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In this chapter, we lay out the foundations of Lagrangian Mechanics. We introduce the basic concepts of Lagrangian mechanical systems, namely the Lagrangian, the action, and the equations of motion, also known as the Euler–Lagrange equations. We also discuss important examples, such as the free particle, the harmonic oscillator, as well as motions in central force potentials, such as Newton’s theory of gravity and Coulomb’s electrostatic theory. Highlighting the importance of symmetries, we study integrals of motion and Noether’s theorem. As an application, we consider motions in radial potentials and, further specializing to motions in Newton’s gravitational potential, we conclude this section with a derivation of Kepler’s laws of planetary motion.