 OK. Ič nas, dobra, in ne zelo, da sem lahko predpravila tega dokohto pristosti. OK. Svajnjeli smo pripljati za način način tako dajšnja danamica izvrata. Osev. Zvrata način danamica izvrata. The last part was the invariant measure, and we discussed the importance of invariant measurement in dynamical systems. And I give you also some information about the theoretical definition of the back measure, that is the generalization of the Piano-Jordan integration theory. The back measure has the advantage that lots of sets taj počefič s težkimi zvukami, taj šeščo. Ok. Daniji razgledajte povečanje in zvuk o stroboskobu, ki je zelo palovo početno, da se naredila nekaj početnih povedov, na načinji, težko se dovoljste projekte, pa težko neča bi tako obokrani. Vesel je, da se zelo vse Stroboscopic view and section just to don't have spurious effect in your representation. Ok. Basically, if you make a very naive projection, you can have some bad ideas or wrong ideas on the motion. Ok. Finally, we define Hamiltonian dynamics. And we introduce the simplative formulation of the Hamiltonian dynamics, which is most symmetric with respect to p, the momentum and the coordinates. And this is written in this way, where this is the gradient of the Hamiltonian with respect to the coordinate with x. X is general coordinate, including momentum and coordinates. This matrix is called the simplatic fundamental matrix. This is the null matrix, null matrix of order n. And this is the identity matrix of order n. Ok. Finally, we started to discuss the problem of changing coordinates. Because we discovered that, we ask, basically that if you have an Hamiltonian system, you want that your coordinate change must preserve the Hamiltonian structure. And this implies a constraint on the transformation. And the constraint is written in this formula. So, this is the Jacobian of the transformation of the canonical transformation, simplatic or canonical transformation. The definition, simplatic transformation or canonical transformation preserve the Hamiltonian structure of the equation of motion. Ok. Definition. Then, we discover that the Jacobian of the transformation must satisfy this strange equation that is called simplatic definition of a matrix. So, you say that if you have a canonical transformation, the Jacobian of this canonical transformation must satisfy this identity. I just to give you some possibility to recall, you see the analogy with the orthogonal matrices. Remember that, if you remember that somehow simplatic transformation is something like orthogonal matrices, orthogonal transformation in the real space. Ok. So, ok. And this is the last part. Can I erase the blackboard? Everything? Question? Comments on this concept? Ok. Let me erase. And now, we start in simple example. I try to do is hit the blackboard without doing mistakes, I hope. So, and consider, for example, a very simple Hamiltonian, the monical oscillator. H is p squared divided by 2 plus q squared divided by 2. Ok. And then I make a transformation of coordinates just to simplify the notation, sorry, capital P. This is capital P. Ok. It looks like polar coordinate transformation. I substitute, and my Hamiltonian, if I do not make mistakes, is simply p squared over 2. Ok. Then I write my equation of motion in this new set of coordinates. Hamiltonian equation of motion. So, you have p dot is equal to minus the derivative of q of k, and q dot is the derivative of k with respect to capital P. Ok. Mateo, correct me if I'm wrong. Ok. Ok. So, the equation of motion is very simple in this case. In fact, you have, you are supposed to have chosen a good transformation. In fact, in this case, you have that your equation of motion is p and 0. Sorry, the other one. 0, sorry, and this is p. Ok. So, p is a constant and given by the initial conditions and q, since p is constants, in nothing but that translation, q is 0 plus p is 0 t. Very simple. Ok. We are happy. We are solved. Now we have to invert. Coming back to the original problem. If you do so, for example, and you can see this in your, in the slides, if you try to invert, you obtain this kind of solution, for example, because you use this, you use this transformation, you come back. p of t is equal to p0, capital P0, that is constant, cosine of q, but q is q of 0 plus p0 t and q of t is equal to, of course, p0 sin q0 plus p0 times. Ok. Apparently, we solved the problem. We use a canonical transform. We use a transformation, a transformation and we solved the problem. What's the, but there is a problem now. In fact, if you try to solve the original problem and it's very easy to solve the original problem, since it's an harmonic oscillator, you find a different solution. In fact, just I give you, I let you this an exercise. You can see that the solution is this. It is incompatible with that one. Ok. Are you convinced of that? The problem, where is the problem now? Maybe I did a mistake in the computation, but tell correct me. What's the problem? Any guess? Any intuition? Exactly. Exactly. This one is not a canonical transformation. And then I run in trouble. Ok. In fact, if you consider here, exactly. This is not a canonical transformation. For example, you can see, if you make the, if you take the Jacobian of the transformation, and you take the determinant, you see the determinant is not one, but it's minus p. If this would be a canonical transformation, the determinant of the Jacobian would be one, instead in this case is p. Ok. And this is why you run often in troubles when you are considering Hamiltonian systems. You need to use canonical transformations. Otherwise, your solutions or your simplification of the model, because