Transformation Properties of the Lagrange— D’Alembert Variational Principle: Conservation Laws of Nonconservative Dynamical Systems
In this chapter we shall demonstrate that the Lagrange-D’Alembert differential variational principle can be used for the study of conser vat ion laws of conservat ive and purely nonconservative dynamical systems. The basic idea of this approach is to consider the transformation properties of the Lagrange-D’Alembert principle with respect to the infinite simaltransform at ion of the generalized coordinates and time. It is of interest to note that for the Lagrangian and Hamiltonian dynamical systems (i.e., for t he systems that are completely described by the Lagrangian or Hamiltonian functions and in which t he nonconservative forces are absent, Q i = 0) the way of obtaining the conservation laws is identical with the famous theory of Emmy Noether , which is based upon the transformation properties of the Hamiltonian action integral ∫ t Ldu. However , the approach based upon the Lagrange-D’Alembert differential variational principle admits the possibility to include into consideration purely nonconservative dynamical systems for which Q i ≠0.
KeywordsTransformation Property Gauge Function Duffing Oscillator Infinitesimal Transformation Nonconservative Force
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