ABOUT THE COURSE:This physics course on classical mechanics uses Lagrangian and Hamiltonian theory which is essential for competitive exams such as JAM, JEST, NET, and GATE, as it cultivates an increased understanding of mechanics beyond Newtonian formulations. Understanding Lagrangian and Hamiltonian mechanics is crucial for tackling advanced physics problems, such as variational principles, constrained motion, and canonical transformations, which are commonly assessed in research-oriented and higher-level competitions. Furthermore, the mathematics involved in Lagrangian mechanics enhances the problem-solving abilities needed for fields such as theoretical physics, quantum mechanics, and astrophysics.INTENDED AUDIENCE:U.G. + P.G. (First semester) studentsPREREQUISITES: Knowledge of Elementary calculus, and B.Sc (I-semester) Grade Mechanics and B.Sc (II-semester) Grade MechanicsINDUSTRY SUPPORT:None

Syllabus

Week 1

Review of Newtonian's dynamics
Constraints
Classification of constraints
Velocity dependent constraints
Degrees of freedom
Generalized coordinates
Generalized velocities
Virtual displacement
Principle of virtual work
D'Alembert's Principle
Lagrange's equation from D'Alembert's Principle.

Week 2

Velocity-dependent potential and Lorents-force
Kinetic energy in terms of generalized coordinates
Applications of Lagrangian formalism to simple mechanical systems.

Week 3

Variational principle
Technique of the calculus of variation and its applications
Hamilton's variational principle
Geodesics
Minimum surface of revolution
Brachistochrone problem.

Week 4

Lagrange equations using Hamilton's principle
Generalized momenta
Cyclic co-ordinates
Definition of energy function and Hamiltonian and its physical significance
Conservation of energy
Linear momenta and angular momenta

Week 5

Hamiltonian dynamics
Hamilton's equation of motion from variational principle
Physical significance
Hamiltonian for simple mechanical systems
Conservation laws and cyclic coordinates
Hamiltonian as a constant of motion

Week 6

Two-body problem-central force problem
The equivalent one-dimensional problem and classification of orbits
Differential equation nfor the orbit and integrable power-law potentials
Conservation of angular momentum and Kepler's second law.

Week 7

Kepler problem-inverse square law of force
Kepler's first and third laws
Condition for closed orbit
Bertrand's problem
Virial theorem and its simple applications.

Week 8

Two-body collisions—scattering by a central force
Rutherford scattering formula; transformation of the scattering problem from center of mass to laboratory coordinates.

Week 9

Equations of canonical transformations
Properties of four special type of canonical transformations
Poisson bracket (PB) and its properties
Equation of motion
Infinitesimal canonical transformations
Conservations theorems in PB formalism

Week 10

Invariance of PB under canonical transformations
Hamilton-Jacobi equation for Hamilton's principal and characteristic functions and its application to simple harmonic oscillator.

Week 11