### Classical mechanics Lecture 14 of 16

October 12, 2012 by Multimedia Publications and Printing Services

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).

Hamilton's Principal( The Principal of Least Action)

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

Lecture 2/9 | Modern Physics: Classical Mechanics (Stanford)

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.

Lecture 3/9 | Modern Physics: Classical Mechanics (Stanford)

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.

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