The Inverse Variational Problem in Classical Mechanics

The following sections are included:

• Foreword

• Contents

The following sections are included:

• Preliminary notions of Kinematics. Translations, proper rotations SO(3), Galilei group transformations

• Preliminary notions of Analytical Dynamics. Newton's Equations

• Constraints. Work. The Principle of Least Action. Euler-Lagrange Equations

• Constants of motion

• Poincaré Lemma and its converse

• Linear partial differential equations. The method of characteristics

• The Inverse Variational Problem.Helmholtz's Conditions

• Theorem of Henneaux

• The matrix σ. Tr σ^α is conserved quantity

• Instructive example of Cisło

• Instructive example of Douglas

• Instructive example of Pardo

• Construction of an autonomous one-particle Lagrange function in (3+1) space-time dimensions yielding rotationally covariant Euler-Lagrange Equations coinciding with the Newton Equations

• Canonical variables. Equivalence problem of the Lagrange functions

• All Lagrange functions, s-equivalent to the Lagrange function L = 1/2ẋ² - U(|x|), in (3+1) space-time dimensions

• The case U(|x|) ≠ αx² + β, where α, β – constants

• The case U(|x|) = αx² + β

• The Hamilton formalism for the model investigated in Subsections 8.1 - 8.3. Equivalence sets of Lagrange functions

• Examples
  • Example of Henneaux and Shepley
  • Example of Stichel
  • Example of Rañada
  • Second example of Ranãda
  • Example of Cisło

• The model of Subsections 8.1 - 8.4 for n ≠ 3

• All autonomous s-equivalent one-particle Lagrange functions for (1+1) space-time dimensions

• All s-equivalent one-particle Lagrange functions for (1+1) space-time dimensions

• Construction of the most general autonomous one-particle Lagrange function in (3+1) space-time dimensions giving rise to rotationally covariant Euler-Lagrange Equations

• Evaluation of the function G_ij

• Symmetry of G_ij and evaluation of the Lagrange function

• Symmetry properties of the Lagrange function

• The largest set of Lagrange functions of one-particle system in a (3+1) dimensional space-time, s-equivalent to a given Lagrange function yielding rotationally forminvariant Equations of Motion (formulation of the problem)

• Construction of the most general two-particle Lagrange function in (1+1) space-time dimensions giving rise to Euler-Lagrange Equations covariant under Galilei transformation

• Galilei forminvariance of the Euler-Lagrange Equations for two particles in (1+1) space-time dimensions

• Construction of the most general two-particle Lagrange function in (3+1) space-time dimensions giving rise to Euler-Lagrange Equations covariant under Galilei transformations

• Galilei forminvariance of the Euler-Lagrange Equations for two particles in (3+1) space-time dimensions

• All two-particle Lagrange functions s-equivalent to a given autonomous Lagrange function yielding Galilei forminvariant Equations of Motion in (1+1) space-time dimensions

• All Euler-Lagrange Equations, forminvariant under the Galilei transformations
  • Case g ≠ 0
  • Case g = 0

• Examples

• Generalization of the set of Lagrange functions (admission of Euler-Lagrange Equations not covariant under Galilei transformations)

• Case of Galilei forminvariant Newton's Equations corresponding to Euler-Lagrange Equations which are not Galilei covariant. (formulation of the problem)

• An Outlook. Application in the Feynman Approach to Quantum Mechanics

The following sections are included:

• Index