Serial of year 28

• Write down the equations for a throw in a homogeneous gravitational field (you don't need to prove them but you need to know how to use them). Design a machine that will throw an item and determine the angle of approach and the velocity. You can throw with the item with a spring, determine its spring constant, mass of the object and calculate the kinetic energy and thus the velocity of the item. What do you think is the precision of the your value of the velocity and angle? Put the boundaries determined by this error into the equations and show in what boundaries we can expect the distance of the landing from the origin to be.Throw the item with your device at least five times and determine the distance of the landing and what are the boundaries within which you are certain of your distance? Show if your results fit into your predictions. (For a link to video with a throw you get a bonus point!)
• Tie a pendulum with an amplitude of $x$, which effectively oscillates harmonically but the frequency of its oscillations depends on the maximum displacement $x_{0}$

$$x(t) = x_0 \cos\left[\omega(x_0) t\right]\,, \quad \omega(x_0) = 2\pi \left(1 - \frac{x_0^2}{l_0^2}\right)\,,$$

where $l_{0}is$ some length scale. We think that are letting go of the pendulum from $x_{0}=l_{0}⁄2$ but actually it is from $x_{0}=l_{0}(1+ε)⁄2$. B By how much does the argument of the cosine differ from 2π after one predicted period? How many periods will it take for the pendulum to displaced to the other side than which we expect? Tip Argument of the cosine will in that moment differ from the expected one by more than π ⁄ 2.

• Take a pen into your hand and let it stand on its tip on the table. Why does it fall? And what will determine if it will fall to the right or to the left? Why can't you predict a die throw even though the laws of physics should predict it? When you play billiard is the inability to finish the game only due to being incapable of doing all the neccessary calculations? Write down your answers and try to enumerate physics phenomenons that occur in daily life which are unpredictable even if we know the situation well.

• We give length values in metres, time values in seconds and mass values in kilograms. Angular velocity $Ω$ we give in radians per second. If you take the equations for the movement of balls from the series, there are three more parameteres included: $α$, $β$, $γ$. What are their dimensions?
• Consider a freefalling ball with $Ω=0$ and $v_{x}=0$. There then exists a terminal velocity $v_{z}^{t}$, at which the frictional force and and gavitational force are equally matched and the fall of the ball isn't accelerating anymore.
• Determine this velocity from the equations for the movement o a ball.
• Change this equation so that it will express $β$. $v_{z}^{t}$ can be easily measured and for ourfootball of mass $m=0,5\;\mathrm{kg}it$ is typically around 25 m\cdot s^{ −1}. Then what is $β?$
• Express the initial $v_{x}$ and $v_{z}$ using the angle at which it was shot out $φ$ with a fixed initial velocity $v=10\;\mathrm{m}\cdot \mathrm{s}^{-1}$. Write a program according to the series and try changing the initial conditions and the following parameters
• Choose some positive $β$, turn off the rotation $Ω=0$ and find out, if the angle under which the the ball reach the farthest is bigger or smaller than 45°. Demonstrate your finding with graphs of the trajectories.
• Choose a positive non-zero $α$ with a numerical value in the given units the same as $β$, $γ=0,01$ (in the given units) and $Ω=±5rad\cdot \;\mathrm{s}^{-1}.How$ will in these specific cases the optimal angle of the shot change?
• *Bonus:** How far would you throw with a cricket ball? Is our model good enough to make such predictions?

• Look at the equations of the Lorenz model and write a script to simulate them in Octave (maybe even refresh your knowledge of the second part of series). Together with the sketching command your script should have the following form: …

function xidot = f(t,xi)

…

xdot=…;

ydot=…;

zdot= …;

xidot = [xdot;ydot;zdot];

endfunction

config = odeset('InitialStep', 0.01,'MaxStep',0.1);

initialCondition=[0.2,0.3,0.4];

solution=ode45(@f,[0,300],initialCondition,config);

plot3(solution.y(:,1),solution.y(:,2),solution.y(:,3)); </pre> Just instead of three dots fill in the rest of the code (just as in the second part of the series) and use $σ=9,5$, $b=8⁄3.Then$ figure out with a precision of at least units for what positive $r$ the system goes from asymptomatic stopping to chaotic oscillation(it is independent of the initial conditions).

• Here is the full text of the Octave script for simulating and visualising the movement of a particle in a gravitational field of a massive object in the plane $xy$, where all the constants and parameters are equal to one: clear all

pkg load odepkg

function xidot = f(t,xi)

alfa=0.1;

vx=xi(3);

vy=xi(4);

r=sqrt(xi(1)^2+xi(2)^2);

ax=-xi(1)/r^3;

ay=-xi(2)/r^3;

xidot = [vx;vy;ax;ay];

endfunction

config = odeset('InitialStep', 0.01,'MaxStep',0.1);

x0=0;

y0=1;

vx0=…;

vy0=0;

initialCondition=[x0,y0,vx0,vy0];

solution=ode45(@f,[0,100],initialCondition,config)

plot(solution.y(:,1),solution.y(:,2));

pause()</pre>

• Choose initial conditions $x0=0,y0=1,vy0=0$ and and a nonzero initial velocity in the direction $x$ such that the particle will be bound (ie. it won't escape the center.)
• Add to the gravitational force the following force $-α\textbf{r}⁄r^{4}$, where $αis$ a small positive number. Choose gradually increasing $α$ beginning with $α=10^{-3}$ and and show that they cause quasiperiodic movement.