generally you make a change of coordinate to simplify the model, is not correct. Ok. Ok. And so there is a technique to avoid these wrong mistakes. Ok. And the technique is using a generating function when you are dealing with Hamiltonian systems. So canonical transformation are, I repeat, canonical transformation are the only admitted change of variable in Hamiltonian systems. Canonical transformation are hard to be found, because in this case you see that I did a mistake soon. I soon I did a mistake. And so we find more solution than the correct one. No, we find the wrong solution, I would say. Something that is not as unphysical basically. Do I reply? Mateo? Yes, thank you. Thank you very much. Yes, ok. Something that is unphysical. If I don't use, if I don't use canonical transformation, I could have unphysical solutions. Ok. And now I would explain so generating function method is a method to avoid simplity violations. Ok. Example. Suppose you have you want to change only the positions, for example, you decided for symmetry, you know that your system has some symmetry and you want to make the transformation from q to q in order to exploit the symmetry of your system, simplify the formulation. No? So once you have, once you ask this, once you define this transformation, for example, q, the question is how do I transform the momentum? Hmm? It's preserving the canonical condition. This is defined the symmetry of your system. But what about momentum? Ok. And the answer is given by the generating function. Ok. Generating function. So consider this systematic technique to obtain canonical transformation. And I give you just a simple demonstration of how one can recover the generating function. Suppose you have, no, suppose, this is the basic notion of preserving the volume of Hamiltonian systems. Hamiltonian systems generally preserve these quantities. We work in two dimension, but can be generalized. Ok. This identity is true. Hmm? Because we saw that the divergence of Hamiltonian flow is zero. So there is a conservation of volumes, but even canonical transformation should preserve volumes. Otherwise it's not a canonical transformation. The determinants should be one. Ok. Now let me apply this can be done for a domain d on the phase space, d. Ok. Are you familiar with the Stokes theorem or Green theorem? Hmm? Ok. So I will repeat if you want. Stokes theorem says that if you have a vector v and you have a curve c, the circulation the line integral of v is equal to the flux of the curve of v over any surface insisting so with a border of c. Ok. And now apply here the Stokes theorem in two dimension that is called Green theorem actually. And ok. So I would like to transform this so due to the in a line integral dp up plus dq uq. So this is the vector my vector u with component u p and uq. But here is a flux. And this is a flux of the curve. The curve in two dimension is something like this. dq dp dq up and uq. Ok. This means that d uq with p derive the respect to q. Matel? Ok. Thank you. But this should be one. So one possibility to transform this in this is to consider that up the vector u is equal to q 0. Sorry. No. It's p 0. And you can write this object into the integral over a line over the border slide p dq. Ok. You can do, I don't know if maybe ok. The idea is this. P No. Yes, q. You want to be here. So this should be, this is 0 and this is q. And this is p. Ok. In fact the curve is 1. Any mistake? I recap. I use the strokes, the greens the green theorem to transform this object on the surface into a line integral. And the line integral I would like to have this form. So my vector is this one and its Carl is this one. Maybe the vector mu should be q 0. So the first component is mu p, right? Oh no, sorry. Because you want the d mu q. I would like this one. To remain with this one. So uq should be p. I think it's ok. I leave you just an exercise. Ok. Professor, only a question. We impose that it must be 1 or it is a consequence because we impose to the line integral. No, no, we impose because it's 1 because here you have the flux of the curve, but the flux of the curve is 1 here. Ok. So modulus errors, but you can repeat your definition. We use the strokes or green theorem. You arrive at this kind of identity p dq plus also in this part I can do this and so I have pdq. Ok. And this should be equal to 0 if I put this on this side of the equation. Ok. With a minus sign? There is a minus sign. Yes, of course. Thank you. A bit lost why does it require that the integral over d of dq dp must be equal to the integral over d of dq uppercase, dp uppercase? Because you want that your canonical transformation satisfy this condition, but this condition means that the determinant of your transformation is 1. So it means that if you change variables from q dp you have of course dq dp multiplied by the determinant of the Jacobian but this Jacobian is 1 because the definition of simplistic transformation This is exactly saying that the simplifying form stays the same over the transformation. Ok. It says that you don't make canonical transformation if you don't preserve the areas because Hamiltonian systems preserve areas. So canonical is a consequence of the fact that you are dealing with canonical systems. Ok. Hamiltonian systems. Ok. Thank you. Ok. At this point can someone tell me the meaning of this identity of this equation? And this is valid for every curve for every curve which bounds the domain D. The domain D is arbitrary. Can you say something about this? What's the definition of that? An exact differential. Perfect. Thank you. This is an exact differential. So it means that f this function is a function