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 → Lagrangian mechanics ↓


Constraints and normalized coordinates


Lagrangian mechanics presents an elegant reformulation of classical mechanics, through which we can effectively describe complex mechanical systems. One of the main components of this framework involves understanding constraints and choosing appropriate generalized coordinates. These concepts allow us to significantly simplify the problem by reducing the number of degrees of freedom, focusing only on the key variables that are necessary to describe a system without redundancy.

Understanding the odds

In classical mechanics, various types of constraints can govern the motion of a system. These constraints can be physical limitations on the motion of particles or objects within a system. They help simplify a real-world problem by defining boundaries and rules, thereby reducing the complexity of the equations governing the motion.

Constraints in mechanical systems are typically classified as holonomic or non-holonomic, and further divided into scleronomic and rheonomic.

  • Holonomic constraints: These are constraints that can be expressed as equations directly involving coordinates. For example, consider a bead sliding on a wire of a given shape. This constraint can be expressed through an equation involving x, y and z coordinates.
  • Example equation:
    x^2 + y^2 + z^2 = R^2
    This equation represents a spherical restriction where a particle is restricted to move on the surface of a sphere of radius R.

An important property of holonomic constraints is that they are integrable, that is, they can be expressed as functions involving only coordinates and times, but not velocities.

  • Non-holonomic constraints: These constraints cannot be expressed in terms of coordinates alone. Instead, they involve inequalities or differential relations involving derivatives. An example is a rolling wheel that can slip but is generally constrained by a non-holonomic relation connecting its velocity and the relative contact speed.
  • Example equation:
    v_x - rw_θ = 0
    This equation expresses the constraint between the translational velocity (v_x) and the rotational speed (θ) of a rotating wheel, where (r) is the radius.

Both holonomic and non-holonomic constraints can significantly influence the potential evolution of a system, and can influence which aspects of the system require special attention due to functional dependencies or limitations.

Exploring generalized coordinates

To efficiently analyze systems subject to constraints, we use a set of independent variables, known as generalized coordinates, denoted as (q_1, q_2, ldots, q_n). These coordinates serve as the minimum set necessary to describe the configuration of the system, effectively capturing all possible states while greatly simplifying the mathematical treatment of the system dynamics.

Let's consider the simple system of a double pendulum - a complex and interesting physical system consisting of two arms, each of which is of a fixed length and attached to each other. The key is to find a set of variables that succinctly describes the position of both bobs in the pendulum. Instead of using Cartesian coordinates (x,y) for each bob, we may find it more effective to describe the system using angular displacements:

  • (θ_1): Angle of the first pendulum arm relative to the vertical.
  • (θ_2): Angle of the second side with respect to the first side.

The equations of motion using these angles are much simpler than using Cartesian coordinates. We capture all the complex movements of the arms using only two variables and our understanding of angular relationships within the limits of the pendulum rods.

Mathematical formulation of generalized coordinates

We consider a system with N degrees of freedom, represented by N generalized coordinates. Let's look at how these are formulated mathematically within Lagrangian mechanics:

The Lagrangian (L) of a system is given as the difference between its kinetic energy (T) and potential energy (V):

L(q, dot{q}, t) = T(q, dot{q}, t) - V(q, t)

Here, (q) denotes the vector of all normalized coordinates, and (dot{q}) denotes their time derivatives.

normalized coordinates

Once this Lagrangian is established, we apply the Euler-Lagrange equations, which provide a set of differential equations that describe the dynamics of the system:

frac{d}{dt} left ( frac{partial L}{partial dot{q}_i} right ) - frac{partial L}{partial q_i} = 0

This equation applies to each generalized coordinate (q_i). The choice of generalized coordinates often determines the structure and simplicity of these equations, highlighting their importance in mechanical analysis.

Advantages and examples of generalized coordinates

A major advantage of using generalized coordinates is their flexibility and adaptability to suit the nature of the system being analyzed. They often correspond closely to the geometric or internal variables of the system, making them physically intuitive and mathematically convenient.

Example 1: Bead on a circular wire
Consider a bead that moves on a circular wire. Instead of specifying the position of the bead using Cartesian coordinates (x, y), we can use the angle (θ) as a generalized coordinate that is more natural and directly related to the constraints of the system.

θ

Example 2: Double pendulum
As discussed earlier, this system can be effectively described using two angular coordinates, greatly reducing the complexity of its analysis compared to using Cartesian coordinates for both pendulum bobs.

Conclusion

In short, constraints and generalized coordinates form the backbone of simplifying and analyzing complex mechanical systems in Lagrangian mechanics. By choosing appropriate generalized coordinates, we can take into account the constraints imposed on a system, focusing on the essential variables that efficiently describe the dynamics of the system. This not only aids theoretical understanding but also facilitates practical problem solving in diverse fields ranging from mechanical engineering to robotics and beyond.

Understanding the delicate interrelationship between constraints and judicious choice of generalized coordinates enhances our ability to manipulate the equations of motion, making otherwise difficult problems in classical mechanics easier to solve.


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Constraints and normalized coordinates

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