Inertia tensor of a rigid body on the Lobachevsky plane and in pseudo-Euclidean space

Abstract

The paper studies the inertia tensor of a rigid body in three-dimensional (pseudo-)Euclidean space (𝑉, 𝑔). The configuration manifold 𝑄 of the system is the six-dimensional Lie group E(𝑉, 𝑔) ∼= 𝑉 ⋋ Aut(𝑉, 𝑔) of isometries of this space, and the kinetic energy is a quadratic form 𝑇(𝑤, 𝑎) on the Lie algebra e(𝑉, 𝑔) ∼= 𝑉 + g where g = aut(𝑉, 𝑔). This allows one to define a symmetric operator 𝐽 : g → g* with the property 𝑇(0, 𝑎) = 1/2 (𝐽𝑎, 𝑎), referred to as the (covariant) inertia tensor of the rigid body. To compute this tensor, a “pseudo-Euclidean vector cross product” [, ]𝑔 is introduced in the (pseudo-)Euclidean space (𝑉, 𝑔), and an isomorphism 𝜇 : 𝑉 → g is constructed using this operation. It is proved that this isomorphism transforms the operation [, ]𝑔 into the Lie bracket on the Lie algebra g, and the scalar product into the Cartan–Killing form, up to a scalar factor. Explicit formulas for the operation [, ]𝑔 are obtained.
Using the operation [, ]𝑔, the operator ̃︀ 𝜔 = 𝜇𝜔 ∈ g of instantaneous rotation with angular velocity 𝜔 ∈ 𝑉 is defined. For any point 𝑞 ∈ 𝑉 , the vector 𝑣 = ̃︀ 𝜔𝑞 = [𝜔, 𝑞]𝑔 ∈ 𝑉 of instantaneous velocity, the vector 𝑀(𝑞) = [𝑞,𝑚𝑣]𝑔 ∈ 𝑉 of angular momentum and the inertia operator
̂︀ 𝐽(𝑞) : 𝑉 → 𝑉 , 𝜔 ↦→𝑀(𝑞), are defined. The symmetricity of the inertia operator ̂︀ 𝐽(𝑞) is proved, along with the formula 𝑇(𝑞) = 1/2 𝑔( ̂︀ 𝐽(𝑞)𝜔,𝜔) for the kinetic energy of the point.
Geometric properties of the inertia operator ̂︀ 𝐽 are studied for single- and multi-point bodies.
In particular, in the pseudo-Euclidean case, the restriction of the corresponding quadratic form to the interior of the light cone is shown to be non-negative. Examples of two- and three-point
bodies are constructed showing that there are no additional restrictions on the signature of the inertia operator. All possible signatures of the inertia operator ̂︀ 𝐽 for a rigid body in threedimensional pseudo-Euclidean space are found. It is proved that, for bodies located within the light cone (e.g., “plates” in the Lobachevsky plane), the inertia operator has a signature
of (−,+, +) or (0,+, +). For bodies located outside the light cone, signatures of (−, 𝑠,−) are possible for all 𝑠 ∈ {0,+, −}. The remaining signatures (−,+, 0) and (−, 0, 0) are also realized by two- and three-point bodies.