of q is a function of q and I have more of course because p is nothing but the derivative of f1 with respect to q and p is nothing but the derivative of f1 with respect to q. And now I arrived at the canonical transformation. I can invert this and minus sorry. Thank you. And this is your canonical transformation because you have p as a function of q and q p as a function of q and capital q. You invert of course and you have. Yes, exactly. In fact now he says it's possible to exchange for example the coordinate. I would like to have not for example capital q q and capital q but he has I think capital p and capital p for example sorry. Do we need to define a second function to obtain p and big p or can we do obtain a relation from f1 to obtain a transformation on p2 p as well. So you need another function you know. We obtain a canonical transformation from q, from small q to big q. Do we need a second function to obtain a transformation from p to pb? No, you have here. It's written here because p is a function of q of p and q of q and here is p q and q capital q. But baby you have for example at the beginning you have for example a transformation q of q so you want to construct this but if you want to also that your transformation is preserved so preserved the canonical. Thank you. This is the first part of the story. There are four kinds of generating function and this generating function can be obtained by the original by the original for example consider this this one if you want to do an example you can use this canonical transformation you see what kind of transformation is in terms of p, q and capital q capital p and capital q try to exercise for this one but if I start from the definition of f1 I can also generate other canonical transformation and the way is very simple because I have only to play with the differentials in some sense is f1 is equal to remember p df 1 is p dq minus p pq, pdq but if I write for example this in this way dpq minus d minus q dp for example and this is again differential so an exact differential I put this with this and I get pdq minus q minus probably this is minus and this is pure correct and this is minus p should be plus yes should be plus ok and now this function 2 that we call f2 is exact differential which depends on smaller p and smaller q smaller q and capital p ok and again the transformation can be given by p is nothing but the derivative of f2 with respect to q and q is nothing but the derivative of f2 with respect to capital p ok and so on maybe can I make a question so does this remind you of anything and our agent transform yeah it's very similar no yes in principle yes yes yes the structure is the agent transform basically you pass from one maybe you're right you can pass from one generating function to the other using the agent transform I would say correct and ok you have of course here the scheme through which you can mix all these quantities and you have 5 kind of sorry 4 kind of generating transformation and I leave you an exercise suppose that suppose that you have this transformation q is equal to some j aj the matrix aj qj where a a transpose is 1 so is an orthogonal an orthogonal transformation basic rotation for example in ispace is a rotation the question is p how p transform ok the question is this the starting point for this exercise I would like that you understand how which is the transformation on p on the momentum the capital momentum using generating functions ok the generating function is a very powerful tool because there is the master of all generating functions there is the best one the optimal one and the optimal one is the one that allows you to solve the problem for example suppose you are you are you have this amytonian at the beginning you want me to make to pass on another amytonian whose equation of motion are very simple just like in the example that I showed you the wrong example ok if you want to if you are able to make this transformation where p dot is 0 and of course q dot now evolves according to the constant this constant and this constant of course depending on k on nu k then it is possible to solve the motion ok because the motion in this case is nothing but the translation of the torus in fact if you remember the example that I I show you the last lecture there is something that it was at the moment was very confusing I told that the transformation of the torus actually are related to the integrable systems in fact you say that a system a system is integrable if your dynamics sooner or later can be considered a composition of torus and this is the way you solve a typical Hamiltonian system if you are able to find a transformation of the Hamiltonian giving this kind of equation very simple your problem is solved there is a theorem in fact Ljuvil Arnold theorem but not Ljuvil theorem about the phase phase but it is Ljuvil Arnold theorem on integrability which says that a necessary and sufficient condition for integrability of any degree a system of any degree of freedom is that your Hamiltonian for any degree of freedom Hamiltonian system is the instance of an independent integral of motions it means that you have n functions preserved by the dynamics and generally f1 is the Hamiltonian if the Hamiltonian is independent on time in other words it means that actually the system is integrable if it has an integral of motion in evolution what means in evolution they are Poisson commuting each other in pairs so this is the Poisson parenthesis have you ever seen Poisson parenthesis but here there is the definition of the Poisson parenthesis and this is the definition in a simplistic formulation of Poisson parenthesis if this