• Assume a pen of length 10 cm with a center of mass precisely in the middle and $g=9.81\;\mathrm{m}\cdot \mathrm{s}^{-2}.Now$ imagine that you put the pen on the table with a null deviation $δx$ with an accuracy of $ndecimal$ places and with a null velocity. How long after making the pen stand can you be sure with just $n-decimal$ places of the nullness of the displacement?

• Consider a model of weather with the biggest Ljapun's exponent $λ=1.16\cdot 10^{-5}s^{-1}$. The weather forecast stops being useful if its error becomes bigger than 20 %. If you had determined the state of the weather with an accuracy of 1 %, how long do you estimate that your forecast would be good for? Give the answer in days and hours.

• Take Lorenz's model of convection from the last part, copy the function $f(xi,t)$ amd simulate and draw the values of the parameters $X(t)$ for two different trajectories using the commands X01=1;

Y01=2;

Z01=5;

X02=…;

Y02=…;

Z02=…;

nastaveni = odeset('InitialStep', 0.01,'MaxStep',0.1);

pocPodminka1=[X01,Y01,Z01];

reseni1=ode45(@f,[0,45],pocPodminka1,nastaveni);

pocPodminka2=[X02,Y02,Z02];

reseni2=ode45(@f,[0,45],pocPodminka2,nastaveni);

plot(reseni1.x,reseni1.y(:,1),reseni2.x,reseni2.y(:,1));

pause()

</pre> Instead of three dots $X02,Y02,Z02you$ have to give the initial conditions for the second trajectory. Run the code for at least five different orders of magnitude that are all still small and note the time, in which the second trajectory shall differ qualitatively from the first(ie will go in the opposite direaction). Don't decrease the deviation under cca 10^{$-8}$, because then the imprecision's of numerical integration start to show. Chart the dependency of the ungluing time on the order of magnitude of the deviation.

Bonus: Attempt to use the gained dependency of the ungluing time on the size of the deviation estimate Ljapun's exponent. You will need more than five runs and you can assume that at the moment of ungluing it will always overcome some constant $Δ_{c}$.

• Show that for arbitrary values of parameters $K$ and $T$ you can express the Standard map from the series express as

$$x_{n} = x_{n-1} y_{n-1},$$

$$\\ y_n = y_{n-1} K \sin(x),$$

where $x$, y$ are somehow scaled d$φ⁄dt,φ$. Show that the physical parameter $K$, x, y$$.

• Look at the model of the kicked rotor from the series and take this time the passed impuls$I(φ)=I_{0}$, after the period $T$ then $I(φ)=-I_{0}$, after another one $I_{0}$ and this way keep on kicking the rotor on and on.
• Make a map $φ_{n},dφ⁄dt_{n}$ on the basis of values $φ_{n-1},dφ⁄dt_{n-1}$ before the doublekick ± $I$ Why not?
• Solve $φ_{n},dφ⁄dt_{n}$ on the basis of some initial conditions $φ_{0},dφ⁄dt_{0}$ for an arbitrary $n$.
• *Bonus:** Try using the ingeredients from this series to design kicking which $will$ result in chaotic dynamics. Take care though because $φ$ is periodic with a period 2π and shouldn't d$φ⁄dt$ unscrew forever through kicking.

Copy the function $iterace_stanMap$ from the series and using the following commands choose ten very close initial conditions for some $K$.

K=…;

X01=…;

Y01=…;

Iter1 = iterace_stanMap(X01,Y01,1000,K);

…

X10=…;

Y10=…;

Iter10 = iterace_stanMap(X10,Y10,1000,K);

</pre> Between $Iter1$ and $Iter10$ there are hidden a thousand iterations of given initial conditions using the Standard map. As to see how the ten points look after the $nth$ iteration, you have to write

n=…;

plot(Iterace1(n,1),Iterace1(n,2),„o“,…,Iterace10(n,1),Iterace10(n,2),„o“)

xlabel („x“);

ylabel („y“);

axis([0,2*pi,-pi,pi],„square“);

refresh;

</pre> we write $"o"$ into $plot$ so that the points will draw themselves as circles. The rest of the commands is then included so that the graph will include the whole square and that it would have the correct labels.

• Set some strong kicks, $Kat$ least approx. -0,6, and place the 10 initial conditions very close to each other somewhere in the middle of the chaotic region (ie for example „on the tip of a pen“). How do the ten iteration's distances with respect to each other change? Document on graphs. How do the ten initially very close initial conditions change after 1 000 iterations? What can we learn from this about the „willingness to mix“of the given area?
• Take again a large kick and set your ten initial conditions along the horizontal equilibrium of the rotor ie $x=0$, $y=0$. How will these ten initial conditions change in time with respect to each other? What can we say about their distance after a large amount of kickso?

• *Bonus:** Try to code and plot the behaviour of some other map. (For inspiration you can look at the sample solution of the last series.)