Proof of Theorem A. N. Kolmogorov on the preservation of conditionally periodic movements with a small change in the function of the Hamilton” by Vladimir Igorevich Arnold is not just a scientific work, but a real pearl in the world of mathematical physics and the theory of dynamical systems. Arnold, a prominent Russian mathematician known for his in-depth research in mechanics, differential equations, and topology, offers the reader a unique look at one of the key theorems that is essential for understanding the behavior of dynamical systems. The book focuses on the theorem proposed by A. N. Kolmogorov, who claims that with a small change in the function of the Hamiltonian, conditional-periodic movements are preserved. This statement is not only theoretical, but also practical, as it allows you to explore the stability of dynamical systems, which is an important aspect in various fields of science - from physics to biology and economics. Arnold, using his unique style and approach, parses the proof of this theorem in detail, focusing on intuitive explanations and visualization of complex mathematical concepts. The book will be of interest not only to professional mathematicians and physicists, but also to students, graduate students, as well as anyone who is fond of mathematics and wants to deepen their knowledge in the field of dynamical systems. If you study topics such as theoretical mechanics, nonlinear systems or mathematical physics, then this work will be a real discovery for you. Arnold masterfully combines rigor and accessibility, making his work attractive to a wide audience. The topics raised in the book cover not only mathematical aspects, but also philosophical questions about the nature of motion and stability. Arnold explores how small changes in a system can lead to significant consequences, which is an important lesson for understanding complex systems in the real world. This book makes you think about how we perceive change and sustainability, and how these concepts can be applied to different areas of knowledge. Arnold's style is clear and logical, which makes his works not only scientific, but also literary works. He skillfully uses examples and visualizations to make complex ideas more accessible to understanding. In his other well-known works, such as Geometry and Dynamics and Ordinary Differential Equations, one can also notice this approach, which combines the rigor of mathematical analysis with artistic expressiveness. This book not only deepens knowledge about Kolmogorov's theorem, but also opens up new horizons for further research. Readers interested in mathematics, physics and chaos theory will find in it many ideas and inspirations for their own research. Proof of Theorem A. N. Kolmogorov on the preservation of conditional-periodic movements with a small change in the function of the Hamilton” is not just a book, but a key to understanding complex dynamical systems, which can change your idea of mathematics and its application in the real world. If you are looking for a deep and fascinating reading that can broaden your horizons and deepen your understanding of dynamic processes, this book will become your faithful companion in the world of science and mathematics.