happens it exists a canonical transformation such that your Hamiltonian depends only of this constant of motions and the q coordinates evolves linearly in time linearly in time means they are q q0 plus omega i q0 i plus t why? the momentum are constant and this is an integrable system and yes in the previous slide yes in the previous slide you used the concept of best canonical transform what does it mean best means best canonical transformation allows you to solve the problem so to arrive at this kind of Hamiltonian equation that are solvable in fact this is constant and this is a shift in time of the coordinate this is solved the solution of this problem is very simple but you have to come back to the original transformation the difficulty is there that means if you succeed to have linear in time that means the best one linear because the simplest the simplest curve that you can have for example the straight line is simple in this way so the difficulty is in coming back coming back to the original to the original to the original coordinate another question yes about coming back to the original coordinates are there cases in which I can't I mean in which there is no general general method and of course this is I'm not working in Hamiltonian system but I think that it's a big problem for people working so this is only half solution of your model the coming back is the most difficult one I mean but there are also some there are also some simplistic transformations be active yes you have to invert the simplicity transformation of course yeah but I mean there are also transformations which are not invertible that are still solutions so I can't come back generally generally you try to don't invert about transformation so they are not because when generally you try to making it's very it's a hard work because the way you solve any Hamiltonian problem in principle there are problems that can be solved other that cannot be solved like in quantum mechanics there are problems that can be solved other not so but does the fact that you preserve volume implies that it must be objective maybe yes maybe you are right probably so even though yes probably the fact that the determinant is one helps a lot to find so don't you have singular singular transformation so maybe you are right yes good okay can I go home yes okay sorry sorry the fact that you are able to find the solutions in that form is called action variable action angle transformation in the books generally you find that if a system is integrable you can find a transformation that is for I think historical reasons action angle angle action angle transformation okay in this case in this case the momentum are called the action and the coordinates are considered angles because of course the solution are periodic no because you can consider your solution on a torus okay and this is for basic historical reason action angle variables okay and I think I skipped some consider for example this is very simple Hamiltonian hfq this is a simple one-dimensional one degrees of freedom of systems the system can be this system can be easily integrated no because of course according to Liouville theorem you have one integral of motion and one degrees of freedom so according to Liouville this can be integrated and the integration is very simple because you write Hamiltonian equations your Hamiltonian equation that is p dot minus u q and q dot is equal to p okay but you have also the constraint of the energy conservation p in q u of q then you can extract for example p from this equation that is 2m e minus u of q write and then you substitute this one sorry can you read write larger can you read okay and people people are so people are sorry yes you are right so I use this one and I substituted you have q dot is equal to square root plus minus 2m e minus q write and this is the typical you can integrate like this the q divided by for the moment leave the double the double sign the ambiguity of the double sign the ambiguity of the double sign can be removed by the choosing of the initial condition in fact this potential you have this potential and this is your energy level which select your orbits no and this for example and your orbit is in the phase space could be let me try this one this way and this is the plus solution and this is the minus solution problem the sign is not a problem can be removed this integration can be done on the circle this one and this one but there is another method to solve in this this called action angle and I can stop here shall I stop pose relax we take 5 minutes break guys we are assuming the lecture can you hear me and this is the so called quadrature integration methods for one dimensional system that is very simple but there is another method to solve this problem using exactly the action angle variables in this case you define the action the action I in this way that remind you something that remind you something like this you see and you integrate on the cycles on the cycles and you see why now your variable is considered an angle because is going on a circle of something that is topological similar to a circle and then you arrive a very simple Hamiltonian and you can use the trick so you define this canonical transformation I and the angle phi and so you have the solution of the model in an alternative way you can use directly the quadrature approach but even if you try to make it the exercise you can use also action angle variables and just few pictures about what was going on you can use action variable angle angle action variables you have this is the solution and your solution lives on for example two dimensional torus or d dimensional