# Applications of the Lie Symmetries to Solve Mechanical Problems # Objectives

• At the end of the course, the student should be able to obtain the Lie symmetries of a problem encountered in engineering and be able to reduce the order of the systems of differential equations to obtain the analytical solution.

This course was offered in February of 2021 to the Programa de Pós-Graduação em Engenharia Mecânica of Unesp-Ilha Solteira with the support of Prof. Dr. Adrián Ruiz and Prof. Dr. Samuel da Silva.

## Program content

1. Introduction to the Lie symmetries

• The importance of Lie symmetries and their uses;

• Objectives of the course;

• Infinitesimal transformations;

• Infinitesimal generators;

• Prolongations and their generators;

• Lie theorem;

• Order reduction.

2. Computation of the Lie symmetries for mechanical systems

• Modeling of mechanical systems;

• How to obtain the Lie symmetries of mechanical systems;

• Application of the Lie theorem.

3. Resolution of mechanical systems via Lie symmetries

• Reduction of order;

• Change of variables;

• Analytical solution;

• Interpretation of the physical meaning of the infinitesimal generators.

4. Use of the Maple software in the context of the Lie symmetries

• Basic commands;

• Libraries used for Lie symmetries use;

• Infinitesimal transformation;

• Infinitesimal generators;

• Prolongation of the transformations and their generators;

• Lie theorem;

• Order reduction.

## support material

Class 0 - Program content and motivation

Class 1 - Introduction to the Lie Symmetries

Class 2 - Computation of the Lie symmetries for mechanical systems

Class 3 - Resolution of mechanical systems via Lie symmetries

Class 4 - Use of the Maple software in the context of the Lie symmetries

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