In this one hour lecture, we will understand the two important formalisms in Classical Mechanics, Lagrangian and Hamiltonian. We will start with a review on Newtonian Mechanics and then discuss the need of the two formalisms. We will introduce the concepts like configuration space, constraints,  generalized coordinates and momenta. Then, we will develop the mathematical foundation of Lagrangian Mechanics via the variational principle, which is fundamental law the nature follows. We will then apply the Lagrangian Dynamics to a few standard problems.

Hamiltonian formalisms will be derived in two way: from variational principle and Legendre Transforms. Before applying the Hamiltonian Dynamics to a few standard problems, we will discuss the important concepts like phase space, symmetries & conservation laws. What Canonical Transformations are and why are they important will be discussed to highlight the importance of Hamiltonian Dynamics. In the end, how useful the Hamiltonian Formalisms is to develop the theories like Quantum Mechanics and Statistical Mechanics will be discussed.