torus of course if you have two n dimensional phase space you have n dimensional torus and you see if omega this quantity omega your integration is a rational they are in rational rational ratio you have a periodic solution on the torus while if for example you have imagine that you have two two angle variables for example theta 1 is theta 1 0 if you have one of t is equal to theta 1 0 plus omega 1 t and you have the second angle t t1 0 plus omega 2 t if you look at this number if this number are in this ratio so these are natural number they are rational you have a periodic solution on the torus in this one if this object is different from that so is not rational is irrational you can have that your solution is dense covers densely the torus okay and here you see some the square that discover is completely shaded by your solution okay in some sense you see something that is in irrational case your torus is completely filled densely by your solution okay you see a spot on your square okay now the other example that we can make is the discrete dynamical systems no, for example apart from Poincare map and the flow map which are discretization of your dynamics there are situations in which evolution law are intrinsically discrete in this case and I have already discussed this problem in this case you have a map in the dimensional space for example and typically systems real systems that can be described but these maps for example is the generation of biological spaces one generation, next generation third generation and so on and for example even in the algorithm on computer on computers are generally discrete if you want to integrate for example differential equation you end up to a map you end up with a map on the computer iteration procedure of computer science for example are discrete map and also seasonal phenomena that repeats for example they are distributed over for example days you can decide the day is your measure of time step then n is an integer denoting the iteration the generation or the discrete time in general nothing difficult as I told you there are two possibilities these are a couple basically there are area preserving and area contracting maps and it depends of course on the Jacobian, on the transformation because the map is nothing but a transformation in the phase space so is a transformation here and you want to know how an infinitesimal set of your phase space omega change due to the fact that this transformed ok and since the transformation is a map you can have that volume transformation is the volume transformation is defined by the determinant of the Jacobian of the map the Jacobian of the map I recall you is nothing but if this is the map the Jacobian of course if this is the map ok in n dimension the Jacobian is the matrix L that is dfi with respect to xj ok or if you consider the map as n plus 1 for example this is nothing but the xi at time n plus 1 divided xj at time n and this is the Jacobian ok and it tells you how volumes preserve or change in time equal 1 is conserved minus 1 the map is dissipative ok very simple ok now let me discuss some example of maps in general in 2 dimension for example in 2 dimension ok a typical map for example is the geometrical transformation it's a translation rotation, dilatation or combination of them for example and this is the first example that you can imagine but there are more general transformation in dynamic system especially in chaotic system and one of the best known transformation is the Hennon map it was introduced in the 76 by Hennon and inspired basically he took inspiration from Poincaré section of some Hamiltonian systems of the laurel maps and so Hennon proposed a mapping on the plane which is a composition of 3 different transformation you are composing and you see what happens at the end and the the composition is that first this is a nonlinear folding in the x direction for example consider this is the starting set I take an ellipsis my starting set is the set of point contained ok my starting point is this the ellipsis all the points inside this object then apply this transformation and the transformation transform this ellipsis into something that is like this ok the first t1 the equation of t1 is given in these in these formulas then we apply you see this folding because I make an horse show this is an horse show horse show is another word that is in dynamical systems chaotic systems is important is recurrent and I will let you remember and ok and then we make this linear contraction of this object so is only basically you are only contract you reduce the dimension the dimension of that is shrinking this is l and this is l prime that is less than ok and finally a rotation or 2 pi take this and put in this in this orientation ok very simple if you combine them all together and you start iterate this transformation this one is this is the formula ok you have the square of the point time n plus xn plus 1 so this is basically quadratic transformation and this is instead the second one is basically a linear transformation so if you compose those transformation you arrive at the final at the final formula that is this one I think that the battery is going down Mateo, the battery is going down can you maybe I have my myself my own I have my own ok, that's I will start with me ok thank you ok the map of course is dissipative how do we know that this is dissipative how do we know that the map is dissipative what operation I have to do this map is dissipative or conservative for example what shall I do exactly the Jacobian I have to compute the Jacobian and if you make the if you compute the Jacobian by exercise you observe that