analytical mechanics an introduction jun 2006

[...]... 16 Lagrangian formalism in continuum mechanics 16.1 Brief summary of the fundamental laws of continuum mechanics 16.2 The passage from the discrete to the continuous model The Lagrangian function 16.3 Lagrangian formulation of continuum mechanics 16.4 Applications of the Lagrangian formalism to continuum mechanics ... coordinate lines u = constant and v = constant in the (x1 , x2 ) plane are not tangent to each other (Fig 1.11) It follows that Definition 1.12 is equivalent to the following x3 v = constant u = constant x2 x1 v = constant u = constant Fig 1.11 1.6 Geometric and kinematic foundations of Lagrangian mechanics Definition 1.13 If the surface S is point P is called non-singular if ⎛ ∂x1 ⎜ ∂u rank ⎝ ∂x1 ∂v 19 given... Geometric and kinematic foundations of Lagrangian mechanics 1.6 / / The analogous analysis can be performed if (∂F/∂x2 )P = 0, or (∂F/∂x1 )P = 0 Equation (1.24) highlights the fact that the points of a regular surface are, at least locally, in bijective and continuous correspondence with an open subset of R2 It is an easy observation that at a non-singular point x0 there exists the tangent plane, whose... the case of a plane curve, it is easy to verify that db/ds = 0, and hence that the binormal unit vector is constant and points in the direction orthogonal to the plane containing the curve Hence the derivative db/ds quantifies how far the curve is from being a plane curve To be more precise, consider a point x(s0 ) on the curve, and the pencil of planes whose axis is given by the line tangent to the curve... ϕ; 1.5 Geometric and kinematic foundations of Lagrangian mechanics 15 x3 n t x2 w A x1 Fig 1.10 yielding for the torsion χ=− R2 λ + λ2 Curvature and torsion are the only two geometric invariants of a curve in space Namely we have the following Theorem 1.4 Let k(s) > 0 and χ(s) be two given regular functions There exists a unique curve in space, up to congruences (rotations and translations), which... −k(s)t ds (1.16) Proof The first formula is simply equation (1.10) The second can be trivially derived from d (n · n) = 0, ds d (n · t) = 0 ds 1.3 Geometric and kinematic foundations of Lagrangian mechanics 11 We end the analysis of plane curves by remarking that the curvature function k(s) completely defines the curve up to plane congruences Namely, ignoring the trivial case of zero curvature, we have... 12 Analytical mechanics: canonical perturbation theory 12.1 Introduction to canonical perturbation theory 12.2 Time periodic perturbations of one-dimensional uniform motions 12.3 The equation Dω u = v Conclusion of the previous analysis 12.4 Discussion of the fundamental equation of canonical... this plane is the one whose normal vector is precisely the unit vector b(s0 ) 1.4 Geometric and kinematic foundations of Lagrangian mechanics 13 b n t osculating plane Fig 1.9 Definition 1.8 The plane normal to b(s0 ) is called the osculating plane to the curve at the point x(s0 ) (Fig 1.9) Hence the osculating plane has parametric equation y = x(s0 ) + λt(s0 ) + µk(s0 )n(s0 ) (1.19) In the case of curves... theorem 12.7 Adiabatic invariants 12.8 Problems 466 471 477 480 481 487 487 499 502 507 516 522 529 532 Contents xi 12.9 Additional remarks and bibliographical notes 534 12.10 Additional solved problems 535 13 Analytical mechanics: an introduction to ergodic theory and to chaotic motion 13.1...x Contents 10.8 10.9 10.10 10.11 10.12 10.13 10.14 Infinitesimal and near-to-identity canonical transformations Lie series Symmetries and first integrals Integral invariants Symplectic manifolds and Hamiltonian dynamical systems Problems Additional remarks and bibliographical notes Additional solved problems . y0 w0 h0" alt="" Analytical Mechanics This page intentionally left blank Analytical Mechanics An Introduction Antonio Fasano University of Florence Stefano Marmi SNS, Pisa Translated by Beatrice. Publication Data Fasano, A. (Antonio) Analytical mechanics : an introduction / Antonio Fasano, Stefano Marmi; translated by Beatrice Pelloni. p. cm. Includes bibliographical references and index. ISBN-13:. Analytical mechanics: canonical formalism 331 10.1 Symplectic structure of the Hamiltonian phase space 331 10.2 Canonical and completely canonical transformations 340 10.3 The Poincar´e–Cartan