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krishnaswami g classical mechanics from particles continua chaos 2024

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Textbook in PDF format This well-rounded and self-contained treatment of classical mechanics strikes a balance between examples, concepts, phenomena and formalism. While addressed to graduate students and their teachers, the minimal prerequisites and ground covered should make it useful also to undergraduates and researchers. Starting with conceptual context, physical principles guide the development. Chapters are modular and the presentation is precise yet accessible, with numerous remarks, footnotes and problems enriching the learning experience. Essentials such as Galilean and Newtonian mechanics, the Kepler problem, Lagrangian and Hamiltonian mechanics, oscillations, rigid bodies and motion in noninertial frames lead up to discussions of canonical transformations, angle-action variables, Hamilton-Jacobi and linear stability theory. Bifurcations, nonlinear and chaotic dynamics as well as the wave, heat and fluid equations receive substantial coverage. Techniques from linear algebra, differential equations, manifolds, vector and tensor calculus, groups, Lie and Poisson algebras and symplectic and Riemannian geometry are gently introduced. A dynamical systems viewpoint pervades the presentation. A salient feature is that classical mechanics is viewed as part of the wider fabric of physics with connections to quantum, thermal, electromagnetic, optical and relativistic physics highlighted. Thus, this book will also be useful in allied areas and serve as a stepping stone for embarking on research. : Preface Mechanical systems with one degree of freedom Newton's laws and qualitative characterization of motion on a line Time period of conservative oscillations between turning points Inverse problem: determination of potential from time period Time delay in unbounded `scattering' trajectories Simple pendulum: basic properties and small oscillations Problems Kepler's gravitational two-body problem Inverse problem: universal law of gravity from Kepler's laws Direct problem: center of mass & relative vectors and conservation laws Planetary orbits Time period of elliptical orbits LRL vector and relations among conserved quantities Collision of two gravitating point masses Rutherford scattering cross section The three-body problem: Euler and Lagrange solutions Problems Newtonian to Lagrangian and Hamiltonian mechanics Time, space, light, simultaneity, causality, homogeneity and isotropy Degrees of freedom and instantaneous configurations Newton's laws and Galileo's relativity and equivalence principles Phase space, dynamical variables, conserved quantities, collisions Principle of extremal action and Euler-Lagrange equations Nonuniqueness of Lagrangian Conjugate momenta, their geometric meaning and cyclic coordinates Coordinate invariance of the form of Lagrange's equations Hamiltonian and its conservation Symmetries to conserved quantities: Noether's theorem Noether's theorem when Lagrangian changes by a time derivative Inertial frames of reference and Galilean invariance Polar vectors, axial vectors, true scalars and pseudoscalars Hamilton's equations Legendre transform: Hamiltonian from Lagrangian Lagrange multipliers and constrained extremization Singular Lagrangians and constraints Action as a function along a trajectory Variational principles for Hamilton's equations Coordinate invariance of Lagrange and Hamilton equations Canonical Poisson brackets Properties of the Poisson bracket Canonical formulation of charged particle in electromagnetic field Poisson algebra of conserved quantities in the Kepler problem Functional independence of conserved quantities Noncanonical Poisson brackets, Poisson and symplectic manifolds Free particle trajectories as geodesics on configuration space Euler-Maupertuis principle and the Jacobi-Maupertuis metric Problems Introduction to special relativistic mechanics Difficulties with Newtonian mechanics Postulates of special relativity Synchronization of clocks and simultaneity Lorentz transformations Time dilation, length contraction, proper length and time Space-like, time-like and light-like intervals and causality Relativistic addition of velocities Relativistic momentum from two particle collision Relativistic energy and energy-momentum dispersion relation Minkowski space-time and relativistic dynamics Problems Dynamics viewed as a vector field on state space Vector fields from Newtonian and Hamiltonian dynamics Vector fields in one dimension Existence and uniqueness of solutions Vector fields on the phase plane Problems Small oscillations for one degree of freedom Linear harmonic oscillator in 1d and neutral stability Linear vector fields on the phase plane Phase portrait from spectrum of coefficient matrix Damped harmonic oscillator: view from the phase plane Critically damped oscillator: deficient coefficient matrix Trace-determinant classification of linear fixed points Robustness of the linear theory Driven or forced oscillations Driven damped oscillations Parametric oscillations and resonant amplification Problems Nonlinear oscillations: pendulum and anharmonic oscillator Simple pendulum: view from phase space Introduction to Jacobi elliptic functions Time-dependence of pendulum in terms of elliptic functions Anharmonic oscillations: quartic double-well potential Quartic oscillator: exact solution and Lindstedt-Poincaré method Problems Rigid body mechanics Lab and comoving frames Configuration space and degrees of freedom Infinitesimal displacement and angular velocity of rigid body Kinetic energy and inertia tensor Types of rigid bodies Angular momentum of a rigid body Equations of motion of a rigid body Force-free motion of rigid bodies Euler angles and rotations Angular velocity and kinetic energy in terms of Euler angles Euler equations for a rigid body in body-fixed frame Ellipsoid of inertia and qualitative description of free motion Solution of force-free Euler equations via elliptic functions Poisson bracket formulation of Euler's equations Motion of a heavy symmetrical (Lagrange) top Problems Motion in noninertial frames of reference