** **Variational Principle and Lagrange’s Equation :** **Hamilton’s Principle , Some Techniques of the Calculus of Variation – Derivation of Lagrange’s Equation from Hamilton’s Principle- Extension of Hamilton’s Principle through non Halonomic Systems, Conservation Theorem and Symmetric Properties.
**Unit – III (18 Hours)** ** **Two Body Central Force Problem :** **Reduction to one body Problem-Equations of motion and first integral- The Equivalent One body Problem and Classification of Orbits- Virial theorem- The differential equation of the Orbit and integrable power law potentials- The kepler problem and inverse square law of force- The motion in time in the kepler problem – Laplace- Runge- Lenz vector
**Unit – IV**** ****(18 Hours) **The Hamilton’s equations of motion :** **Legendre’s transformation and the Hamilton’s equation of motion- Cyclic Co-ordinates and Conservation Theorem- Derivation of Hamilton’s equation from a variational principle- The principle of least action.
**Unit – V**** ****(18 Hours)**** **Canonical transformations:** **Equation of canonical transformations- examples of Canonical Transformation- simple approach to Canonical transformation- Poisson brackets and other Canonical invariants. **Text Book**: Classical Mechanics ## - Herbert Goldstein-2^{nd} Edition, Narosa Publishing House, 1986.
**Unit** |
**Chapter/Section** | I | Chapter 1 | II | Chapter 2 (except 2.5) | III | Chapter 3 (except 3.10, 3.11) | IV | Chapter 8 : 8.1, 8.2, 8.5, 8.6 only | V | Chapter 9 : 9.1 , 9.2 , 9.4 only |
## ## ## Reference Book: Classical Mechanics - V.B.Bhatia ## Narosa Publishing House – 1998
**THIAGARAJAR COLLEGE (AUTONOMOUS), MADURAI – 625 009** (Re- Accredited with 'A' Grade by NAAC) **Department of Mathematics** (From 2011 – 2013 batch onwards) Course : M.Sc. Code No. : 3PM2 Semester : III No. of hours allotted : 6 Paper : Core 12 No. of credits : 4 Title of the Paper **: ****Functional Analysis**
**Course objective: **To develop the skills in theorem through geometry and to get knowledge in using classes of functions rather than individual functions.
**Unit – I (18 Hours)**
** **Fundamentals of Normed Linear Spaces:** **Normed Linear Spaces – Continuity of Linear Maps – Hahn-Banach Theorems – Completeness of a Norm.
**Unit – II (18 Hours)**
** **Bounded Linear Maps on Banach Spaces:** **Uniform Boundedness Principle – Closed Graph Theorem – Open Mapping Theorem – Spectrum of a Bounded Operator.
**Unit – III (18 Hours)**
** **Spaces of Bounded Linear Functionals:** **Duals and Transposes – Duals of L^{p}([a,b]) and C([a,b]) – Weak and Weak* Convergence.
**Unit – IV (18 Hours)**
** **Geometry of Hilbert Spaces: Inner Product Spaces – Orthonormal Sets – Projection and Riesz Representation Theorems.
**Unit – V (18 Hours)**
** **Bounded Operators on Hilbert Spaces:** **Bounded Operators and Adjoints – Normal, Unitary and Self-Adjoint Operators – Spectrum and Numerical range – Compact self-adjoint operators.
**Text Book : **Functional Analysis - Balmohan Vishnu Limaye – Wiley eastern Limited, 1989.
**Unit** |
**Chapter/Section** | I | Chapter II (Except Section 7.14) | II | Chapter III (Except Section 9.5, 9.6, 9.8, 11.4, 11.5, 11.6) | III | Chapter IV (Except Section 14.9, 15.9 & 16) | IV | Chapter VI (Except Section 23) | V | Chapter VII |
**Reference Books: **1)** **Introductory Functional Analysis with Applications - Erwin Kreyszig John Wiley & Sons, Third Print – 2007. 2) Functional Analysis – George Bachman Lawrence Narici Academic Press International, New York, 1996
**THIAGARAJAR COLLEGE (AUTONOMOUS), MADURAI – 625 009** (Re- Accredited with 'A' Grade by NAAC) **Department of Mathematics** (From 2011 – 2013 batch onwards) Course : M.Sc. Code No. : 3PM3 Semester : III No. of hours allotted : 4 Paper : Core 13 No. of credits : 3 Title of the Paper **: ****Number Theory**
**Course objective: **To study some importance tools in number theory and study of distributions of primes
**Unit - I (12 Hours)**
** **Congruences: Congruences – Solutions – Congruence of degree 1 – The function (n)
**Unit - II (12 Hours)** ** **Quadratic reciprocity:** **Quadratic residues – Quadratic reciprocity – The Jacobian symbol
**Unit - III (12 Hours)**
** **Some functions of Number Theory:** **Greatest integer function – Arithmetic function – The Mobius formula.
**Unit - IV (12 Hours)** ** **Diophantine equations:** **Diophantine equations – The equation ax + by = c – Positive solutions – Other linear equations – The equation x^{2} + y^{2} = z^{2} – The equation x^{4} + y^{4} = z^{4 }– Sums of four squares and five squares.
**Unit - V (12 Hours)**
** **Distribution of primes: The function (x) – The sequence of primes – Betrand’s postulate
**Text Book: ** An introduction to Number Theory – Ivan Niven and Zuckerman, Wiley Eastern, Reprint 1989
**Unit** |
**Chapter/Section** | I | 2.1 – 2.4 | II | 3(Full) | III | 4.1 – 4.3 | IV | 5.1 – 5.7 | V | 8(Full) |
**Reference book: ** Introduction to Analytic Number Theory – Martin Erickson and Anthony Vazzana, Chapman and Hall /CRC publications,2009.
**THIAGARAJAR COLLEGE (AUTONOMOUS), MADURAI – 625 009** (Re- Accredited with 'A' Grade by NAAC) **Department of Mathematics** (From 2011 – 2013 batch onwards) Course : M.Sc. Code No. : 4PM1 Semester : IV No. of hours allotted : 6 Paper : Core 14 No. of credits : 4 Title of the Paper **: ****Statistics**
**Course objective: **To present the importance of theoretical approach in statistical methods.
**Unit – I (18 Hours)**
** **Theoretical Distributions: Binomial, Trinomial distributions, - Poisson distributions, Gamma and Chi-square distribution – Normal distribution
**Unit – II (18 Hours)** ** **Distributions Functions of Random Variables:** **Sampling theory-Transformation of variables of discrete type - Transformation of variables of continuous Type – The t and F distributions- The moment generating function technique- The Distributions of X and n S^{2}/s^{2}
**Unit – III (18 Hours)** |