Suppose that we have a dynamical system specified in terms of m independent variables x = ( x 1 , … … , x m ) , n dependent variables u = ( u 1 , … … , u n ) , and a Lagrangian function L ( x , u , u ( 1 ) … … , u ( r ) ) of some finite order r . Here u ( k ) = ( u i 1 . . . i k σ σ ) = ( d i 1 … … d i k u σ σ ) is the collection of all k th order partial derivatives of the dependent variables. As a general rule, latin indices i , j , k , … … from the middle of the alphabet take the values 1 , … … , m , greek indices take the values 1 , … … , n , and the summation convention apply to them. Multiindex notation for the latin indices is also introduced as follows. A multiindex I of length k is an ordered list I = ( i 1 , … … , i k ) of k ordinary indices. The length is denoted as | I | = k . The summation convention does not directly apply to multiindices since the summation over lengths needs to be displayed explicitly, e.g. ∑ ∑ | I | = 0 r f I g I = f g + f i g i + f i j g i j + ⋯ ⋯ + f i 1 . . . i r g i 1 . . . i r . The variation of the Lagrangian with respect to an arbitrary variation δ δ u σ σ of the independent variables is δ δ L = ∂ ∂ L ∂ ∂ u σ σ δ δ u σ σ + ∂ ∂ L ∂ ∂ u i σ σ δ δ u i σ σ + ⋯ ⋯ + ∂ ∂ L ∂ ∂ u i 1 . . . i r σ σ δ δ u i 1 . . . i r σ σ = ∑ ∑ | I | = 0 r ∂ ∂ L ∂ ∂ u I σ σ δ δ u I σ σ , and applying the inverse product rule of differentiation we get δ δ L = E σ σ δ δ u σ σ + d i ( ∑ ∑ | I | = 0 r − − 1 P σ σ i I δ δ u I σ σ ) where E σ σ = ∂ ∂ L ∂ ∂ u σ σ − − d i ∂ ∂ L ∂ ∂ u i σ σ + ⋯ ⋯ + ( − − 1 ) r d i 1 … … d i r ∂ ∂ L ∂ ∂ u i 1 . . . i r σ σ = ∑ ∑ | I | = 0 r ( − − 1 ) | I | d I ∂ ∂ L ∂ ∂ u I σ σ are the Euler-Lagrange expressions of the Lagrangian, and the coefficients P σ σ I (Lagrangian momenta) are given by P σ σ I = ∑ ∑ | J | = 0 r − − | I | ( − − 1 ) | J | d J ∂ ∂ L ∂ ∂ u I J σ σ