the Jacobian is determinant of the Jacobian is L is B, sorry, B, this parameter here is B generally is positive but is considered is taken less than 1 so the map is dissipative but the map is invertible yes, of course in fact is very simple to make the inversion of this of this map and I leave you this exercise also anyway you start with the inversion of this and then you you try this is very simple and you can invert the other one, you give the transformation inversion means that you have to write xn in function as a function of n minus s plus 1 and of course x1 in function these two quantities in function of these two quantities you have inversion, this map is invertible in fact the nice thing is one of the few quadratic maps that can be invertible and so it's a one-to-one mapping of the plane onto itself and generally Hannon studied the iteration for several range of parameters and finding lots of behaviors but what he found very interesting was a chaotic motion on a strage attractor and if you want an idea what the strage attractor is and is this one you see you see that it is folding of the points but also the stretching this is the typical effect of chaos, I repeat the stretching and folding is a typical mechanism through which chaos develops Lodzi, another mathematician, introduced in the 78 another map that is somehow a simplification of Hannon map he plays instead of square here he put the modulus but basically again the map is contracting 2 if beta sorry is B is less than 1 the map is again invertible and the attractors there is some similar to the Hannon attractor with the fact that there are some kind of casps here and probably the casps is due to the fact that you are considering the modulus probably, the fact that you are considering the modulus here this is another famous and this is the example of a strage attractor in the lecture of Angelou Piani you will see how to characterize strage attractors in the way to characterize them in terms of fractal geometry instabilities because here there are lots of instabilities in the motion due to the fact that the system stretch and folds continually your your points Angelou Piani tells you something about that how to characterize to study this kind of object just for completeness I would say that there are maps for example that are simplatic too basically they are the discrete version of the Hamiltonian systems consider a simplative map is nothing but the discrete version of Hamiltonian systems so the properties of the can be translated into the of the maps again this is the map the map has a Jacobian so the Jacobian is the the important object of the maps because it define how volumes the form in the phase space changing the phase space and if the map is simplatic of course it's Jacobian should be simplatic but by the rule that I already show you for Hamiltonian canonical transformations and this is M if L, I use L if L is the Jacobian I expect if the map is is simplatic that it satisfies this equation again I'm so boring but I have to repeat this one because nemonically you remember that this is the same like if you want a nemonical rule to understand to recall to remind the the simplatic maps of course simple version of the simplatic map is this one is a linear map you see but linear not so linear because at some point you take the modulus the modulus is a nonlinear operation so the structure is linear but as soon as you take the modulus is the transformation becomes nonlinear otherwise would be very trivial of course ok, this maps acts on the torus 00 yes? ah, what's the modulus? ah ok, the modulus is some function let's say is that if you have a number x that is in 01 it remains x otherwise is outside you have to make this operation if x does not belong to 01 you have to make this transformation x is x minus integer of x and this is ok k integer for example you need for so you are folding so if you have 1 if the point is here you remain here is the point is here you have to compute how many intervals are distance from from this and you take this point and you fold in this point it's a folding basically you report you take your point let's say a here of course even in negative geometrically is this ok you refold you try to refold all your real axis on 2 on 201 this is the modulus ok yes, exactly on the other side you sum in the other side you sum instead of subtracting ok if you want to know the action of the map you do the same you take I know an ellipsis of point and you see that the first iteration these ellipsis are structured but the modulus operation take the part of the ellipsis into the because transformation is a root a if you make this for example you have the ellipsis you take the ellipsis the ellipsis is transformed no this is 0 sorry the plot is this you have this is the ellipsis and this is the torus you apply the transformation the ellipsis is the form ok and you take the part of the ellipsis you take this part and you put here for example you take here and you have here so basically you have to fold your ellipsis into the square and the branch corresponds to this branch another branch corresponds to this branch also all into the same the same square ok if you make another iteration you see that another iteration you see nothing because of the of the reproduction of the graphics ok but more if you take at least for example ten iteration just you see that these ellipsis is scrambled over the square and you are not able to recognize the form if I give you the last picture you don't know which is the source of the beginning was this walls you don't know you are not able to recognize because of the fact that chaos acts in this way the