IIT Madras Course , Prof. V. Balakrishnan

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IIT Madras Course , Prof. V. Balakrishnan

Mechanics of a system of particles in vector form. Conservation of linear momentum, energy and angular momentum - Degrees of freedom, generalized coordinates and velocities. Lagrangian, action, and the principle of extremal action. Euler-Lagrange equations. Constraints. Applications of the Lagrangian formalism - Legendre transform, generalized momenta. Hamiltonian and Hamilton's equations of motion. Phase space, phase trajectories. Applications to systems with a small number of degrees of freedom - Scattering in a central potential, Rutherford formula, scattering cross section - Charged particle in an electromagnetic field - Non-inertial frames of reference and pseudo-forces: centrifugal, Coriolis and Euler forces - Rotations in three dimensions, Euler angles. Angular momentum. Elements of rigid-body dynamics. The symmetric top - Small oscillations. Normal mode analysis. Normal modes of simple molecules and harmonic chains - Elementary ideas on general dynamical systems: conservative versus dissipative systems. Hamiltonian systems and Liouville's theorem - Constants of the motion and symmetries in classical mechanics. Noether's Theorem and the connection between symmetry, invariance and conservation principles - Canonical transformations, Poisson brackets. Action-angle variables. Integrable Hamiltonians. Non-integrable systems and elements of chaotic motion - Special relativity: Inertial frames. Principle and postulate of Relativity. Lorentz transformations. Length contraction, time dilation and the Doppler effect. Velocity addition formula. Rapidity - Four-vectors and tensors. Energy-momentum of a free particle. Relativistic invariance of physical laws. The wave equation. Transformation properties of electric and magnetic fields. The homogeneous and inhomogeneous Lorentz groups

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2.3 (7 Ratings)

Topics in Nonlinear Dynamics by Prof. V. Balakrishnan,Department of Physics,IIT Madras.For more details on NPTEL visit httpnptel.ac.in

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- 1.Overview
- 2.Critical points of a dynamical system
- 3.Two-dimensional flows
- 4.Stable and unstable manifolds
- 5.Hamiltonian dynamics (Part I)
- 6.Hamiltonian dynamics (Part II)
- 7.Hamiltonian dynamics (Part III)
- 8.Hamiltonian dynamics (Part IV)
- 9.Hamiltonian dynamics (Part V)
- 10.Elementary bifurcations
- 11.Limit cycles
- 12.PoincarÂ´e index
- 13.Illustrative examples
- 14.Quiz 1. Questions and answers
- 15.Bead on a rotating hoop
- 16.Types of dynamical behaviour
- 17.Discrete time dynamics (Part I)
- 18.Discrete time dynamics(Part II)
- 19.Discrete time dynamics (Part III)
- 20.Discrete time dynamics (Part IV)
- 21.Coarse-grained dynamics in phase space (Part I)
- 22.Coarse-grained dynamics in phase space (Part II)
- 23.Stochastic dynamics (Part II)
- 24.Stochastic dynamics (Part III)
- 25.Coarse-grained dynamics in phase space (Part IV)
- 26.Discrete time dynamics (Part V)
- 27.Quiz 2. Questions and answers
- 28.Stochastic dynamics (Part V)
- 29.Stochastic dynamics (Part VI)

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