2.1. Observing Phase Space: What We Now See

One of the striking results from this new era has been the detection of internal rotation in many star clusters. Far from being classified as non-rotating systems, many GCs now show ordered rotational patterns (see, e.g., Bianchini et al. Reference Bianchini, van der Marel and del Pino2018; Kamann et al. Reference Kamann, Husser and Dreizler2018; Ferraro et al. Reference Ferraro, Mucciarelli and Lanzoni2018). Recent studies, for instance, have revealed evidence of differential rotation in well-known clusters such as 47 Tucanae (see, e.g., Bellini et al. Reference Bellini, Bianchini and Varri2017) and Omega Centauri (e.g., see Haberle et al. 2025, also in this volume).

Velocity anisotropy – the degree to which stellar motions differ along radial and tangential directions – provides another important window into cluster dynamics. Some star clusters display complex anisotropy profiles, with tangentially biased motions in the very outer regions, likely due to the preferential loss of stars on radial orbits through tidal perturbations (see, e.g., Tiongco et al. Reference Tiongco, Vesperini and Varri2016).

These kinematic signatures are powerful because they encode the balance between relaxation, rotation, and external tidal influences. They are also sensitive to the presence of dark remnants (black holes, neutron stars) and binaries, which contribute to the cluster’s energy budget and heat the core.

The phase space mapping does not rely on astrometry alone. High-resolution spectroscopy from surveys such as Gaia-ESO, and forthcoming WEAVE, MOONS, and 4MOST provides line-of-sight velocities and chemical abundances. Together with proper motions, this enables a full three-dimensional velocity reconstruction for thousands of stars per cluster. The chemical information, meanwhile, allows the study of multiple stellar populations, revealing whether stars with different compositions have different dynamical properties (see, e.g., Cordero et al. Reference Cordero, Hénault-Brunet and Pilachowski2017; Cordoni et al. Reference Cordoni, Casagrande and Milone2025, also in this volume).

At even higher redshifts, the James Webb Space Telescope (JWST) and upcoming Extremely Large Telescopes (ELTs) are beginning to resolve the much anticipated “proto-globular clusters” in the early universe (see, e.g., Bouwens et al. Reference Bouwens, Illingworth and Oesch2017; Vanzella et al. Reference Vanzella, Calura and Meneghetti2017). These observations let us study the birth environments of GCs directly, probing whether cluster formation at high redshift followed the same physics we see in the local universe.

2.2 Computational Advances: Towards Realistic Simulations

For decades, the study of cluster dynamics relied heavily on equilibrium models, such as King (Reference King1966) distribution function-based equilibria, and approximate mathematical techniques, such as gaseous, Fokker-Planck, and Monte Carlo methods. While these approaches remain extremely valuable, they necessarily involve simplifying assumptions, such as isotropy or spherical symmetry. The past decade has seen the emergence of direct N-body simulations that fully capture the gravitational interactions of up to a million stars. For instance, the DRAGON simulations (Wang et al. Reference Wang, Spurzem and Aarseth2016) represented a major step forward, allowing the evolution of clusters with realistic initial mass functions, stellar evolution prescriptions, binary fractions, and galactic tidal fields.

One robust set of results from these early theoretical developments and numerical simulations concerned the evidence of a natural growth of radial anisotropy. Even systems that start perfectly isotropic spontaneously develop anisotropy in their outer regions as two-body relaxation pumps stars onto more eccentric orbits (Giersz & Heggie Reference Giersz and Heggie1994). When a tidal field is included, the anisotropy profile becomes more complex: outermost stars become preferentially tangential due to the loss of radial orbits (Baumgardt & Makino Reference Baumgardt and Makino2003), creating a signature that closely depends on a cluster’s initial tidal filling factor and subsequent tidal evolution (see, e.g., Tiongco et al. Reference Tiongco, Vesperini and Varri2016).

