Lecture

Constraints and Lagrange Formalism

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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 by the instructor covers the concept of constraints in physics, focusing on degrees of freedom and Lagrange formalism. The slides discuss generalized coordinates, holonomic constraints, and the Lagrange representation of systems. Emphasis is placed on mathematical representations and the principle of d'Alembert.

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

Related concepts (18)

Holonomic constraints

In classical mechanics, holonomic constraints are relations between the position variables (and possibly time) that can be expressed in the following form: where are n generalized coordinates that describe the system (in unconstrained configuration space). For example, the motion of a particle constrained to lie on the surface of a sphere is subject to a holonomic constraint, but if the particle is able to fall off the sphere under the influence of gravity, the constraint becomes non-holonomic.

Canonical coordinates

In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of classical mechanics. A closely related concept also appears in quantum mechanics; see the Stone–von Neumann theorem and canonical commutation relations for details.

Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules.

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).

Irreducible representation

In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation or irrep of an algebraic structure is a nonzero representation that has no proper nontrivial subrepresentation , with closed under the action of . Every finite-dimensional unitary representation on a Hilbert space is the direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.

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