 I would like to say that my longstanding collaboration with Tybal had a significant impact on my scientific career. I have visited Tybal several times at the IHS and each visit was very, say, intensive. So I learned a lot from Tybal, Tybal Tenkyu and Plurimos Annos. Because almost all my collaboration with Tybal was also a collaboration with Gerhard Schaefer, in my talk I will discuss some results achieved by the triplet DJS at the turn of the millennium and a bit later. And these achievements were related to the field of computation of high-order post-Newtonian dynamics of compact binary systems. I will not talk about any spin-dependent effects, so I have displayed here orders related with the pure orbital motion of two structurally spined masses. Red color means orders which are completely worked out. By orange I denoted, as far as I know, those orders which are almost completely worked out. I would like to remark on the five post-Newtonian order. Namely, most probably next week in the archive there will be a new paper by the DESI group in which this order will be completely done. Now I would like to make some historical remarks on the computation of the two-and-a-half post-Newtonian equations of motion. I don't want to repeat what you have already heard from Luik on Tuesday. I put in red color the first complete derivation of equations of motion at individual p-and-orders. And maybe I should explain that by complete I mean here something to which nothing else one can add. So all terms are derived, all numerical coefficients of these terms have done unique numerical values. Of course the result is free of errors and free of any omissions. So the first complete derivations at the two p-and-and-two-and-a-half p-and order was done by Tybal. More importantly, in 1983 he wrote a seminal and elegant paper in which he derived the rate of the decay of the orbital period of the two-body system directly from two-and-a-half p-and equations of motion. Directly means here not using balanced equation in which one equates the time of the derivative say binding energy of the system to the gravitational wave luminosity. And this work ended at least to a large extent the so-called quadrupole formula controversy, which was very vividly discussed in the 70s and 80s of the 20th century. Still higher order post-Newtonian equations of motion where and still are important for the needs of gravitational wave astronomy. And because of this, at the turn of the millennium, DJS made their tour de force. That is, they computed three p-and-and-four p-and conservative equations of motion which are derivable from two-point mass Hamiltonian computed within the ADM approach of general relativity. And so here I will very briefly sketch some point of this derivation together with some technical issues which are quite important to make the very last statement of my today's talk more credible. So the story begins from the constrained equations, which are written here for the system of two-point masses. The ADM transverse traceless TT gauge in which the space like metric is the sum of the TT part and some isotropic part which is parameterized by the single function phi. The field momentum with this conjugate to the space like metric has its Euclidean trace equal to zero. This is the second gauge condition. Afterwards, one splits the field momentum into its longitudinal and TT part and the longitudinal part can be expressed in terms of one vectorial quantity. TT means here application of the d-dimensional spatially non-local TT projection operator defined formally here. The next step is to express say the longitudinal degrees of freedom of the gravitational field that is the functions phi and vi in terms of the dynamical degrees of freedom that is particles, positions, momentum and TT variables of the gravitational field. This is done by a perturbative solving the constraint equation within the post Newtonian setting. That is, we expand the functions phi and vi into post Newtonian series displayed here. Please note that the numbers in parentheses denote the formal order in the inverse of speed of light. And at the end of the day, the whole multiple field dynamics is described by the so-called reduced form of the ADM Hamiltonian given here. It is just the volume integral from the flat Laplacian of the function phi. We are interested here only with the conservative section of the dynamics and we would like to have the Hamiltonian which depends only on the matter variables. To do this, we first make the Legend transformation with respect to the field variables, what leads to the Routhian. And after this, we eliminate the field variables by replacing them by time symmetric solutions of the field equations. At the end one obtains the high order Hamiltonian which depends also on the derivatives of the particle position and the momentum and these derivatives can be eliminated through the use of lower order equations of motion. Now it is a good point to use the important result of Luke and Thibaut from 1988. They prove that starting exactly at the four PN level, the metric is the sum of two pieces. The first pieces and instantaneous functional of the source variables. And then second pieces, non-local and time-tailed contribution. Consequently to obtain the complete conservative Hamiltonian, one has to add to the usual near-zone Hamiltonian time symmetric part of the tail contribution. The near-zone contribution is computed in an unusual post-Newton and way that is one takes the field equation of the function for the function HTT which has the form of the non-linear wave equation. And one take the time symmetric and the near-zone solution of this equation by using time symmetric green function that is half advanced, half retarded green function, which is formally expanded into the post-Newtonian series. After making the post-Newtonian expansion also of the source terms, one obtain the local time symmetric solution in the form of the post-Newtonian series for which individual terms fulfilled Poisson equations which depends for higher orders on the lower order solutions of these equations. Finally, after replacing HTT by its local and time symmetric near-zone solution, one obtains the near-zone conservative Hamiltonian. We used, we, I mean, DJS used the result of Luke and Thibault from 1988 to show that the time symmetric part of the tail metric contributes to the equations of motion through this non-local and time Hamiltonian. And please note that