Daddy Balondo Iyela, UNIKIN, DRC), Jan Govaerts, Univ. cath. Louvain, UCLouvain, Louvain-la-Neuve, Belgium)

When discussing consequences of symmetries of dynamical systems based on Noether's first theorem, most standard textbooks on classical or quantum mechanics present a conclusion stating that a global continuous Lie symmetry implies the existence of a time independent conserved Noether charge which is the generator of the action on phase space of that symmetry, and which necessarily must as well commute with the Hamiltonian. However this need not be so, nor does that statement do justice to the complete scope and reach of Noether's first theorem. Rather a much less restrictive statement applies, namely that the corresponding Noether charge as an observable over phase space may in fact possess an explicit time dependency, and yet define a constant of the motion by having a commutator with the Hamiltonian which is nonvanishing, thus indeed defining a dynamical conserved quantity. Furthermore, and this certainly within the Hamiltonian formulation, the converse statement is valid as well, namely that any dynamical constant of motion is necessarily the Noether charge of some symmetry leaving the system's action invariant up to some total time derivative contribution. The present contribution revisits these different points and their consequences, straightaway within the Hamiltonian formulation which is the most appropriate for such issues. Explicit illustrations are also provided through three general but simple enough classes of systems.

Mordehai Milgrom

It is pointed out that MOND defines a fiducial specific angular momentum (SAM) for a galaxy of total (baryonic) mass $\mathcal{M}$: $j_M(\mathcal{M})\equiv\mathcal{M}^{3/4}(G^3/a_0)^{1/4}\approx 383(\mathcal{M}/10^{10}M_\odot)^{3/4}{\rm kpc~km/s}$. It plays important roles in disc-galaxy dynamics and evolution: It underlies scaling relations in virialized galaxies that involve their angular-momentum. I show that the disc SAM should be $j_D\approx[\langle r\rangle/r_M(\mathcal{M})]j_M(\mathcal{M})=[\Sigma_M/\langle \Sigma\rangle]^{1/2}j_M(\mathcal{M})$, with $\langle r\rangle$ the mean radius of the disc, $\langle \Sigma\rangle=\mathcal{M}/2\pi\langle r\rangle^2$ some mean surface density of the galaxy, $r_M=(\mathcal{M} G/a_0)^{1/2}$ is the MOND radius of the galaxy, and $\Sigma_M=a_0/2\pi G$ is the (universal) MOND surface density. So, e.g., for a fixed $\langle \Sigma\rangle$, $j_D\propto \mathcal{M}^{3/4}$, while for a fixed $\langle r\rangle$, $j_D\propto \mathcal{M}^{1/4}$. Furthermore, $j_M(\mathcal{M})$ is a reference predictor of the type of galaxy a protogalaxy will settle into, if it evolves in isolation: A protogalaxy of mass $\mathcal{M}$, and SAM $j\gg j_M(\mathcal{M})$ should settle into a low-surface-density disc -- with mean acceleration $\langle a\rangle/a_0\approx j_M/j\ll 1$. While a protogalaxy with $j\lesssim j_M(\mathcal{M})$ should end up with a disc of mass $\mathcal{M}_D\approx j\mathcal{M}/j_M(\mathcal{M})$, having a SAM $j_D\approx j_M(\mathcal{M})$, which is tantamount to $\langle a\rangle\approx a_0$ (i.e., at the `Freeman limit'); it should also develop a low-SAM bulge, taking up the rest of the mass $\mathcal{M}_B\approx\mathcal{M}-\mathcal{M}_D$.

Xiaoju Xu, Saurabh Kumar, Idit Zehavi, Sergio Contreras

Understanding the impact of halo properties beyond halo mass on the clustering of galaxies (namely galaxy assembly bias) remains a challenge for contemporary models of galaxy clustering. We explore the use of machine learning to predict the halo occupations and recover galaxy clustering and assembly bias in a semi-analytic galaxy formation model. For stellar-mass selected samples, we train a Random Forest algorithm on the number of central and satellite galaxies in each dark matter halo. With the predicted occupations, we create mock galaxy catalogues and measure the clustering and assembly bias. Using a range of halo and environment properties, we find that the machine learning predictions of the occupancy variations with secondary properties, galaxy clustering and assembly bias are all in excellent agreement with those of our target galaxy formation model. Internal halo properties are most important for the central galaxies prediction, while environment plays a critical role for the satellites. Our machine learning models are all provided in a usable format. We demonstrate that machine learning is a powerful tool for modelling the galaxy-halo connection, and can be used to create realistic mock galaxy catalogues which accurately recover the expected occupancy variations, galaxy clustering and galaxy assembly bias, imperative for cosmological analyses of upcoming surveys.