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Rigid body with rotors is a widespread mechanical system modeled after the direct product $SO(3)\times {𝕊}^1\times {𝕊}^1\times {𝕊}^1$, which under mild assumptions is the symmetry group of the system. In this paper, the authors present and compare different Lagrangian reduction procedures: Euler–Poincaré reduction by the whole group and reduction by stages in different orders or using different connections. The exposition keeps track of the equivalence of equations as well as corresponding conservation laws.

The present paper describes the application of methods of qualitative analysis in the classical problem of rigid body motion in medium. Three particular problems are used as examples: bolides flight, galloping of aeroelastic constructions, and behavior of horizontal axis wind turbine. It is shown that this approach reveals the general properties of the behavior of objects studied, such as impossibility of translational deceleration of bolides and the presence of hysteretic phenomena in the operation of wind turbines.

Applying the Damour–Soffel–Xu framework of general-relativistic celestial mechanics, the theory of relativistic rigid body presented by Thorne and Gürsel is extended and developed in this paper. We successfully construct a quasi-rigid body model in the full post-Newtonian framework for the first time. This model has some simple properties in a similar way to the Newtonian rigid body, and it could be applied in geodynamics and astronomy, for example, to solve problems on rotation or precession of celestial bodies when relativistic effects are not negligible.

An integrable variant of a nonholonomic Suslov problem with compact and noncompact levels of the energy integral is investigated. The integral manifolds can be both compact and noncompact two-dimensional surfaces, in particular, a torus with four holes, a sphere with four holes, and a sphere with a number of handles and a number of holes. The surgeries of noncompact integral manifolds of integrable Hamiltonian systems are also considered. A generalization of the Reeb graph is proposed for noncompact case.

We consider kinematics of a rigid body rotation about a fixed point. An angular velocity is considered to be a sum of deterministic and random vector-functions. We suppose that the components of the latter are statistically independent small stationary processes. The problem is, for example, important for the theory of a rotational Brownian motion or an error analysis of strapped-down inertial navigation systems. We construct a multiplicative perturbation theory so that the desired matrix is represented as a product of deterministic matrix by random perturbation one. The perturbation matrix is approximately found by the first terms of a matrizant. As example, we consider the constant deterministic velocity and the components of the colored random velocity to be white noises (the Brownian motion), the Ornstein-Uhlenbeck processes, or the time derivative of the Ornstein-Uhlenbeck processes ("green" noises). In the second case it is shown that a diffusion of an instanta-neous rotation angle appears only if the deterministic rotation is not zero.

A very important property in the study of rigid body dynamics, moment of inertia describes the resistance of an object to any change in its angular velocity, given a certain amount of torque. Although many novel methods have been developed to simplify its calculation, this paper presents a remarkable theorem in moment of inertia that has never been widely used, the three-axis theorem. The theorem provides an alternative way for moment of inertia computation and better visualization in integrating each infinitesimal constituent mass element of a rigid body. The key idea is to focus on the distance from this infinitesimal mass to the intersection of the three axes, instead of its distance to a certain rotational axis.

There is an ever growing number of proposals for high precision experiments to measure gravitational effects, from simple Newtonian gravity to gravitational waves and even precision tests of general relativity (GR). In particular, more and more researchers from the fields of quantum optics and quantum opto-mechanics are becoming interested in GR and propose metrological experiments. Usually, such proposals rely heavily on a notion of length. However, in GR, as coordinates have no physical meaning, there is no unique concept for the length of a matter system. In this proceedings article, we summarize the article, where the conceptual problem of length is addressed for a subset of experimental proposals. In particular, the effect of gravitational fields and acceleration on the frequency spectrum of an optical resonator is discussed in the framework of GR. The optical resonator is modeled as a deformable rod of matter connecting two mirrors. Explicit expressions for the frequency spectrum are given for the case of a small perturbation. Example situations are discussed and a connection is obtained to a relativistic concept of rigidity.