Physlib.ClassicalMechanics.RigidBody.Basic

For a rigid body $R$ in $d$-dimensional Euclidean space $\text{Space } d$, the total mass is the real number defined by the action of its mass distribution $\rho$ on the constant function $f(x) = 1$.

For a rigid body $R$ in $d$-dimensional Euclidean space $\text{Space } d$, the center of mass is a vector in $\text{Space } d$ whose $i$-th component (for $i \in \{0, \dots, d-1\}$) is defined as: \[ (\text{centerOfMass})_i = \frac{1}{M} \rho(x_i) \] where $M$ is the total mass of the rigid body and $\rho(x_i)$ is the result of the mass distribution $\rho$ acting on the $i$-th coordinate function $f(x) = x_i$. In standard physical notation, this corresponds to the integral calculation of the mean position: \[ \mathbf{r}_{\text{cm}} = \frac{1}{M} \int \mathbf{x} \, dm \]

For a rigid body $R$ in a $d$-dimensional Euclidean space, the inertia tensor is a $d \times d$ real matrix. Its entries $I_{ij}$ for $i, j \in \{0, \dots, d-1\}$ are defined by the action of the mass distribution $\rho$ of the body on the function: \[ f(x) = \delta_{ij} \left( \sum_{k=0}^{d-1} x_k^2 \right) - x_i x_j \] where $\delta_{ij}$ is the Kronecker delta (which is $1$ if $i = j$ and $0$ otherwise) and $x_k$ denotes the $k$-th coordinate of the position vector $x$. In standard physical notation, this represents the integral $I_{ij} = \int (\delta_{ij} \sum_k x_k^2 - x_i x_j) \, dm$.

For a rigid body $R$ in $d$-dimensional space, the entries $I_{ij}$ of its inertia tensor satisfy $I_{ij} = I_{ji}$ for any indices $i, j \in \{0, \dots, d-1\}$.

The kinetic energy $T$ of a rigid body is the sum of its translational kinetic energy and its rotational kinetic energy. It is defined by the expression: $$T = \frac{1}{2} M |\mathbf{v}|^2 + \frac{1}{2} \sum_{i,j} I_{ij} \omega_i \omega_j$$ where $M$ is the total mass of the body, $\mathbf{v}$ is the velocity of the center of mass, $\mathbf{I}$ is the inertia tensor (represented as a symmetric matrix $I_{ij}$), and $\boldsymbol{\omega}$ is the angular velocity vector of the body.

The motion of a rigid body is described using two coordinate systems: a fixed inertial coordinate system $(X, Y, Z)$ and a moving coordinate system $(x_1, x_2, x_3)$ that is rigidly attached to the body.

A rigid body in three-dimensional space $\mathbb{R}^3$ has $6$ degrees of freedom, which consist of $3$ translational degrees of freedom corresponding to the position of its center of mass and $3$ rotational degrees of freedom corresponding to its orientation.

In a rigid body, the velocity $\mathbf{v}$ of any point at position $\mathbf{r}$ (measured relative to the origin of the moving body-fixed coordinate system) is given by the sum of the translational velocity $\mathbf{V}$ of that origin and the velocity due to rotation, $\boldsymbol{\omega} \times \mathbf{r}$, where $\boldsymbol{\omega}$ is the angular velocity of the body. That is, $\mathbf{v} = \mathbf{V} + \boldsymbol{\omega} \times \mathbf{r}$.

The angular velocity vector $\boldsymbol{\omega}$ of a rigid body is independent of the choice of the coordinate system, provided that the coordinate system is fixed relative to the body.

The motion of a rigid body can be decomposed into a translation of a reference point and a rotation about that point. Specifically, there exists a time-dependent translational velocity vector $\mathbf{V}(t)$ and an angular velocity vector $\boldsymbol{\omega}(t)$ such that the velocity $\mathbf{v}$ of any point in the body at position $\mathbf{r}$ (measured relative to the reference point) is given by $\mathbf{v}(\mathbf{r}) = \mathbf{V} + \boldsymbol{\omega} \times \mathbf{r}$.

For a rigid body with total mass $M$, the center of mass moves as if the entire mass were concentrated at that single point and acted upon by the resultant external force $\sum \mathbf{F}_{ext}$. This motion is described by the equation: $$M \mathbf{a}_{CM} = \sum \mathbf{F}_{ext}$$ where $\mathbf{a}_{CM}$ is the acceleration vector of the center of mass.

