The Euler-Lagrange equations are fundamental equations in the field of calculus of variations, which is concerned with finding the path or function that minimizes or maximizes a certain quantity. These equations provide a systematic way to derive the equations of motion for physical systems, particularly in classical mechanics. In essence, the Euler-Lagrange equations relate the derivatives of a function, known as the Lagrangian, to the variations of that function. By applying these equations, one can determine the optimal trajectory of a system, making them crucial for understanding dynamics in physics and engineering.