In these “Classical DynamicsNotes PDF”, you will study the problem solving ability and develops understanding of physical problems. The emphasis of these notes is to enhance the understanding of Classical Mechanics (Lagrangian and Hamiltonian Approach).
The topics we will cover will be taken from the following list:
Classical Mechanics of Point Particles: Review of Newtonian Mechanics; Application to the motion of a charge particle in external electric and magnetic fields- motion in uniform electric field, magnetic field- gyroradius and gyro-frequency, motion in crossed electric and magnetic fields. Degrees of freedom of a system, Generalized coordinates and velocities. Hamilton’s Principle, Lagrangian and Lagrange’s equations of motion of one- dimensional simple harmonic oscillators, falling body in uniform gravity. Cyclic coordinates. Canonical momenta & Hamiltonian. Hamilton’s equations of motion. Comparison of Newtonian, Lagrangian and Hamiltonian mechanics. Applications of Hamiltonian mechanics: Hamiltonian for a simple harmonic oscillator, solution of Hamilton’s equations for simple harmonic oscillations (1-D), particle in a Central Force Field – conservation of angular momentum and energy.
Small Amplitude Oscillations: Minima of potential energy and points of stable equilibrium, small amplitude oscillations about the minimum, normal modes of longitudinal simple harmonic oscillations (maximum 3 masses connected by 4 springs). Kinetic energy (T) and potential energy (V) in terms of normal co-ordinates. T and V matrices: finding eigen-frequencies and eigen-vectors using these matrices.
Special Theory of Relativity: Postulates of Special Theory of Relativity. Lorentz Transformations. Minkowski space. The invariant interval, light cone and world lines. Space-time diagrams: Time-dilation, Length contraction, Simultaneity.
Four -vectors: space-like, time-like and light-like. Four-displacement four velocity, four acceleration four-space. Four-momentum and energy-momentum relation. Doppler effect from a 4-vector perspective. Application to two-body decay of an unstable particle. Metric tensor and alternating tensors and their properties.
Fluid Dynamics: Density ρ and pressure P in a fluid, an element of fluid and its velocity, continuity equation and mass conservation, stream-lined motion, laminar flow, Poiseuille’s equation for flow of a liquid through a pipe. Analogy between liquid flow and current flow, rate of liquid flow through capillaries in series and in parallel combination. Navier Stoke’s equation, Reynolds number.
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