PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → Classical mechanics → Newtonian mechanics ↓


Central force motion


In the field of classical mechanics, the concept of central force motion plays a vital role in understanding the interactions and motion of objects under the influence of forces emanating from a central point. Central force motion is particularly important when analyzing systems such as a planet orbiting a star, an electron orbiting the nucleus of an atom, or any scenario where the force is directed toward or away from a central point. In this detailed discussion, we will delve deep into the framework of central force motion, providing a comprehensive explanation that is both theoretically rich and accessible in terms of simplicity and clarity.

What is centripetal force?

Central force is the force that acts along the line connecting the center of the force field and the point of action (the object experiencing the force). This means that the force vector always points toward or away from a fixed point and its magnitude depends only on the distance from that point.

Mathematically, the central force F can be expressed as:

F = f(r) * r̂

where f(r) is a scalar function of the radial distance r, and r̂ is the unit vector in the direction of the radius. This shows that the force is radially symmetric and does not depend on the angles (such as azimuthal or polar angles) describing the position of the object.

Examples of centripetal forces

To understand central force motion, let's examine some common examples where central forces are applied in physics:

  • Gravitational force: This is a classic example where the force between two point masses is directed along the line connecting them. The force follows the inverse-square law, which is central in nature.
    • F = -G * (m1 * m2) / r² * r̂
  • Coulombic interaction: The force between two charged particles is also a central force, which again follows the inverse-square law.
    • F = k * (q1 * q2) / r² * r̂
  • Elastic force: In a perfectly elastic system such as a simple harmonic oscillator, the restoring force is central and depends linearly on the displacement.
    • F = -k * x

Mathematical framework for central force motion

The dynamics of central force motion can be understood through Newton's laws of motion. For a particle of mass m moving under a central force, Newton's second law can be expressed in radial coordinates (assuming only radial dependence) as follows:

m * d²r/dt² = f(r)

Since the central force does not act in any direction other than the radial direction, the angular momentum L is conserved. Therefore, we have:

L = r × mv = constant

where v is the tangential velocity. Conservation of angular momentum implies that the momentum depends primarily on the radius and the force. We can analyze this further using effective potential energy.

Effective potential energy

The concept of effective potential energy helps us analyze radial momentum by incorporating angular momentum into the potential energy framework. The total energy E of the system can be written as:

E = 1/2 * m * (dr/dt)² + L²/(2mr²) + U(r)

Here, U(r) represents the potential energy corresponding to the radial force f(r). The L²/(2mr²) term is derived from the angular momentum and serves as an additional potential energy term, known as the centrifugal potential.

Visualization of central force motion

Let us imagine paths or orbits generated by central forces. Consider a scenario where the force is gravity, obeying the inverse-square law.

Center

In this visual example, the black dot represents a central point, such as the Sun in our solar system. The blue dot is an orbiting body, such as a planet. The gray line represents the radius vector, and the red curve represents an elliptical orbit, a typical path for central forces such as gravity.

Kepler's laws of planetary motion

As a result of central force motion, particularly gravity, Johannes Kepler formulated three laws of planetary motion that are important in understanding celestial mechanics:

  1. Law of Orbits: All planets move in elliptical orbits with the Sun at one focus.
  2. Law of Area: The line segment joining a planet and the Sun covers equal area in equal interval of time.
  3. Law of Period: The square of the period of revolution of a planet is proportional to the cube of the semi-major axis of its orbit.

Stability in central force motion

The stability of orbits in central force systems depends on the nature of the potential function. For potentials such as the gravitational or electrostatic inverse-square law, orbits are stable, whereas, in other cases, they may be unstable.

Conic sections and central force

The trajectory or path of a body subject to a central force can take the form of conic sections, depending on the energy and angular momentum of the system:

  • Elliptical: This occurs when the total energy is negative, which is typical of bound systems such as planets in the Solar System.
  • Parabola: This occurs when the total energy is zero, it is a marginal escape scenario.
  • Hyperbola: It arises when the total energy is positive, it is representative of disjoint systems.

The concept of reduced mass

In a two-body central force problem, such as the Earth and the Moon, it is convenient to use the concept of a reduced mass μ, rather than analyzing each body independently. This simplifies the two-body problem into a one-body problem:

μ = (m1 * m2) / (m1 + m2)

By using reduced masses and focusing on the motion of only one particle relative to the other, we can simplify the complexities involved in dealing with the two-body central force problem.

Lagrangian and Hamiltonian formulation

For a deeper exploration of central force motion, Lagrangian and Hamiltonian mechanics can be used:

Lagrangian mechanics:

This involves considering a function called the Lagrangian L, which is expressed as the difference between kinetic and potential energy:

L = T - U

The action principle leads to the Euler–Lagrange equations, which govern the dynamics of the central motion.

Hamiltonian mechanics:

The Hamiltonian H is the total energy of the system expressed in terms of coordinates and conjugate momenta:

H = T + U

By reformulating the momentum equations via Hamilton's equations, an alternative powerful insight into central force dynamics can be obtained.

Conclusion

The study of central force motion in Newtonian mechanics provides invaluable insight into the behavior of particles and bodies subjected to forces directed toward a central point. Through the conservation of angular momentum and energy and the analytical frameworks available in advanced mechanics, such as the Lagrangian and Hamiltonian methods, central force problems are solved with astonishing precision. Whether investigating planetary orbits, atomic structures, or any radially symplectic systems, the principles of central force motion form the basis from which a deeper understanding develops.


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Central force motion

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