Constrains and constrained motion

• A constrained motion is a motion which can not proceed arbitrary in any manner.
• Particle motion can be restricted to occur (1) along some specified path (2) on surface (plane or curved) arbitrarily oriented in space.
• Imposing constraints on a mechanical system is done to simplify the mathematical description of the system.
• Constraints expressed in the form of equation f(x₁,y₁,z₁,......,x_n,y_n,z_n :t)=0 are called holonomic constraints.
• Constraints not expressed in this fashion are called non-holonomic constraints.
• Scleronomic conatraints are independent of time.
• Constraints containing time explicitely are called rehonomic.
• Therefore a constraint is either

          "Scleronomic where constraints relations does not depend on time or rheonomic where constraints relations depends explicitly on time "
          and either
           "holonomic where constraints relations can be made independent of velocity or non-holonomic where these relations are irreducible functions of velocity"

Constraints types of some physicsl systems are given below in the table

Wave velocity in a continuous system

• Any system whose particle motion are governed by classical wave equation is a system in which harmonic waves of any wavelength can travel with the speed v
  
  
• The value of v depends on the elastic and inertial properties of the system under consideration.
  

(1) Transverse wave on a stretched string

• Displacement of the string is governed by the equation
  
  
  
  
  
  
  Where T is the tension and µ is the linear density (mass per unit length of the string)
  
• Velocity of wave on the string is
  v=√(T/μ)
  v is the velocity of the wave.
  
• Medium through which waves travel will offer impedance to these waves.
  
• If the medium is loss less i.e., it does not have any resistive or dissipative components, the impedance is solely determined by its inertia and elasticity.
  
• Characteristic impedance of string is determined by
  Z=T/v=√(μT)=μv
  
• Since v is determined by the inertia and elasticity this shows that impedance is also governed by these two properties of the medium.
  
• For loss-less medium impedance is real quantity and it is complex if the medium is dissipative.
  

(2) Longitudinal waves in uniform rod

• Equation for longitudinal vibrations of a uniform rod is
  
  
  
  
  
  
  
• where ξ (x,t )→displacement
  Y is young’s modulus of the rod
  ρ is the density
  
• Velocity of longitudinal wave in rod is
  v=√(Y/ρ)
  

(3) Electromagnetic waves in space

• When electric and magnetic field vary in time they produce EM waves.
  
• An oscillating charge has an oscillating electric and magnetic fields around it and hence produces EM waves.
  
• Example: - (1) Electrons falling from higher to lower energy orbit radiates EM waves of particular wavelength and frequency. (2) The motion of electrons in an antenna radiates EM waves by a process called Bramstrhlung.
  
• Propagation of EM waves in a medium is also due to inertial and elastic properties of the medium.
  
• Every medium (including vacuum) has inductive properties described by magnetic permeability µ of the medium.
  
• This property provides magnetic inertia of the medium.
  
• Elasticity of the medium is provided by the capacitive property called electrical permittivity ε of the medium.
  
• Permeability µ stores magnetic energy and the permittivity ε stores the electric field energy.
  
• This EM energy propagates in the medium in the form of EM waves.
  
• Electric and magnetic fields are connected by Maxwell’s Equations (dielectric medium)
  ∇×H =ε (∂E )/∂t
  ∇×E = - μ (∂H⃗)/∂t
  ε(∇∙E⃗)=ρ
  ∇∙H =0
  
• Here in above equations E ⃗ is electric field , H ⃗ is the magnetic field and ρ is charge density
  

SYLLABUS FOR PHYSICAL SCIENCES PAPER I AND PAPER II

The full Syllabus for Part B of Paper I and Part B of Paper II.

The syllabus for Part A of Paper II comprises Sections I-VI.

I. Mathematical Methods of Physics

Dimensional analysis; Vector algebra and vector calculus; Linear algebra, matrices, Cayley Hamilton theorem, eigenvalue problems; Linear differential equations; Special functions (Hermite, Bessel, Laguerre and Legendre); Fourier series, Fourier and Laplace transforms; Elements of complex analysis: Laurent series-poles, residues and evaluation of integrals; Elementary ideas about tensors; Introductory group theory, SU(2), O(3); Elements of computational techniques: roots of functions, interpolation, extrapolation, integration by trapezoid and Simpson’s rule, solution of first order differential equations using Runge-Kutta method; Finite difference methods; Elementary probability theory, random variables, binomial, Poisson and normal distributions.

II. Classical Mechanics

Newton’s laws; Phase space dynamics, stability analysis; Central-force motion; Two-body collisions, scattering in laboratory and centre-of-mass frames; Rigid body dynamics, moment of inertia tensor, non-inertial frames and pseudoforces; Variational principle, Lagrangian and Hamiltonian formalisms and equations of motion; Poisson brackets and canonical transformations; Symmetry, invariance and conservation laws, cyclic coordinates; Periodic motion, small oscillations and normal modes; Special theory of relativity, Lorentz transformations, relativistic kinematics and mass–energy equivalence.