this Hamiltonian is written here in already regularized form in three dimensions and it depends on one specified length scale S. 4PN-accurated conservative Hamiltonian is the sum of the Newtonian, 1PN, 2PN, 3PN and 4PN pieces. Newtonian, 1PN, 2PN and 3PN Hamiltonians develop only UV divergences whereas the 4PN Hamiltonian is the sum of three pieces. The first piece is UV divergent but infrared convergent. The second is UV convergent by infrared divergent and the third is related to the tail contribution. The near-zone Hamiltonian density contains contact and field-like terms. The contact terms are proportional to Dirac-Delta distributions and they are regularized by means of Hadamard-Partifini's recipe how to compute the finite value of the function at its singular point. And now an important technical issue, namely in three dimensions and at least up to the force-pose Newtonian order but probably also to high orders. Using Prolet's pharaohidal coordinates, one can show that the most general integrand of the field-like type is very simple. It is exactly of this form. We have devised with Gerhard's analytical formula to compute all this type of integrands and therefore our working course of the regularization was the three-dimensional regularization performed on recent Hadamard. That is, it boils down to the multiplication of the integrand by two regularization factors which I displayed here and then looking for the limit when epsilon one and epsilon two both go to zero. Unfortunately, the result of this regularization is non-unique. Therefore, we computed the dr-dimensional regularization correction to this in the following way. We take the contribution coming from two small balls surrounding the particle positions and we consider that here only contributions with only terms which develop logarithmic divergences, that is, which behave like one over distance to the singularity to the power of three. We just replace these contributions by their d-dimensional counterparts. And at the very end, the result is as if all computations were fully done in d-dimensions. The situation is different with infrared regularization because, as I have already mentioned, we use the tail Hamiltonian in already regularized three-dimensional form. So we had to devise some new methods to regularize infrared near-zone divergences. In fact, we use two methods. We had to introduce a new regularization length scale as bar and both methods yielded the same result, modulo total time derivative and a change in the constant C, which ended the Hamiltonian in a very specific way, displayed here. So you see that an additive change in the constant C is equivalent to the multiplicative rescaling of the scale as bar. In the next step, we identified the length scale used in the near-zone infrared regularization with the length scale employed to define the tail contribution. This identification, the dependence on S, cancels between the local and non-local and time contribution, so the total Hamiltonian depends only on the constant C. And then we determined this constant using, say, beyond near-zone information taken from self-gravity computations of Binay-Idamur. They computed in the linear or symmetric mass ratio of the masses and for circular orbits, the gauge invariant link between the binding energy and the angular momentum of the system. Now, some more historical remarks on the derivation of the 3 pn and 4 pn equations of motion. I displayed here the first four independent and mutually compatible derivation. The first one was done by DJS and published in 2001. The second one was published in 2003 by Itoch and Futamaze. It is an interesting derivation because it is the only one performed in three dimensions and using some extended body model with so-called strong field point particle limits together with a surface integral approach. The derivation was by Binay-Idamur-Esposite-Fares and Phi, published in 2004. And finally, the first effective field theory approach derivation was done by Fofa and Sturani in 2011. For p-conservative two-body equations of motion, we have up to now four independent and again mutually compatible derivations. All derivations use delta sources and dimensional regularization. The first one was by DJS published in 2014. The second one was by Group of Luke published in 2017. In fact, they publish almost at the same time two derivations. The first one used some beyond-the-zone information and the second one was beyond this because of a new treating of infrared divergences by means of dimensional regularization only. And we have still two more independent derivation boats within the realm of effective field theory approach by Fofa, Porto, Rothstein and Sturani and by Blumlein, Meyer, Valkar and Schaefer. We have only time to shortly denumerate other application of the ADM Hamiltonian formalism and to point out some DJS contribution to this. First of all, the formalism can be successfully used to compute spin-dependent effects in dynamics of compact binaries. This DJS contributed to here by another derivation of next-to-leading order spin orbit corrections. Then conservative Hamiltonians, as we know, are one of the key ingredients of the effective one-body formalism. And here DJS contributed, say, new important building blocks to the existing formalism by adding information about the orbital first 3 pn dynamics, then 4 pn dynamics and also by adding next-to-leading order spin orbit interactions for the first time. It is worth to mention the implementation phase space of the puan-care invariance of the two-body dynamics described by the ADM Hamiltonian, namely the fulfillment of the full puan-care algebra relations in which ADM Hamiltonian operates as a generator of time translation is a really powerful check of the correctness of the ADM Hamiltonian. And finally, there are some results for radiation reaction effects in equations of motion and gravitational wave luminosities. Namely, DJS computed two and a half and three and a half O's Newtonian dissipative Hamiltonians for many body point mass systems. And to use these Hamiltonians to derive at the leading order and also at the next-to-leading order gravitational wave luminosities of the two-body systems in quasi elliptical motion. Also the leading order and next to leading, sorry, also the leading order spin orbit at spin one spin two dissipative Hamiltonians were derived within this approach. And now the future of the post-Newtonian two-body problem first result to