stretching is due to the fact that the ellipsis is structured is rotated and structured but the folding the modulus is the folding operation no the folding makes this the behavior very very difficult to be understood ok and this is called the cat map and I'm sorry for the quality of the picture sorry anyway the cat map because someone says this is the sea properties automorphism on torus because in mathematical language basically the transformation is on automorphism of the torus but someone says that cat comes from the fact that Arnold who was the mathematician who introduced a pictorial representation the face of a cat and so we see how the transformation works and this is the result of the of the modulus for example you have stretching the formation and stretching of your cat but the cat after the first step becomes like this poor cat ok so the cat map has the properties of randomizing any regular spot of point point which starts very close to each other quickly separate this is important to important concept first ok but the second thing is important that point which starts very close to each other for example here on the lips day after just in a first iteration they becomes very far one here and one here for example ok and this is the origin of chaos another interesting this is of course a simplistic map because the determinant is one of course I miss this this is a simplistic map you can prove this another simplicity map that you can encounter if you read a book of chaos it is important to discuss it is called I can skip this one I can skip this is the Chirikov map or standard map standard map ok, finish with standard map ok, hopefully the standard map standard map ok or Chirikov map Chirikov from the Russian mathematician who first introduced this one Chirikov ok, Chirikov Chirikov map and this was introduced for explaining some very strange effect in probably acceleration particle physics some strange behavior of the accelerators accelerators ok this is the of course the map is defined like this ok basically can be put in action angle representation where the momentum is the action and p and q is the angle in the next slide I will use this notation ok and the map can be defined for example using this very strange amytonian that is amytonian of rotator which is subject to infinite action of resonances ok, this is amytonian of this is the the momentum ok, the action and this is the potential the potential is a strange potential because it depends on time but it also has some kind of forcing in fact we say that this is a kicker the rotator every period t your system, your rotator has a kick is kicked in this sense so there are lots of resonances an infinite number of resonances ok this amytonian can be put in a very in a different way that is more easy to deal with and this can be done by this simple transformation for example if you have the cosine if you have the sum of cosine theta t plus n over t this is the period of your forcing of your kick then if you use the euler representation 1 over 2 is correct now consider that I can change the sum here n to n to minus n so I have I have changed ok then I can factorize this and so and I can use this exponential to recover the cosine I can write cosine theta the sine is minus but not important because if let me use the sinus and this is this formula basically I rewrite this formula the fact that this is an expansion of what is called the Dirac combo have you ever seen the Dirac combo? no? the Dirac combo is nothing but this the sum Dirac combo is nothing but like I will call Dirac delta of period t but the sum minus infinity to infinity n of delta t plus or minus nt ok this is a periodic function of course can you say why this is a periodic function this is infinity if I make a translation yes so I can redefine this one and this is delta because I have shift to the sum but since the sum goes from minus infinity to infinity I can reabsorb this one into the sum this is a period but any period function in an interval it's amenable to representation for your representation no? if I make a Fourier representation of this because it's periodic I can have and this I leave you this exercise I can have this so this is the representation of this object the Fourier representation of the Dirac combo then I arrived to this very simple Hamiltonian that I write now you see I'm using the actual angle representation I passed to this ok and then I write the equation of motion ok I write the equation of motion of this Hamiltonian and the equation of motion are quite simple because equation is I dot is equal to minus dh with respect to theta and theta dot is equal to the derivative of h with respect to 2i I dot is equal to k sinus that is minus sign maybe cos sign is minus sign no we have cos sorry I don't remember we use cos I don't remember the original yes we use cos so there is minus sign minus sign and this is equal to i ok now I have to integrate them but I have a problem I have delta ok generally when you want to solve a problem differential equation with some strange distribution the delta what you have to do is ignore for the moment the delta you know that the delta is here you separate your region region 1 region 2 here the delta is not acting and so you can solve the equation in a very simple way ok but since you have a delta you have a discontinuity in the derivatives for example and you have to introduce the matching condition so at this point you have one solution another solution that does not take care to account the delta but if you have a delta there is a jump for example in the derivatives for example and you have to introduce this matching the boundary