By Harrison D.M.

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Each class can be characterized by the pair (x, y) ∈ 2 , and therefore the manifold X is isomorphic to the Euclidean plane 2 . Given two arbitrary points in this plane (x 1 , y 1 ) and ( x 2 , y 2 ), they correspond respectively to the sets of group elements ( β, x 1 , y 1 ) and ( β ' , x 2 , y 2 ) of ε 2 , and therefore . Thus , there exists at least a group leads to element that links both points of the manifold 2 . It is in fact a homogeneous space of ε 2 . 7. GENERALIZED NOETHER’S THEOREM Noether’s analysis for generalized Lagrangian systems also states the following Theorem: To every one-parameter group of continuous transformations that transform the action function of the system in the form leaving dynamical equations invariant, there is associated a classical observable N, which is a constant of the motion.

11) of the Lagrangian. 54) in terms of the functions F i and their time derivatives. 55) We see that the Noether constants of the motion N α are finally expressed in terms of the functions Fi and their time derivatives, of the (s ) functions M i α that represent the way the different kinematical variables transform under infinitesimal transformations, and of the functions B α which, as we shall see below, are related to the exponents of the group GENERAL FORMALISM 31 G. Functions Fi and their time derivatives are homogeneous functions of zero degree in terms of the derivatives of the kinematical variables i.

35) In particular, when acting with the operator on j the coordinate x we get The operators X σ are called the generators of the infinitesimal transformations. 36) The coefficients are a set of real constant numbers, called the structure constants of the group, and the vector space spanned by the generators is named the Lie algebra L (G), associated to the Lie group G. 36) are the commutation relations that characterize the structure of the Lie algebra of the group. If a finite group transformation of parameters g σ can be done in n smaller steps of parameters g σ /n, with n sufficiently large, then a finite transformation U (g) h(x) can be obtained as 21 GENERAL FORMALISM This defines the exponential mapping and in this case the group parameters g σ are called normal or canonical parameters.