The total angular momentum $\mathbf{L}$ of a rigid body about a fixed point $O$ is defined as the integral $\mathbf{L} = \int \mathbf{r} \times \mathbf{v} \, dm$. By decomposing the velocity of each point as $\mathbf{v} = \mathbf{V} + \boldsymbol{\omega} \times \mathbf{r}'$, where $\mathbf{V}$ is the velocity of the center of mass and $\mathbf{r}'$ is the position relative to the center of mass, the total angular momentum can be expressed as: $$\mathbf{L} = \mathbf{R} \times (M \mathbf{V}) + \mathbf{I}_{CM} \boldsymbol{\omega}$$ where $\mathbf{R}$ is the position vector of the center of mass relative to $O$, $M$ is the total mass of the body, $\boldsymbol{\omega}$ is the angular velocity, and $\mathbf{I}_{CM}$ is the inertia tensor of the body calculated relative to its center of mass.

In an inertial frame, the translational equation of motion for a rigid body states that the rate of change of the total linear momentum $\mathbf{P}$ with respect to time is equal to the total external force $\mathbf{F}$ acting on the body: $$\frac{d\mathbf{P}}{dt} = \mathbf{F}$$ where $\mathbf{P}$ is the total linear momentum and $\mathbf{F}$ is the total external force.

In an inertial frame, the rotational equation of motion for a rigid body about its center of mass states that the rate of change of the total angular momentum $\mathbf{M}$ with respect to time is equal to the total external torque $\mathbf{K}$: $$\frac{d\mathbf{M}}{dt} = \mathbf{K}$$ where $\mathbf{M}$ is the total angular momentum and $\mathbf{K}$ is the total external torque.

The total kinetic energy $T$ of a rigid body can be decomposed into translational and rotational components relative to its center of mass: $$T = \frac{1}{2} M |\mathbf{V}|^2 + \frac{1}{2} \boldsymbol{\omega} \cdot \mathbf{I}_{CM} \boldsymbol{\omega}$$ where $M$ is the total mass of the body, $\mathbf{V}$ is the velocity of the center of mass, $\boldsymbol{\omega}$ is the angular velocity vector, and $\mathbf{I}_{CM}$ is the inertia tensor calculated with respect to the center of mass.

The parallel-axis theorem states that the inertia tensor $\mathbf{I}_{O'}$ of a rigid body about a point $O'$ is related to the inertia tensor $\mathbf{I}_O$ about its center of mass $O$ by the expression: $$\mathbf{I}_{O'} = \mathbf{I}_O + M (|\mathbf{a}|^2 \mathbf{1} - \mathbf{a} \otimes \mathbf{a})$$ where $M$ is the total mass of the body, $\mathbf{a}$ is the displacement vector from the center of mass $O$ to the point $O'$, $\mathbf{1}$ is the identity tensor, and $\otimes$ denotes the outer product.

The principal axes of inertia of a rigid body are defined as the directions of the vectors forming an orthonormal basis in which the inertia tensor \( I \) is diagonal. Given that the inertia tensor is a real symmetric tensor, such an orthonormal basis always exists.

For a rigid body with principal moments of inertia denoted by $I_1, I_2,$ and $I_3$, the value of any one principal moment cannot exceed the sum of the other two. Mathematically, this is expressed by the triangle inequalities: $$I_1 \le I_2 + I_3$$ $$I_2 \le I_1 + I_3$$ $$I_3 \le I_1 + I_2$$

An asymmetrical top is defined as a rigid body whose three principal moments of inertia, denoted by $I_1, I_2,$ and $I_3$, are all distinct from one another. This condition is expressed as $I_1 \neq I_2$, $I_2 \neq I_3$, and $I_3 \neq I_1$.

A symmetrical top is defined as a rigid body for which exactly two of its three principal moments of inertia, denoted as $I_1, I_2,$ and $I_3$, are equal to each other. This implies that the third principal moment is distinct from the other two, a condition expressed as $(I_1 = I_2 \neq I_3)$ or its permutations.

A spherical top is a rigid body whose three principal moments of inertia, denoted by $I_1, I_2,$ and $I_3$, are all equal. This condition is expressed as $I_1 = I_2 = I_3$.

A rotating body-fixed frame is a coordinate system attached to a rigid body that rotates with the body relative to an inertial (fixed) reference frame. This frame is mathematically characterized by its angular velocity vector $\boldsymbol{\Omega}(t)$ at time $t$.

Let $\mathbf{A}$ be a vector. The time derivative in the rotating frame, denoted as $\frac{d'\mathbf{A}}{dt}$, is the derivative of the components of $\mathbf{A}$ with respect to time when expressed in the rotating (body-fixed) reference frame.

For any vector field $\mathbf{A}(t)$, the relationship between its time derivative in an inertial frame and its time derivative in a frame rotating with angular velocity $\boldsymbol{\Omega}$ is given by: $$\left(\frac{d\mathbf{A}}{dt}\right)_{\text{inertial}} = \left(\frac{d\mathbf{A}}{dt}\right)_{\text{rotating}} + \boldsymbol{\Omega} \times \mathbf{A}$$ where $\left(\frac{d\mathbf{A}}{dt}\right)_{\text{rotating}}$ denotes the derivative as observed from within the rotating frame, also commonly denoted as $\frac{d'\mathbf{A}}{dt}$.

