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 ↓


Advanced Kinematics


Kinematics, an important branch of classical mechanics, delves deeply into the details of motion, focusing only on aspects of motion such as displacement, velocity, and acceleration, without considering the forces or masses that affect it. Advanced kinematics extends the basic principles learned in undergraduate physics by exploring complex systems and introducing more sophisticated mathematical tools for analyzing motion in three dimensions.

Kinematic equations for linear motion

In one-dimensional linear motion, the relationship between displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t) can be expressed by the following equations:

        v = u + at s = ut + 1/2at^2 v^2 = u^2 + 2as

These equations are fundamental to analyzing motion in a straight line. When advanced kinematics is considered, it extends these basic concepts to multidimensional motion.

Vector kinematics

In advanced kinematics, vectors are necessary to describe motion in two or three dimensions. Velocity and acceleration become vector quantities, described as:

        (vec{v} = frac{dvec{r}}{dt}) (vec{a} = frac{dvec{v}}{dt})

Here, (vec{r}) is the position vector. In the Cartesian coordinate system, vectors can be written as:

        (vec{r} = xhat{i} + yhat{j} + zhat{k}) (vec{v} = frac{dx}{dt}hat{i} + frac{dy}{dt}hat{j} + frac{dz}{dt}hat{k}) (vec{a} = frac{d^2x}{dt^2}hat{i} + frac{d^2y}{dt^2}hat{j} + frac{d^2z}{dt^2}hat{k})

Projectile motion

Projectile motion is a type of motion experienced by an object thrown near the surface of the Earth, moving along a curved path under the action of gravity only. Two-dimensional motion can be described without air resistance as follows:

        x = u_xt y = u_yt - frac{1}{2}gt^2

Here, (u_x) and (u_y) are the initial velocity components in the (x) and (y) directions, respectively. The path of the projectile is a parabola, which is the primary characteristic of projectile motion.

Nonlinear and curvilinear motion

In advanced kinematics, we consider motion that is not in a straight line. This study involves using curvilinear coordinates to describe motion along a curve in the plane or in space.

When dealing with curvilinear motion, it is useful to work with polar coordinates ((r, theta)). The position in polar coordinates is:

        (vec{r} = rhat{e}_r)

Velocity and acceleration in polar coordinates can be obtained as follows:

        (vec{v} = frac{dr}{dt}hat{e}_r + rfrac{dtheta}{dt}hat{e}_theta) (vec{a} = (frac{d^2r}{dt^2} - r(frac{dtheta}{dt})^2)hat{e}_r + (rfrac{d^2theta}{dt^2} + 2frac{dr}{dt}frac{dtheta}{dt})hat{e}_theta)

Rotational motion

Rotation around a fixed axis is an intrinsic part of advanced kinematics. Angular displacement, angular velocity and angular acceleration correspond to their linear counterparts:

        (theta = int omega dt) (omega = frac{dtheta}{dt}) (alpha = frac{domega}{dt})

Here, (theta) represents angular displacement, (omega) is angular velocity, and (alpha) represents angular acceleration.

Circular motion

Circular motion requires an understanding of centripetal acceleration, which is directed toward the center of the circular path. It can be given as follows:

        a_c = frac{v^2}{r} = romega^2

In this case, v is the linear velocity, r is the radius of the circle, and (omega) is the angular velocity.

Visual example: circular motion

Below is a visual depiction of a simple circular motion:

        
        
        
        Center
        R

In this example, the small circle at the center represents the center of the circular path. The radius r is represented as a line from the center to the edge of the circle.

Relative speed

Relative motion involves the motion of an object relative to another moving object. The velocity of object A relative to object B is expressed as:

        (vec{v}_{AB} = vec{v}_A - vec{v}_B)

This concept is extremely important when analyzing dynamic systems relative to different reference frames, such as observing the motion of a car from different vehicles.

Visual example: relative motion

Here's an example of relative motion:

        
        
        
        Car A
        Car B

In this view, car A moves from left to right as shown by the red arrow, and car B also moves in the same direction as shown by the blue arrow. Relative motion is the analysis of the motion of these objects relative to each other.

Dynamics in non-inertial frames

Non-inertial reference frames introduce pseudo forces into the kinetic equations. These forces, such as the centrifugal and Coriolis forces, arise from the acceleration of the reference frame itself.

The Coriolis force for an object moving with velocity ( vec{v'} ) in a frame rotating with angular velocity ( vec{Omega} ) is given by:

        (vec{F}_{Coriolis} = -2m(vec{Omega} times vec{v'}))

This idea is particularly important in aviation, meteorology, and astrophysics, where different frames of reference are often used.

Conclusion

Advanced kinematics forms an important foundation for explaining complex motion in multiple dimensions and in different frames of reference. Using vector calculus, polar coordinates, and non-inertial frames, advanced kinematics provides a comprehensive framework for understanding sophisticated dynamic systems. These tools and techniques are invaluable to engineers, physicists, and mathematicians who work with dynamic and complex systems in their respective fields.


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Advanced Kinematics

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