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We study the stability of symmetric trajectories of a particle on the Lie group $SO(3)$ whose motion is governed by an $SO(3)\times SO(2)$ invariant metric and an $SO(2)\times SO(2)$ invariant potential. Our method is to reduce the number of degrees of freedom at {\em singular} values of the $SO(2)\times SO(2)$ momentum map and study the stability of the equilibria of the reduced systems as a function of spin. The result is an elementary analysis of the fast/slow transition in the Lagrange and Kirchhoff tops. More generally, since an $SO(2)\times SO(2)$ invariant potential on $SO(3)$ can be thought of as ${\bf Z}_2$ invariant function on a circle, we get a condition on the second and fourth derivatives of the potential at the symmetric points that guarantees that the corresponding system gains stability as the spin increases.

We study the stability of symmetric trajectories of a particle on the Lie group $SO(3)$ whose motion is governed by an $SO(3)\times SO(2)$ invariant metric and an $SO(2)\times SO(2)$ invariant potential. Our method is to reduce the number of degrees of freedom at {\em singular} values of the $SO(2)\times SO(2)$ momentum map and study the stability of the equilibria of the reduced systems as a function of spin. The result is an elementary analysis of the fast/slow transition in the Lagrange and Kirchhoff tops. More generally, since an $SO(2)\times SO(2)$ invariant potential on $SO(3)$ can be thought of as ${\bf Z}_2$ invariant function on a circle, we get a condition on the second and fourth derivatives of the potential at the symmetric points that guarantees that the corresponding system gains stability as the spin increases.

We study the stability of symmetric trajectories of a particle on the Lie group $SO(3)$ whose motion is governed by an $SO(3)\times SO(2)$ invariant metric and an $SO(2)\times SO(2)$ invariant potential. Our method is to reduce the number of degrees of freedom at {\em singular} values of the $SO(2)\times SO(2)$ momentum map and study the stability of the equilibria of the reduced systems as a function of spin. The result is an elementary analysis of the fast/slow transition in the Lagrange and Kirchhoff tops. More generally, since an $SO(2)\times SO(2)$ invariant potential on $SO(3)$ can be thought of as ${\bf Z}_2$ invariant function on a circle, we get a condition on the second and fourth derivatives of the potential at the symmetric points that guarantees that the corresponding system gains stability as the spin increases.

This textbook on mathematical analysis is based on many years’ experience of lecturing at a higher technical college. Its aim is to train the students in active approach to mathematical exercises, as is done at a seminar. Much attention is given to problems improving the theoretical background. Therefore standard computational exercises are supplemented by examples and problems explaining the  theory, promoting its deeper understanding and stimulating precise  mathematical thinking. Some counter examples explaining the need for  certain conditions in the formulation of basic theorems are also  included. The book is designed along the following lines. Each section opens with a concise theoretical introduction containing the principal definitions, theorems and formulas. Then follows a detailed solution of one or more typical problems. Finally, problems without solution are given, which are similar to those solved but contain certain peculiarities. Some of them are provided with hints. These sections should prove of interest to the inquiring student, and possibly also to lecturers in selecting material for classwork or seminars. This book was translated from the Russian by Leonid Levant . The book was published by first Mir Publishers in 1973.

This textbook on mathematical analysis is based on many years’ experience of lecturing at a higher technical college. Its aim is to train the students in active approach to mathematical exercises, as is done at a seminar. Much attention is given to problems improving the theoretical background. Therefore standard computational exercises are supplemented by examples and problems explaining the  theory, promoting its deeper understanding and stimulating precise  mathematical thinking. Some counter examples explaining the need for  certain conditions in the formulation of basic theorems are also  included. The book is designed along the following lines. Each section opens with a concise theoretical introduction containing the principal definitions, theorems and formulas. Then follows a detailed solution of one or more typical problems. Finally, problems without solution are given, which are similar to those solved but contain certain peculiarities. Some of them are provided with hints. These sections should prove of interest to the inquiring student, and possibly also to lecturers in selecting material for classwork or seminars. This book was translated from the Russian by Leonid Levant . The book was published by first Mir Publishers in 1973.

xiii, 785 p. 24 cm

This book can help in IIT JEE and other competitive engineering/science exam's preparation.

This book can help in IIT JEE and other competitive engineering/science exam's preparation.

Problems in Calculus of One Variable covers the in-depth study of mathematical analysis based on many years of the author’s experience. The author has presented this book with an aim to train the students in active approach to mathematical exercises, as is done at a seminar. The present book has been divided into eight chapters namely Introduction to Mathematical Analysis, Differentiation of Functions, Application of Differential Calculus to investigation of Functions, Indefinite Integrals, Basic Method of Integration, Basic Classes of Integrable Functions, The Definite Integrals, Applications of the Definite Integral and Improper Integrals. Each chapter in the book begins with a theoretical introduction of the topics containing the principal definitions, theorems and formulas followed by solved examples based on the topics discussed. The solved examples have been covered topic-wise along with the explanation of concepts for immediate and effective understanding of the concepts.

Problems in Calculus of One Variable covers the in-depth study of mathematical analysis based on many years of the author’s experience. The author has presented this book with an aim to train the students in active approach to mathematical exercises, as is done at a seminar. The present book has been divided into eight chapters namely Introduction to Mathematical Analysis, Differentiation of Functions, Application of Differential Calculus to investigation of Functions, Indefinite Integrals, Basic Method of Integration, Basic Classes of Integrable Functions, The Definite Integrals, Applications of the Definite Integral and Improper Integrals. Each chapter in the book begins with a theoretical introduction of the topics containing the principal definitions, theorems and formulas followed by solved examples based on the topics discussed. The solved examples have been covered topic-wise along with the explanation of concepts for immediate and effective understanding of the concepts.

532 p. 25 cm

532 p. 25 cm