We aim to investigate the connections existing between the density profiles of the stellar populations used to define a gravitationally bound stellar system and their star formation history: we do this by developing a general framework accounting for both classical stellar population theory and classical stellar dynamics. We extend the work of Pasetto et al. (2012) on a single composite-stellar population (CSP) to multiple CSPs, including also a phase-space description of the CSP concept. In this framework, we use the concept of distribution function to define the CSP in terms of mass, metallicity, and phase-space in a suitable space of existence $mathbb{E}$ of the CSP. We introduce the concept of foliation of $mathbb{E}$ to describe formally any CSP as sum of disjointed Simple Stellar Populations (SSP), with the aim to offer a more general formal setting to cast the equations of stellar populations theory and stellar dynamics theory. In doing so, we allow the CSP to be object of dissipation processes thus developing its dynamics in a general non-Hamiltonian framework. Furthermore, we investigate the necessary and sufficient condition to realize a multiple CSP consistent with its mass-metallicity and phase-space distribution function over its temporal evolution, for a collisionless CSP. Finally, analytical and numerical examples show the potential of the result obtained.
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