Lecturer: M. Fabbrichesi , SISSA

In this lesson the professor continues with the study of the **Hamiltonian** approach to classical mechanics and he shows how it can be derived from a **variational principle**. Then he introduces the **canonical transformation** and the canonical equations. As a first example for a **canonical transformation** he use the problem of the **Harmonic oscillator**. In the second part of the lesson he introduces the **Poisson bracket** and their properties like that they are invariant under **canonical transformation**, then he writes the equation of motion in Poisson notation. In this lesson he left the Homework number 5 (Homework.5).

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In this video it is discussed the Calculus of variations and **Hamilton's Principal.**

In this lesson the professor continues with the study of the **Newton's laws of motion** and it consequences. Then with the help of the **parametric equations** for the **trajectory** of the motion of a particle introduce the **Principle of least action **in a rigorous mathematical way.

In this lesson continues with the study of **Principle of least action,** but before to start he recalls some mathematical theorems like the **Integration by parts** and the fact that is the integral of a function A(t) multiplied by an arbitrary function f(t) is zero then the function A(t)=0 for all t. He introduced the concept **action **in a very general way as the integral over the trajectory of any function L(q,dq/dt) that depend on the position and the velocity of the particle then he applies the **variational principle** to the **action** and obtain the **Euler–Lagrange equations** as a condition for the minimal **action** and it shown that the **Lagrangian** L=T-U satisfied the **Euler–Lagrange equations**, where T is the kinetic energy and U the potential energy. Finally some conservations laws are obtained.