2.1 General aspects of conserved quantities Taking only the linear terms we can write the variation of the Lagrangian as 6L= (8) We now assume that the variation of the action can be written as the integral of the variation of the Lagrangian 0=6S= a∑ aL d aLl δqi +, dt agi| (9) In the last equality we used integration by parts and that the variation is zero at the endpoints of integration.By the fundamental lemma of the calculus of variations this integral is zero only if the integrand is zero.Also we assume that the coordinates are independent,which implies that the coefficient in front of each Ogi is zero,and thus we obtain the Euler-Lagrange-equations ∂Ld∂L =0. (10) Oqi dt aqi These are one equation for each coordinate gi.By the Euler-Lagrange-equations we can write ∑- 三 d aL Zdt∂4 (11) By inserting this expression in the expression for the variation of the Lagrangian we can write =∑ d aL sqs dt aq 0q (12)2.1 General aspects of conserved quantities Taking only the linear terms we can write the variation of the Lagrangian as δL = X i ∂L ∂qi δqi + X i ∂L ∂q˙i δq˙i . (8) We now assume that the variation of the action can be written as the integral of the variation of the Lagrangian 0 = δS = Z t2 t1 dtδL = Z t2 t1 dtX i  ∂L ∂qi δqi + ∂L ∂q˙i δq˙i  = Z t2 t1 dtX i  ∂L ∂qi − d dt ∂L ∂q˙i  δqi . (9) In the last equality we used integration by parts and that the variation is zero at the endpoints of integration. By the fundamental lemma of the calculus of variations this integral is zero only if the integrand is zero. Also we assume that the coordinates are independent, which implies that the coefficient in front of each δqi is zero, and thus we obtain the Euler-Lagrange-equations ∂L ∂qi − d dt ∂L ∂q˙i = 0. (10) These are one equation for each coordinate qi . By the Euler-Lagrange-equations we can write X i ∂L ∂qi = X i d dt ∂L ∂q˙i . (11) By inserting this expression in the expression for the variation of the Lagrangian we can write δL = X i  d dt ∂L ∂q˙i δqi  + X i ∂L ∂q˙i δq˙i = X i d dt ∂L ∂q˙ δqi  . (12) 6