Fokker-Planck and N-body simulations have also explored the impact of internal rotation on the dynamical evolution of collisional systems (see, e.g., Einsel & Spurzem Reference Einsel and Spurzem1999; Ernst et al. Reference Ernst, Glaschke and Fiestas2008, and also Bianchini in this volume). It is still being investigated whether rotating clusters experience core collapse earlier than non-rotating counterparts with the same initial relaxation time, likely because rotation increases the central density and enhances energy transport (see, e.g., Inagaki & Hachisu Reference Inagaki and Hachisu1978, and also Bissekenov and Kim in this volume). Similarly, tangential anisotropy leads to earlier core collapse, while strong radial anisotropy can delay it, as noted by Breen et al. (Reference Breen, Varri and Heggie2017). These insights are crucial for both our fundamental understanding of key collisional processes and their applications to interpreting observed phase space information. They suggest that kinematic signatures today can be hold some memory of the evolutionary path of a cluster, provided we account for the combined effects of tides, anisotropy, and rotation.

3.1 Goal 1: Primordial vs. Evolutionary Features

A major priority is to disentangle primordial features (i.e., those imprinted at formation) from evolutionary features (i.e., those generated dynamically). This is not just an academic question – it directly affects how we use GCs as tracers of galaxy assembly. If a cluster’s rotation is primordial, it tells us something about the angular momentum of its natal gas cloud. If it is evolutionary, it may be the result of long-term dynamical processes such as angular momentum transport, spin-orbit coupling, or even mergers with other clusters. Phase space data, together with simulations, allow us to answer these questions. For instance, if two clusters with similar present-day masses, relaxation times, and tidal evolution show very different anisotropy profiles, that difference might be a fossil signature of distinct initial conditions.

3.2 Goal 2: Bridging the Initial Conditions “Reality Gap”

There is a significant conceptual gap between the initial conditions used in many direct summation N-body simulations (idealised, isotropic, single-population clusters) and what the community increasingly believes to be true: that GCs formed in turbulent, gas-rich environments, yielding multiple stellar generations as a result of complex star formation histories. Recent studies are tackling this goal from two directions. On the one hand, an indirect approach can be deployed: studying how long clusters can retain some “memory” of their initial conditions, especially kinematics. Recent work shows that rotation and anisotropy can persist for a large fraction of a cluster’s lifetime, especially in the outer regions (see, e.g., Tiongco et al. Reference Tiongco, Vesperini and Varri2019). This means present-day observations can, in principle, help constrain formation conditions. On the other hand, the direct approach is now becoming feasible: observing the first instances of proto-cluster populations in the early universe with JWST. The discovery of star-forming “clumps” in gravitationally lensed galaxies provides empirical input on cluster birth masses, spatial distributions, and dynamical states (see, e.g., Adamo et al. Reference Adamo, Bradley and Vanzella2024; Mowla et al. Reference Mowla, Iyer and Asada2024). Cosmological simulations are still unable to achieve the appropriate mass and spatial resolution to self-consistently follow the formation of star clusters ab initio, but this line of research is very active and rapidly evolving (see, e.g., Phipps et al. Reference Phipps, Khochfar, Varri and Dalla Vecchia2020; Pascale et al. Reference Pascale, Calura and Vesperini2025, among many others, including Lahén et al. Reference Lahén, Naab, Rantala and Partmann2025, also in this volume).

3.3 Goal 3: Laying Groundwork for Exotic Physics

Phase space richness also opens a new window onto topical issues such as the presence of intermediate-mass black holes (IMBHs), the retention of subsystems of stellar-mass black holes, and even the possibility of dark matter mini-halos in star clusters. New numerical simulations (see, e.g., Fujii et al. Reference Fujii, Wang, Tanikawa, Hirai and Saitoh2024; Rantala et al. Reference Rantala, Naab and Springel2021, also in this volume) are investigating the rich dynamical pathways from seeds to massive black holes. New analytic models have been developed for systems with a central massive object, producing self-consistent equilibria that can be isotropic and spherically symmetric or rigidly rotating and axisymmetric (Bonsor, Varri & Vanneste, in preparation). By comparing these self-consistent models to observed velocity dispersion profiles and proper motion maps, we can test for the presence of IMBHs with renewed confidence. This avenue is particularly timely given the advent of gravitational-wave astronomy. Globular clusters are thought to be prolific factories of black hole binaries (see, e.g., Marin Pina and many others in this volume), and phase space data will help determine how efficiently clusters can form, harden, and eject such binaries – directly affecting the predicted merger rate seen by LIGO, Virgo, and KAGRA.