Runge-Lenz-vector from a symmetry of the action integral 2 Runge-Lenz-vector from a symmetry of the ac- tion integral 2.1 General aspects of conserved quantities Noether's theorem relates conservation laws with symmetries.A somewhat simpli- fied way to state this theorem is to say that if the action integral is invariant under some transformation then there is some conserved quantity.Hamilton's principle says that the physical path of a system is such that the action is stationary.This simply means that the action is invariant under an infinitesimal variation of the path as g(t)(t)=qi(t)+oni.Besides the Euler-Lagrange-equations,this also implies conservations laws.A trivial example is a Lagrangian with some cyclic co- ordinate,then the canonical momentum conjugate to this coordinate is conserved. This follows directly from the Euler-Lagrange-equations and thus this conservation law follows from Hamilton's principle.Noether's theorem can somewhat simplified be stated as,an invariance of the Lagrangian corresponds to a conservation law. To see how this works we vary the path as 9→q=9+0q, (5) where the variation ogi is such that it is zero at the endpoints.The velocity then becomes 4→6=+0： (6) The Lagrangian of the new coordinates can be expanded in a power series as 60=e创+影+∑ 6+ (7)
Runge-Lenz-vector from a symmetry of the action integral 2 Runge-Lenz-vector from a symmetry of the action integral 2.1 General aspects of conserved quantities Noether’s theorem relates conservation laws with symmetries. A somewhat simpli- fied way to state this theorem is to say that if the action integral is invariant under some transformation then there is some conserved quantity. Hamilton’s principle says that the physical path of a system is such that the action is stationary. This simply means that the action is invariant under an infinitesimal variation of the path as q(t) 7→ q 0 i (t) = qi(t) + δηi . Besides the Euler-Lagrange-equations, this also implies conservations laws. A trivial example is a Lagrangian with some cyclic coordinate, then the canonical momentum conjugate to this coordinate is conserved. This follows directly from the Euler-Lagrange-equations and thus this conservation law follows from Hamilton’s principle. Noether’s theorem can somewhat simplified be stated as, an invariance of the Lagrangian corresponds to a conservation law. To see how this works we vary the path as qi 7→ q 0 i = qi + δqi , (5) where the variation δqi is such that it is zero at the endpoints. The velocity then becomes q˙i 7→ q˙ 0 i = ˙qi + δq˙i . (6) The Lagrangian of the new coordinates can be expanded in a power series as L(q 0 i , q˙ 0 i ) = L(qi , q˙i) +X i ∂L ∂qi δqi + X i ∂L ∂q˙i δq˙i + . . . (7) 5
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