Syllabus
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Course Code: MMATH21 -401 Course Name: Mechanics and Calculus of Variations |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
|---|---|---|
| 1 | Moments and products of inertia, The theorems of parallel and perpendicular axes, Angular
momentum of a rigid body about a fixed point and about fixed axes, Principal axes. Kinetic energy of a rigid body rotating about a fixed point, Momental ellipsoid – equimomental system, Coplanar distributions, General motion of a rigid body. Problems illustrating the laws of motion, Problems illustrating the law of conservation of angular momentum, Problems illustrating the law of conservation of energy, Problems illustrating impulsive motion. (Relevant portions from the book ‘Textbook of Dynamics’ by F. Chorlton). |
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| 2 | Euler’s dynamical equations for the motion of a rigid body about a fixed point, Further properties
of rigid motion under no forces, Some problems on general three-dimensional rigid body motion,
The rotating earth. Note on dynamical systems, Preliminary notions, Generalized coordinates and velocities, Virtual work and generalized forces, Derivation of Lagrange’s equations for a holonomic system, Case of conservative forces, Generalized components of momentum and impulse. Lagranges equations for impulsive forces, Kinetic energy as a quadratic function of velocities. Equilibrium configurations for conservative holonomic dynamical systems, Theory of small oscillations of conservative holonomic dynamical systems. |
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| 3 | Lagrange’s equations for potential forces, Variational principles in Mechanics: Hamilton’s principle, The principle of least action. Hamiltonian and canonical equations of Hamilton. Basic integral invariant of Mechanics. Canonical transformations, Hamilton Jacobi equation. | |
| 4 | Functional and its variation, Euler’s (Euler-Lagrange) equations, Variational problems for functionals depending on one independent and one dependent variable(s) and its (i) first derivative (ii) higher derivatives with fixed end conditions, Variational problems for functionals depending on n functions of a single independent variable and functional depending on a function and its n derivatives, Functionals dependent on functions of several independent variables. Variational problems in parametric form. Natural boundary conditions and transition conditions, Invariance of Euler’s equation. Conditional extremum. Variational problem with moving boundaries. Some basic problems in calculus of variations: shortest distance, minimum surface of revolution, Brachistochrone problem, isoperimetric problem and geodesic problems. | |