Uniformly accelerating frames and the equivalence principle Nonuniformly accelerated frames: Lagrangian approach Uniform rotation: Hamiltonian formulation and magnetic analog Precession of Foucault's pendulum Circular restricted three-body problem Problems Canonical transformations From point transformations to canonical transformations Preservation of Hamilton's equations and Poisson brackets Comparison of classical and quantum mechanical formalisms Canonical transformations and area-preserving maps Canonical transformations preserve Poisson tensor Generating function for infinitesimal canonical transformations Symmetries & Noether's theorem in the Hamiltonian framework Liouville's theorem Poincaré recurrence Generating functions for finite canonical transformations Problems Angle-action variables Angle-action variables for the harmonic oscillator Generator of CT to angle-action variables: Hamilton-Jacobi equation Generating function for oscillator angle-action variables Angle-action variables for systems with one degree of freedom Angle-action variables for libration of the simple pendulum Bohr-Sommerfeld quantization rule Liouville integrability and KAM tori Liouville-Arnold theorem Conserved quantities from a Lax pair Harmonic oscillator Lax pair Isospectral evolution and conserved quantities Toda chain: Flaschka's variables and a Lax pair Euler-Poinsot top Lax pair: spectral parameter Problems Hamilton-Jacobi equation Time-dependent Hamilton-Jacobi evolution equation Connection of Hamilton-Jacobi to Schrödinger and eikonal equations Separation of variables in the Hamilton-Jacobi equation Hamilton's principal function as action along a trajectory Geometric interpretation of Hamilton-Jacobi equation Problems Normal modes of oscillation and linear stability Elementary examples of coupled small oscillations Normal modes of two weakly coupled pendula Normal modes of a diatomic molecule Double pendulum: formulation and small oscillations Energy, Lagrangian and equations of motion Normal modes of a double pendulum Normal modes around a static equilibrium: general framework Small perturbations around a periodic solution Formulation as a system of first order equations Time evolution matrix Monodromy matrix Stability of a periodic solution Kapitza pendulum with oscillating support: Mathieu equation Problems Bifurcations: qualitative changes in dynamics Bifurcations of vector fields on the real line Saddle-node bifurcation Transcritical bifurcation Pitchfork bifurcations Supercritical pitchfork bifurcation Subcritical pitchfork bifurcation Bifurcations in two dimensions Saddle-node, transcritical and pitchfork bifurcations Hopf bifurcations Problems From regular to chaotic motion Chaos in iterations of a map Lyapunov exponent and sensitivity to initial conditions Chirikov-Taylor standard map: a kicked rotor Logistic map: period doubling, Cantor dust and Lyapunov exponent Lyapunov exponents for continuous-time dynamical systems Poincaré return map and Homoclinic tangle Hamiltonian chaos: order-chaos-order transition in a double pendulum Poincaré sections and onset of chaos Return to regularity at high energies Understanding the zero gravity double pendulum Chaos in Lorenz's model for convection Problems Dynamics of continuous deformable media Vibrations of a stretched string and the wave equation Wave equation for transverse vibrations of a stretched string Finite differences: wave equation as a system of ODEs Normal modes and solution by Fourier series Right- and left-moving waves and d'Alembert's solution Conserved energy of small oscillations of a stretched string Three local conservation laws for the wave equation Lagrangian and Hamiltonian for stretched string Conserved quantities from Noether's theorem Dispersion relation, phase and group velocities Lax pair for the first order wave equation Problems Heat diffusion equation and Brownian motion Obtaining the heat equation and its basic properties Solution of initial value problem on an interval by Fourier's series Heat kernel: time evolution operator for heat equation From Brownian motion to the diffusion equation Brownian motion and the atomic hypothesis Random walk model and the diffusion equation Problems Introduction to fluid mechanics Fluid element, local thermal equilibrium and dynamical fields Fluid statics: aero- or hydrostatics Flow visualization: streamlines, streaklines and pathlines Material derivative Compressibility, incompressibility and divergence of velocity field Local conservation of mass: continuity equation Euler equation for inviscid flow Ideal adiabatic flow: entropy advection and equation of state Bernoulli's equation Sound waves in homentropic flow Vorticity and its evolution Vortex tubes: Kelvin and Helmholtz theorems Conservation of energy, (angular) momentum and helicity Hamiltonian and Poisson brackets for inviscid flow Clebsch variables and Lagrangian for ideal flow Navier-Stokes equation for incompressible viscous flow Problems Mathematical and kinematical background Vectors in Euclidean space Position coordinates and velocity and acceleration vectors Circular motion: uniform and nonuniform Integration of kinematical equations: uniform acceleration Plane polar coordinates Spherical polar coordinates Taylor series Some vector calculus: grad, div and curl Stokes', Green's and Gauss' integral theorems Vector spaces, matrices and eigenvalue problems Fourier transform Problems Primer on manifolds, tensors and groups The concept of a manifold Submanifolds, connected and simply connected manifolds Smooth functions or scalar fields Vector fields Covector fields or 1-forms Tensors of rank two and 2-forms Higher rank tensor fields and forms Pushforward and pullback of tensors Exterior algebra, exterior derivative and Bianchi's identity Integration on manifolds and Stokes' theorem Covariant derivative Curvature on a Riemannian manifold Riemann-Christoffel curvature tensor Geodesic deviation and Riemannian curvature Groups, Lie groups and their Lie algebras Quaternions and the axis-angle representation of rotations Problems Supplementary reading References Index

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krishnaswami g classical mechanics from particles continua chaos 2024