Classical Mechanics (Theory)

Paper Code: 
CPHY 711
Credits: 
04
Contact Hours: 
60.00
Max. Marks: 
100.00
Objective: 

Course Objectives:

This course will enable the students to -

  1. The student will be able to apply knowledge to calculate the Calculus of variation and its application to simple problems, formulation of Lagrangian and Hamiltonian for a problem, Canonical transformation and Action angle variable.
  2. The student shall acquire ability to differentiate between problems of classical and quantum nature and apply classical methods to solve the relevant problems.

Course Outcomes (COs):

Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

PAPER CODE

Paper Title

 

 

 

 

 

 

 

 

CPHY 711

Classical Mechanics (Theory)

 

 

 

 

 

 

The students will be able to:

 

CO136: understand the basic concepts of coordinate systems and degrees of freedom and apply on various problems.

CO137: learn about Hamilton's and Lagrange's Equations using calculus of variation and apply them to solve problems of mechanics.

CO138: gain Knowledge about Conservation principles, Noether's theorem, Eulerian angles and Euler’s theorem and apply these concepts to relevant problems.

CO139: understand basic Lagrange's-  Poisson’s brackets and Canonical transformation  then analyze their relations.

CO140: learn about the Action angle variables, Lagrangian formulation and their applications.

Approach in teaching:

 

Interactive Lectures, Discussion, problem solving in Tutorials,  Demonstration

 

Learning activities for the students:

Self learning assignments, Effective questions, Simulation, Seminar presentation, Solving numerical.

 

 

 

Class test, Semester end examinations, Quiz, Solving problems, Assignments, Presentations

 

 

CONTENTS

 

12.00
Unit I: 
I
Basic concepts of coordinate systems and degrees of freedom; Holonomic and non-holonomic constraints; D-Alembert's Principle; Generalized coordinates; Principle of virtual work;Lagrangian, Lagrange's equation and its applications; Velocity dependent potential in Lagrangian formulation; Generalized momentum; Hamilton's Principle; Lagrange’s equation from Hamilton's Principle.
 
12.00
Unit II: 
II
Calculus of variation and its application to simple problems: Hamilton's Equations; Applications of Hamilton’s equation; Hamilton’s canonical equation in different coordinate systems; Calculus of Variation and Euler-Lagrange’s equations; Brachistochrone Problem; Derivation of Lagrange's and Hamilton’s canonical equation from Hamilton’s variational principle; Method of Lagrange's undetermined multipliers; Principle  of least action.
 
11.00
Unit III: 
III
Conservation principle and Noether's theorem; Conservation of energy, linear momentum and angular momentum as a consequence of homogeneity of time and space and isotropy of space respectively.
Eulerian angles; Euler’s theorem; Angular momentum and Inertia Tensor; Euler’s equations; Euler’s equation of motion for a rigid body.Levels of biological diversity: genetic, species and ecosystem diversity
13.00
Unit IV: 
IV
Canonical transformation: Legendre transformations; Generating functions; Conditions for Canonical transformation; Bilinear invariant conditions; Lagrange's and Poisson’s brackets and their relations; Angular momentum and Poisson Brackets; equation of motion in Poisson bracket formulation; Invariance of Poisson’s and Lagrange’s Brackets under canonical transformations; Liouville's theorem. 
 
 
 
 
12.00
Unit V: 
V

Action angle variable: Hamilton Jacobi equation and its applications; adiabatic invariance of action variable; The Kepler problem in action angle variables; theory of small oscillations in Lagrangian formulation; theory of small oscillations in normal coordinates and normal modes; Two coupled oscillator and solution of its differential equation; Two coupled pendulum; Double pendulum.

Essential Readings: 
• “Classical Mechanics”, Goldstein, Addison Wesley.
• “Mechanics-VolumeI”,Landau.AndLifshiz. 
• “ClassicalMechanics”,A.Raychoudhary.
• “Classical Mechanics” N.C. Rana and P.S. Joag, Tata Mc Graw Hill
• “Classical Mechanics” J.C.  Upadhyaya, Himalaya Publishing House.
 
Academic Year: 
2023-2024
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