By Nail H. Ibragimov
This graduate textbook starts off via introducing Tensors and Riemannian areas, after which elaborates their software in fixing second-order differential equations, and ends with introducing idea of relativity and de Sitter house. according to forty years of educating event, the writer compiles a well-developed choice of examples and workouts to facilitate the reader's studying.
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Example text
16) Now we use the infinitesimals α α i x ¯i ≈ xi + aξ i , u ¯α ≈ uα + aη α , u ¯α i ≈ ui + aζi , J ≈ 1 + aDi (ξ ) and obtain L(¯ x, u ¯, u ¯(1) )J ≈ L(x, u, u(1) ) + a X(L) + LDi (ξ i ) . This equation implies Eq. 14). 7). 1. 14). Then the vector C = (C 1 , . . , C n ) with the components C i = ξiL + W α ∂L , ∂uα i i = 1, . . e. 7) = 0. In Eqs. 17) and in the expressions of conserved vectors for higher-order Lagrangians given further, I use the following notation: W α = η α − ξ i uα i , α = 1, .
6) are written ∂L δL ∂L ≡ − Di α α δu ∂u ∂uα i = 0, α = 1, . . , m. 6) have the form δL ∂L ∂L ≡ − Di α α δu ∂u ∂uα i + Di Dk ∂L ∂uα ik = 0, α = 1, . . , m. 3 29 Conserved vectors defined by symmetries In the classical mechanics, various mechanical systems are characterized by first-order Lagrangians L = L(t, x, x), ˙ where the independent variable is the time t, the dependent variables are the coordinates x = (x1 , . . , xm ) of particles of the system, and x˙ is the vector with the components x˙ α ≡ dxα , dt α = 1, .
14). Then the conserved vector is written Ai = C i − B i , i = 1, . . 23) where C i are given by Eqs. 21) for the first-order, second-order and higher-order Lagrangians, respectively. 22). 9), ∂L d ∂L − = 0, α ∂x dt ∂ x˙ α α = 1, . . , m, with a first-order Lagrangian L = L(t, x, x). 24) ∂t ∂x be a symmetry generator for Eqs. 9). Then Eqs. 17) give the following conserved quantity: ∂L C = ξL + (η α − ξ x˙ α ) α . 1. The free motion of a particle with a constant mass m provides a simple example for illustrating Noether’s theorem.