In the context of rigid body dynamics, let $\mathbf{P}$ denote the linear momentum and $\boldsymbol{\Omega}$ denote the angular velocity of a rotating reference frame. The relation between the time derivative of the linear momentum in the inertial frame, $(d\mathbf{P}/dt)_{\text{inertial}}$, and its derivative in the rotating frame, $d'\mathbf{P}/dt$, is given by the transport theorem: $$\left(\frac{d\mathbf{P}}{dt}\right)_{\text{inertial}} = \frac{d'\mathbf{P}}{dt} + \boldsymbol{\Omega} \times \mathbf{P}$$ By applying Newton's second law, which states that the rate of change of linear momentum in the inertial frame equals the total external force $\mathbf{F}$ (i.e., $\mathbf{F} = (d\mathbf{P}/dt)_{\text{inertial}}$), the equation of motion in the rotating frame is: $$\frac{d'\mathbf{P}}{dt} + \boldsymbol{\Omega} \times \mathbf{P} = \mathbf{F}$$

In the context of rigid body dynamics, let $\mathbf{M}$ denote the angular momentum and $\boldsymbol{\Omega}$ denote the angular velocity of a rotating reference frame. The relation between the time derivative of the angular momentum in the inertial frame, $(d\mathbf{M}/dt)_{\text{inertial}}$, and its derivative in the rotating frame, $d'\mathbf{M}/dt$, is given by the transport theorem: $$\left(\frac{d\mathbf{M}}{dt}\right)_{\text{inertial}} = \frac{d'\mathbf{M}}{dt} + \boldsymbol{\Omega} \times \mathbf{M}$$ By applying Newton's second law for rotation, which states that the rate of change of angular momentum in the inertial frame equals the total external torque $\mathbf{K}$ (i.e., $\mathbf{K} = (d\mathbf{M}/dt)_{\text{inertial}}$), the equation of motion in the rotating frame is: $$\frac{d'\mathbf{M}}{dt} + \boldsymbol{\Omega} \times \mathbf{M} = \mathbf{K}$$

In the reference frame of a rigid body where the axes are aligned with the principal axes of inertia (such that the inertia tensor is diagonal with principal moments $I_1, I_2, I_3$), the equations of rotational motion, known as Euler's equations, are given by: $$I_1 \frac{d\omega_1}{dt} + (I_3 - I_2) \omega_2 \omega_3 = M_1$$ $$I_2 \frac{d\omega_2}{dt} + (I_1 - I_3) \omega_3 \omega_1 = M_2$$ $$I_3 \frac{d\omega_3}{dt} + (I_2 - I_1) \omega_1 \omega_2 = M_3$$ where $\omega_i$ are the components of the angular velocity vector along the principal axes and $M_i$ are the components of the external torque acting on the body about its center of mass.

A rigid body can perform steady (uniform) rotation about any of its principal axes if the component of the external torque $\mathbf{K}$ about that axis vanishes. Mathematically, if the angular velocity vector $\boldsymbol{\Omega}$ is aligned with a principal axis of the inertia tensor (i.e., $\boldsymbol{\Omega} = \Omega_i \mathbf{e}_i$), the condition for steady rotation $\frac{d\boldsymbol{\Omega}}{dt} = 0$ is satisfied when the torque $K_i = 0$. Furthermore, the stability of such a rotation is determined by the relative ordering of the principal moments of inertia $I_1, I_2$, and $I_3$.

For a rigid body with principal moments of inertia $I_1, I_2, I_3$ (where $I_1 < I_2 < I_3$), steady rotation about the principal axes corresponding to the largest ($I_3$) and smallest ($I_1$) moments of inertia is stable under small perturbations. In contrast, rotation about the principal axis corresponding to the intermediate moment of inertia ($I_2$) is unstable. This phenomenon is known as the tennis-racket effect.

If a rigid body is confined to planar motion, its dynamics reduce to a two-dimensional problem. In this case, the inertia tensor simplifies to a scalar moment of inertia $I$, and the rotation is described by a single angular velocity $\omega$.

The power $P$ delivered to a rigid body by a system of forces $F_i$ applied at points with velocities $v_i$ is given by $P = \sum F_i \cdot v_i$. This power can be decomposed into translational and rotational contributions as $P = F_{\text{tot}} \cdot V + M \cdot \omega$, where $F_{\text{tot}} = \sum F_i$ is the total force acting on the body, $V$ is the velocity of the chosen reference point, $M$ is the total torque (moment of force) about that reference point, and $\omega$ is the angular velocity of the rigid body.

Small oscillations of a rigid body about a stable equilibrium orientation are governed by linearized equations of motion obtained by expanding the system's energy to second order in the angular displacements $\theta_i$. The resulting normal modes and frequencies of oscillation are determined by the body's inertia tensor and the restoring torques.