III. Electromagnetic Theory

Electrostatics: Gauss’ Law and its applications; Laplace and Poisson equations, boundary value problems; Magnetostatics: Biot-Savart law, Ampere's theorem, electromagnetic induction; Maxwell's equations in free space and linear isotropic media; boundary conditions on fields at interfaces; Scalar and vector potentials; Gauge invariance; Electromagnetic waves in free space, dielectrics, and conductors; Reflection and refraction, polarization, Fresnel’s Law, interference, coherence, and diffraction; Dispersion relations in plasma; Lorentz invariance of Maxwell’s equations; Transmission lines and wave guides; Dynamics of charged particles in static and uniform electromagnetic fields; Radiation from moving charges, dipoles and retarded potentials.

IV. Quantum Mechanics

Wave-particle duality; Wave functions in coordinate and momentum representations; Commutators and Heisenberg's uncertainty principle; Matrix representation; Dirac’s bra and ket notation; Schroedinger equation (time-dependent and time-independent); Eigenvalue problems such as particle-in-a-box, harmonic oscillator, etc.; Tunneling through a barrier; Motion in a central potential; Orbital angular momentum, Angular momentum algebra, spin; Addition of angular momenta; Hydrogen atom, spin-orbit coupling, fine structure; Time-independent perturbation theory and applications; Variational method; WKB approximation;

Time dependent perturbation theory and Fermi's Golden Rule; Selection rules; Semi-classical theory of radiation; Elementary theory of scattering, phase shifts, partial waves, Born approximation; Identical particles, Pauli's exclusion principle, spin-statistics connection; Relativistic quantum mechanics: Klein Gordon and Dirac equations.

V. Thermodynamic and Statistical Physics

Laws of thermodynamics and their consequences; Thermodynamic potentials, Maxwell relations; Chemical potential, phase equilibria; Phase space, micro- and macrostates; Microcanonical, canonical and grand-canonical ensembles and partition functions; Free Energy and connection with thermodynamic quantities; First- and second-order phase transitions; Classical and quantum statistics, ideal Fermi and Bose gases; Principle of detailed balance; Blackbody radiation and Planck's distribution law; Bose-Einstein condensation; Random walk and Brownian motion; Introduction to nonequilibrium processes; Diffusion equation.

VI. Electronics

Semiconductor device physics, including diodes, junctions, transistors, field effect devices, homo and heterojunction devices, device structure, device characteristics, frequency dependence and applications; Optoelectronic devices, including solar cells, photodetectors, and LEDs; High-frequency devices, including generators and detectors; Operational amplifiers and their applications; Digital techniques and applications (registers, counters, comparators and similar circuits); A/D and D/A converters; Microprocessor and microcontroller basics.

VII. Experimental Techniques and data analysis

Data interpretation and analysis; Precision and accuracy, error analysis, propagation of errors, least squares fitting, linear and nonlinear curve fitting, chi-square test; Transducers (temperature, pressure/vacuum, magnetic field, vibration, optical, and particle detectors), measurement and control; Signal conditioning and recovery, impedance matching, amplification (Op-amp based, instrumentation amp, feedback), filtering and noise reduction, shielding and grounding; Fourier transforms; lock-in detector, box-car integrator, modulation techniques.

Applications of the above experimental and analytical techniques to typical undergraduate and graduate level laboratory experiments.

VIII. Atomic & Molecular Physics

Quantum states of an electron in an atom; Electron spin; Stern-Gerlach experiment; Spectrum of Hydrogen, helium and alkali atoms; Relativistic corrections for energy levels of hydrogen; Hyperfine structure and isotopic shift; width of spectral lines; LS & JJ coupling; Zeeman, Paschen Back & Stark effect; X-ray spectroscopy; Electron spin resonance, Nuclear magnetic resonance, chemical shift; Rotational, vibrational, electronic, and Raman spectra of diatomic molecules; Frank – Condon principle and selection rules; Spontaneous and stimulated emission, Einstein A & B coefficients; Lasers, optical pumping, population inversion, rate equation; Modes of resonators and coherence length.

IX. Condensed Matter Physics

Bravais lattices; Reciprocal lattice, diffraction and the structure factor; Bonding of solids; Elastic properties, phonons, lattice specific heat; Free electron theory and electronic specific heat; Response and relaxation phenomena; Drude model of electrical and thermal

conductivity; Hall effect and thermoelectric power; Diamagnetism, paramagnetism, and ferromagnetism; Electron motion in a periodic potential, band theory of metals, insulators and semiconductors; Superconductivity, type – I and type - II superconductors, Josephson junctions; Defects and dislocations; Ordered phases of matter, translational and orientational order, kinds of liquid crystalline order; Conducting polymers; Quasicrystals.

X. Nuclear and Particle Physics

Basic nuclear properties: size, shape, charge distribution, spin and parity; Binding energy, semi-empirical mass formula; Liquid drop model; Fission and fusion; Nature of the nuclear force, form of nucleon-nucleon potential; Charge-independence and charge-symmetry of nuclear forces; Isospin; Deuteron problem; Evidence of shell structure, single- particle shell model, its validity and limitations; Rotational spectra; Elementary ideas of alpha, beta and gamma decays and their selection rules; Nuclear reactions, reaction mechanisms, compound nuclei and direct reactions; Classification of fundamental forces; Elementary particles (quarks, baryons, mesons, leptons); Spin and parity assignments, isospin, strangeness; Gell-Mann-Nishijima formula; C, P, and T invariance and applications of symmetry arguments to particle reactions, parity non-conservation in weak interaction; Relativistic kinematics.