conditions of these two solutions ok in this case you see that for example the solution that you have is if you don't consider the the delta is something like this your action are constant of course and you say that is the action on the interval n n minus 1 sorry and this is the action for example in the interval n but of course since you have the delta these terms is acting only when you have the delta so here outside is zero of course so even the equation of motion of theta is very simple is a straight line and you have for example I call theta minus of t and theta and theta plus in this part with a different slope the slope is defined by this one and theta plus of t and this is the solution of the two the straight line and I satisfy this condition where is plus plus is this one and minus plus is this one and here I define what is plus and minus then I consider that the angle is continuing because the angle is not affected by the delta by the peak the sharp peak but is into the action behavior the action behavior if you match at this point this equation you see that you can write your evolution at each forcing you did it it's basically a stroboscopic view it's nothing more you see that you can write your evolution of the map at each interval on each spike in this way and this define the map the shirikov map and this is derived from the equation of the kicker rotator questions domanda there is an alternative derivation if you prefer this is much simple the question is you have again the equation since you have the delta as I told you I have to integrate this object into a period but since I have the this is a shaded I don't know if you are able to see this is a shaded area I integrate the equation of motion over a period but I shifted this period by an epsilon value very small epsilon value that I'm going that will go to zero if I integrate this into this period I arrived to this simple equation consider that the action is constant so in this way the area of the rectangles of these rectangles this is the rectangles here for example this is the interval of time and this is the height of the rectangle in this way you have also to sum since you are integrating on a period you have also to sum this part and again this is the epsilon is the base is the height so this is the integration over your equation of motion over a period you put epsilon to zero and you obtain the same equation is an alternative derivation ok since I give you the slides you can find this computation already done ok so this is a standard map and you can of course study and I leave you this is an exercise the fixed point of the map and also the Jacobian of the map of course is the Jacobian of the map is how much is this the Jacobian of the is one exactly because the area of preserving and not only area of preserving it's implectic that you can study the stability but it's ok something that we will see and for k equal to zero the map is integrable for k equal to zero the map you put here k equal to zero the map is very simple then the map is integral because you have for example that remains remains constant remains the value at the beginning remains at the value at the beginning while theta n is of course of course theta zero plus n i zero ok and again if i zero is a rational the motion of the orbit is periodic if i zero is irrational it feels it's not the motion is quasi periodic ok what I'm saying is that suppose you start this is your zero one or zero two pi square where I fold my map because I forgot here to write that this is modulus two pi so I fold the trajectories into this square if I take for example this is i zero theta if I take for example i zero that is rational you see that the evolution of the integral of map is a set of points and it repeats instead if i zero is an irrational what you see is a continuous line which is the square horizontal horizontal continuous line and this is the value of theta of theta one theta two theta three theta four theta five and they repeat ok theta one you have a repetition periodic motion but if you increase if you increase the values k the values k that is your displacement for integrability you see that the behavior slightly changes here first of all for small k you see that the scenario is very similar to a pendulum in fact as I told you the kicker rotator is nothing but a pendulum one gives a kick every time every second for example you are kicking tongue every second and in fact it represents basically the scenario of the pendulum the deformation of the straight line is due to the fact that you have the cosine so the cosine deform a little bit but the behavior is basically recall a pendulum if you increase again the nonlinear parameter k you see that something new happens so that there are some thin layer of of chaos for example and that are separated by those lines that survive are not destroyed by the perturbation because some lines are only deformed by the perturbation but for example periodic periodic behavior doesn't exist anymore of course but maybe survive some and they are the island for example the island could be the islands for example but anyway between these two deformed lines you can have the development of chaos until you arrive at the values kc kc where larger portion of chaotic behavior are appearing and but there is what is called a survival survival line survival integral ok sorry next time I will explain better because you don't have this is important because maybe Angelo Wumpiani tells you something about camp theory and the standard map is the best like an oscillator as the same importance of harmonic oscillator in physics in quantum physics ok thank you sorry to be late