We find and solve a large class of integrable dynamical systems which includes Calogero-Sutherland models and various novel generalizations thereof. In general they describe N interacting particles moving on a circle and coupled to an arbitrary number, m, of su(N ) spin degrees of freedom with interactions which depend on arbitrary real parameters x j , j = 1, 2, . . . , m. We derive these models from SU(N ) Yang-Mills gauge theory coupled to non-dynamic matter and on spacetime which is a cylinder. This relation to gauge theories is used to prove integrability, to construct conservation laws, and solve these models.Integrable models have always played a central role in classical and quantum mechanics. Most prominent examples, like the Kepler problem, are systems with few (≤ 3) degrees of freedom. An important exception is a class of integrable N-particle models associated with the names Calogero, Moser and Sutherland [1, 2] (for review see [3]). These are models for identical particles moving on one dimensional space and interacting via certain repulsive twobody potentials v(r). A well-known example is v(r) ∝ g 2 / sin 2 (gr) (which includes v(r) ∝ 1/r 2 in the limit g → 0), and we refer to the corresponding model as CS model. It is known that these models allow for interesting generalizations which also have dynamic spin degrees of freedom [4,5]. The CS model and its generalizations have recently received quite some attention in different contexts. Here we only mention their relation to gauged matrix models [6] and gauge theories on a cylinder [7] which will be relevant for us. a a the latter relation is implicit already in earlier work; see e.g. [8]

Yang-Mills gauge theory models on a cylinder coupled to external matter charges provide powerful means to find and solve certain non-linear integrable systems. We show that, depending on the choice of gauge group and matter charges, such a Yang-Mills model is equivalent to trigonometric Calogero-Moser systems and certain known spin generalizations thereof. Choosing a more general ansatz for the matter charges allows us to obtain and solve novel integrable systems. The key property we use to prove integrability and to solve these systems is gauge invariance of the corresponding Yang-Mills model.