Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → Classical mechanics ↓


Rigid body mobility


Rigid body dynamics is a branch of classical mechanics that investigates how solid bodies move under the influence of forces. Unlike the point masses considered in other areas of mechanics, rigid bodies have a definite size and shape, meaning that the relative positions of their constituent particles remain unchanged. This assumption simplifies the analysis of motion because it allows us to focus on rotational dynamics rather than on individual particles. Rigid body dynamics has wide applications, including mechanical engineering, robotics, aerospace, biomechanics, and more.

Basic definitions

A rigid body is defined as an object that has a uniform mass distribution and a defined shape that does not deform when forces are applied. The simplification is made by assuming that no distances change between any two particles inside the body as a result of the applied forces. In practice, objects are rarely perfectly rigid, but this assumption is true for many engineering materials under normal conditions.

Degrees of freedom

A rigid body in space typically has six degrees of freedom: three translational and three rotational. Translational degrees of freedom allow the body to rotate along the x, y, and z axes, while rotational degrees allow the body to rotate around these axes. These concepts are important to understand because they determine the possible movements a rigid body can undergo.

Translational and rotational motion

We can divide the motion of a rigid body into translational and rotational components. The translational component represents the motion of the body's center of mass (COM), while the rotational component deals with the motion of the body about its COM. An analogy to make this clear would be to imagine a frisbee spinning in the air. It moves in space and rotates around an axis.

Translational motion

Translational motion is described in terms of the center of mass. The center of mass can be thought of as the average location of the body's mass distribution. For translational motion:

F = ma

Here, F is the total force acting on the body, m is the mass, and a is the acceleration of the center of mass.

Rotational motion

The rotational motion of a rigid body is often more complex. It is governed by concepts such as moment of inertia and torque. Moment of inertia represents the resistance of the body to a change in rotational motion and depends on the distribution of mass around the axis of rotation. The equation governing rotational dynamics is:

T = Iα

In this equation, T is the torque, I is the moment of inertia, and α is the angular acceleration.

To visualise, imagine a rotating wheel. If the axis is free the wheel does not want to stop rotating instantly because it has moment of inertia.

Equations of motion for a rigid body

To understand the dynamics of a rigid body we use Newton's laws adapted for rotation:

Euler's equations

Euler's equations describe the rotational motion of a rigid body without taking into account translational forces:

I₁ω̇₁ - (I₂ - I₃)ω₂ω₃ = T₁
I₂ω̇₂ - (I₃ - I₁)ω₃ω₁ = T₂
I₃ω̇₃ - (I₁ - I₂)ω₁ω₂ = T₃

Here, I₁, I₂, I₃ are the principal moments of inertia and ω₁, ω₂, ω₃ are the angular velocity components about the principal axes. T₁, T₂, T₃ are the torques, respectively.

Applications in various fields

Robotics

Rigid body dynamics is fundamental in robotics for designing and controlling robotic arms. Understanding dynamics enables robots to perform complex tasks by calculating the torque required to achieve the desired motion.

Aerospace

In aerospace engineering, rigid body dynamics is used to predict the behavior of aircraft and spacecraft. Control systems are designed based on understanding the dynamics to ensure stability and maneuverability.

Mechanical Engineering

Many mechanical systems, such as gears and engines, rely on rigid body dynamics. Incorrect calculations can lead to inefficiency or failure, making this understanding critical in the design and testing phase.

Stability analysis

When analyzing dynamics, it is important to understand stability. Stability refers to the tendency of a body to return to a state after a disturbance. This is often explored through energy methods or linearizing the equations of motion.

Example problem

Consider a solid disk rotating with an initial angular velocity ω₀. If a torque T acts on it for time Δt, then the new angular velocity ω can be found using:

ω = ω₀ + (T/I)Δt

Let ω₀ = 5 rad/s, T = 10 Nm, I = 2 kg · m², and Δt = 3 s. Then:

ω = 5 + (10/2) * 3 = 5 + 15 = 20 rad/s

Conclusion

Rigid body dynamics is a fascinating and important area of study in classical mechanics, making it indispensable in a variety of fields where the motion of solid bodies is analyzed. Despite real materials not being perfectly rigid, the simplifications provided by rigid body assumptions allow engineers, scientists, and researchers to solve practical problems related to motion, leading to advances in technology and the understanding of our physical world.


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Rigid body mobility

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