Mechanics of particles and systems

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• ConceptCentre of mass

  The centre of mass of a body is the average position of its mass, with each point weighted by the density there. For a body with a continuous density it is a quotient of triple integrals, or of double integrals for a plane lamina.

• TheoremShell theorem

  Let $R > 0$ and $M > 0$. Let $\mu$ be the measure on $\R^3$ of total mass $M$ spread uniformly over the sphere $S_R = \set{y : \abs{y} = R}$, that is, $\mu = \frac{M}{4\pi R^2} \, dS$ on $S_R$. Let $U^\mu(x) = \int \frac{d\mu(y)}{\abs{x - y}}$ be its Newtonian potential. Then $$U^\mu(x) = \frac{M}{\abs{x}} \quad (\abs{x} \ge R), \qquad U^\mu(x) = \frac{M}{R} \quad (\abs{x} \le R).$$ Consequently the gravitational force of the shell on a point mass is $0$ inside the shell. Outside the shell it equals the force of a point mass $M$ at the centre.

• Factd'Alembert's principle

  Consider $N$ particles with masses $m_i > 0$, positions $r_i(t) \in \R^3$ and accelerations $a_i = \ddot r_i$, for $i = 1, \dots, N$. Let $F_i$ be the applied force on particle $i$. Let the particles be subject to ideal constraints, that is, constraints whose reaction forces do no work on any virtual displacement. A virtual displacement at time $t$ is a family $(\delta r_1, \dots, \delta r_N)$ of vectors tangent to the constraints at fixed time $t$. Then the motion satisfies $$\sum_{i=1}^{N} (F_i - m_i a_i) \cdot \delta r_i = 0$$ for every virtual displacement $(\delta r_1, \dots, \delta r_N)$.

• FactGauss's principle of least constraint

  Consider $N$ particles with masses $m_i > 0$ and applied forces $F_i$, for $i = 1, \dots, N$, subject to ideal constraints that are holonomic or linear in the velocities. Fix a time $t$ and the positions and velocities of the particles at time $t$. Call a family $(b_1, \dots, b_N)$ of vectors of $\R^3$ an admissible acceleration if it is compatible with the constraints at these positions and velocities. Define the constraint (Gauss's function) $$Z(b_1, \dots, b_N) = \sum_{i=1}^{N} m_i \,\Big\lvert b_i - \frac{F_i}{m_i} \Big\rvert^2.$$ Then the actual accelerations $a_i = \ddot r_i(t)$ are the unique minimiser of $Z$ among admissible accelerations.

• ExampleRotation group SO(3)

  The rotation group is the special orthogonal group in dimension $3$: $$SO(3) = \set{R \in M_3(\R) : R^T R = I_3, \ \det R = 1}.$$ Each $R \in SO(3)$ is a rotation by some angle $\theta \in [0, \pi]$ about some line through the origin. For example, the rotation by $\theta$ about the $z$-axis is $$R_z(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix}.$$ Check: its columns are orthonormal, so $R_z(\theta)^T R_z(\theta) = I_3$, and its determinant is $\cos^2\theta + \sin^2\theta = 1$.

• ConceptScrew theory

  Screw theory describes the velocities of a rigid body and the forces on it by six-dimensional vectors called screws. Twists describe motions, and wrenches describe forces and moments. Its geometric basis is Chasles' theorem. It says that every orientation-preserving rigid motion of space is a screw motion. That is, it is a rotation about a line combined with a translation along the same line. A screw has an axis, a line in space, and a pitch, the ratio of translation to rotation. It is used in the kinematics and statics of mechanisms and robots. There the power of a wrench on a twist is given by a simple pairing.