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

A. Yu. Shubert

Lomonosov Moscow State University (Moscow)

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 $\mathrm{E}(V,g)\cong V\leftthreetimes \mathrm{Aut}(V,g)$ of isometries of this space, and the kinetic energy is a quadratic form $T(\mathbf{w},a)$ on the Lie algebra $\mathfrak{e}(V,g)\cong V+\mathfrak{g}$ where $\mathfrak{g}=\mathfrak{aut}(V,g)$. This allows one to define a symmetric operator $J:\mathfrak{g}\to\mathfrak{g}^*$ with the property $T(0,a)=\frac12(Ja,a)$, 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 $\mu:V\to\mathfrak{g}$ is constructed using this operation. It is proved that this isomorphism transforms the operation $[,]_g$ into the Lie bracket on the Lie algebra $\mathfrak{g}$, 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 $\widetilde{\mathbf{\omega}}=\mu\mathbf{\omega}\in\mathfrak{g}$ of instantaneous rotation with angular velocity $\mathbf{\omega}\in V$ is defined. For any point $\mathbf{q}\in V$, the vector $\mathbf{v}=\widetilde{\mathbf{\omega}}\mathbf{q} = [\mathbf{\omega},\mathbf{q}]_g\in V$ of instantaneous velocity, the vector $\mathbf{M}^{(\mathbf{q})}=[\mathbf{q},m\mathbf{v}]_g \in V$ of angular momentum and the inertia operator $\widehat J^{(\mathbf{q})}:V\to V$, $\mathbf{\omega}\mapsto\mathbf{M}^{(\mathbf{q})}$, are defined. The symmetricity of the inertia operator $\widehat J^{(\mathbf{q})}$ is proved, along with the formula $T^{(\mathbf{q})} =\frac12g(\widehat J^{(\mathbf{q})}\mathbf{\omega},\mathbf{\omega})$ for the kinetic energy of the point.
Geometric properties of the inertia operator $\widehat J$ 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 $\widehat J$ for a rigid body in three-dimensional 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 $s\in\{0,+,-\}$. The remaining signatures $(-,+,0)$ and $(-,0,0)$ are also realized by two- and three-point bodies.

Keywords: inertia tensor, rigid body, pseudo-Euclidean space, Lobachevsky plane, signature, isomorphism.

UDC: 514

Received: 02.01.2025
Accepted: 07.04.2025

DOI: 10.22405/2226-8383-2025-26-2-232-253