be achieved. At the moment, the most accurate templates for inspiring compact binaries are made of three and a half pn order. So after completion of the computation of gravitational wave luminosity of two point mass system at the four and four and a half pn orders, it would be possible to construct 4.5 pn accurate templates. And of course, after completion of even a higher pn, higher pn order computations of two point mass equations of motion, but together with computations at the same orders of gravitational wave luminosities, it would be possible to construct even more accurate templates for the inspiring compact binaries. Here the progress on the level of equation of motion is very, very fast, but not so fast on the level of gravitational wave luminosities. Still, it is very important to compute within the post-Newtonian framework higher order tidal corrections needed to describe accurately dynamics of binaries containing neutron stars. And the last remark is as follows. There are still some very complicated, in fact, analytical results achieved within the post-Newtonian approach, which were computed only once. I mean here, first of all, gravitational wave luminosities computed for two point mass systems by group of luke, it would be very desirable to have at least one more independent derivation of these results. So I think that it would be interesting to have a derivation of 4 pn to body equations of motion using any extended body model to explicitly show validity of the replacement principle at this order. And new refreshed tools which can be used to obtain new results. So I can look for a new treatment of regularization issues related to the usage of Delta sources, which would simplify higher order post-Newtonian computations. First one can think about replacing Delta distribution by some sequences of classical functions, the so called Delta sequences. So there is no need for using distributional derivatives of singular homogeneous functions. The point is that these derivatives at higher pn orders are becoming more and more cumbersome. And actually at the free pn level, DJS successfully recomputed all dangerous that is UV logarithmically divergent terms using the dimension of these kernels to model point masses. The possibility is to look for some extensions or modifications of standard first distribution theory that would be most suitable for purely three dimensional regularization, such a net that was made by blanche and fire, but unfortunately they are extended had a mart regularization cannot be combined with dimensional regularization. And the last issue. Any a project which is just starting. I will try to increase the level of algorithmization and automatization of computation of two point mass IDM Hamiltonians. Recently I have realized that say the way of computing things are DJS is not so terribly complicated compared to the machinery use for example by the realm of effective field theory approach. The first task. So I decided say to improve and refresh my computer algebra programs, which I have been developing for more than 20 years. The first task is the first task is here to read the right for pn two point mass IDM Hamiltonian without usage. Gravitational cell for results and without introducing an ambiguity parameters. And this is this work is no progress the renewal of this problem was inspired by Gerhard and I am collaborating with him here. I think that two point mass Hamiltonians at orders, say, find a half five and five and a half seem to be within reach, and I do hope that before my retirement I will have some new nice results and the very last slide. I want to fully explicitly the hero of my talk that is the conservative for pn accurate two point mass IDM Hamiltonian computed in general that is non center of mass reference reference frame. Thank you for your attention. So we have time for questions. Hi, look at speaking. So my question is related to your last statement on the previous slide. Can you hear us? Yes, yes, I hear. And I wonder how will you proceed in order to be able to compute within the ADM Hamiltonian formalism. This last constant see without relying on cell for calculation and to compute it probably using the complete version of dimensional regularization for the infrared divergences. I mean, this is your last point. Working progress about the end. If you have some comments about that. Okay, you see the situation at the moment is as follows. We have repeated your and people computation of 1988 in the dimensions and this is in fact what you did also in your papers. Yes. So we took the dimensional tail Hamiltonian. And what do we have already obtained. I recomputed infrared divergences pure in the dimensions. And at the moment I obtained the perfect consolation of the polls in the one over D minus three together with the logs which depends on the regularization dimensional regularization scale at zero. That is, but still more work is before because we have some non consistencies, not related to the polls. Yes, yes. You asked for the calculation of support divided by 30 this constant yes within the ADM formalism. Quite recently, I made a suggestion to go to how to calculate it, but it is cumbersome and not yet finished. So there are ideas within idiom formalism to do it. Tt project is very complicated, but it is simple in Fourier space. So it looks similar to the fofa Sturrani integral where they got the 41 divided by 13. And it's definitely different. I thought it is identical, but when I checked it, I found out it's not identical. So this means something has to be done more. But how to do it in principle within ADM, it's clear, but never published, but we can. Maybe we'll do the calculations in the near future. This is the problem of relating the tail turn to the near zone dynamics by matching it. But like fofa Sturrani got to 41 divided by 30 correctly with the integral. And, you know, it's just what you were missing at the beginning of your calculations. Finally, you found exactly this missing and then the collaboration was much more important. So the same should happen in idiom formalism, that if we do it, so to say, full dimension regularization, that something is missing, which just is the 41 or a related number in ADM. We have not yet proven it's 41 divided by 30, but we know how it should look like the integrals will be performed. And this is so to say open to do. Was that a question? Come on. No. No. Let me check. Not okay. The rest of the video again.