, L1/2, 0]);



> T := 1/2*m2*(v2 &o v2) + 1/24*m2*L1^2*q2t^2;

> V := m2*g*(N &to r2)[1][2];

> L := T - V;

> L := subs(q2t=u2, L);

> L_q2 := diff(L, q2);

> L_u2 := diff(L, u2);

> L_u2_t := &dt L_u2;

> eq2 := L_u2_t - L_q2 = 0;


> kde := {q2t=u2};


> eqs := eq2 union kde;

> InitCond := { q2(0)=0.2, u2(0)=0};


> param := {m2=1, L1=1, g=9.8, a=0.05, b=0.1, omega = 50}: eqst := subs(param, eqst);

> ff := dsolve(eqst union InitCond, {q2(t), u2(t)}, type=numeric, maxfun=500000);

> odeplot(ff,[[t,q2(t)]],0..5,thickness=0,numpoints=1000);


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