### Introduction

In classical mechanics, the Lagrangian formulation is a powerful and elegant method for analyzing the dynamics of systems. It provides a framework for dealing with complex systems, especially those with constraints. This tutorial will discuss the concepts of degrees of freedom and equations of constraint, which are essential for understanding the behavior of physical systems within the Lagrangian framework.

### Degrees of Freedom

Degrees of freedom (DOF) refer to the minimum number of independent coordinates required to specify the configuration of a system completely. Each degree of freedom corresponds to a possible independent motion.

For example:

• A single particle moving in three-dimensional space has three degrees of freedom (its coordinates x,y,z).
• A rigid body in three-dimensional space has six degrees of freedom (three translational and three rotational).

The degrees of freedom of a system determine the complexity of its configuration space, which is the space of all possible states of the system.

### Equations of Constraint

Constraints are conditions that limit the motion of a system. They reduce the number of degrees of freedom by establishing relationships between the coordinates. Constraints can be classified into two main types:

Understanding degrees of freedom and constraints is essential for effectively applying the Lagrangian formulation to solve problems in classical mechanics. This approach provides a systematic way to handle complex systems and constraints, making it an invaluable tool for students preparing for the CSIR NET Physics exam.

### Practice Problems

1. Double Pendulum: Derive the equations of motion for a double pendulum system using the Lagrangian formulation.
2. Bead on a Rotating Hoop: A bead is free to slide on a rotating hoop. Determine the equations of motion considering the constraints and degrees of freedom.
3. Atwood’s Machine: Analyze the motion of an Atwood’s machine using the Lagrangian approach and identify the constraints involved.

By working through these problems, students will gain a deeper understanding of the application of the Lagrangian formulation in classical mechanics.