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Lecture

Hamiltonian Mechanics: Analytical Approach

In course

PHYS-202: Analytical mechanics (for SPH)

Ce cours propose une introduction à la mécanique analytique. Il introduit les formalismes lagrangien et hamiltonien, en insistant sur leur capacité à simplifier la description des systèmes contraints, des symétries et des lois de conservation. Il présente ...

Description

This lecture covers the analytical approach to Hamiltonian mechanics, focusing on the harmonic oscillator and the Hamilton-Jacobi equation. It discusses the conservation of quantities, separation of variables, and the quadratic form of kinetic energy. The instructor explains the process of putting everything back together, the bias of examples, and the explicit appearance of certain terms. The lecture also delves into the complexities of potential functions and the ambiguity of certain quantities. It concludes with a detailed exploration of canonical transformations and the hidden effects on the temporal evolution of systems.

Instructors (2)

Paolo De Los Rios

Paolo De Los Rios earned his master in Electronic Engineering at the Turin Institute of Technology (Politecnico di Torino) in May 1993. In November 1993 he moved to Trieste, Italy, to enter the PhD program in Theoretical Condensed Matter Theory at the International School for Advanced Studies (SISSA/ISAS) where he obtained the PhD degree in October 1996 for his work on the statistical physics of disordered systems. After a one year postdoc at the Max-Planck Institute for the Physics of Complex Systems in Dresden, Germany, in November 1997 he moved to the University of Fribourg, Switzerland, to join the group of Prof. Yi-Cheng Zhang. There he has worked on various applications of statistical physics to complex systems. In September 2000 he has been appointed Assistant Professor in Statistical Physics of Living Matter and Complex Systems at the Institute of Theoretical Physics of the University of Lausanne, Switzerland. Since April 2010 he is Associate Professor at the Institute of Theoretical Physics of the Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland.

Philippe René Joseph Jacquod

Official source

Ontological neighbourhood

Physics

Mechanics: Classical mechanics

Related concepts (32)

Hamiltonian mechanics

Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics.

Quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, is a quadratic form in the variables x and y. The coefficients usually belong to a fixed field K, such as the real or complex numbers, and one speaks of a quadratic form over K. If , and the quadratic form equals zero only when all variables are simultaneously zero, then it is a definite quadratic form; otherwise it is an isotropic quadratic form.

Analytical mechanics

In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the 18th century and onward, after Newtonian mechanics. Since Newtonian mechanics considers vector quantities of motion, particularly accelerations, momenta, forces, of the constituents of the system, an alternative name for the mechanics governed by Newton's laws and Euler's laws is vectorial mechanics.

Definite quadratic form

In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every non-zero vector of V. According to that sign, the quadratic form is called positive-definite or negative-definite. A semidefinite (or semi-definite) quadratic form is defined in much the same way, except that "always positive" and "always negative" are replaced by "never negative" and "never positive", respectively.

Isotropic quadratic form

In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More explicitly, if q is a quadratic form on a vector space V over F, then a non-zero vector v in V is said to be isotropic if q(v) = 0. A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or null vector) for that quadratic form. Suppose that (V, q) is quadratic space and W is a subspace of V.

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Canonical Transformations: Phase Portraits and Action Variables

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