Mathematical Methods for Physics and Engineering - Matematica.NET

22.5 PHYSICAL VARIATIONAL PRINCIPLESyOdx l xFigure 22.9 Transverse displacement on a taut string that is fixed at twopoints a distance l apart.states that in moving from one configuration at time t 0 to another at time t 1 themotion of such a system is such as to make∫ t1L = L(q 1 ,q 2 ...,q n , ˙q 1 , ˙q 2 ,...,˙q n ,t) dt (22.21)t 0stationary. The Lagrangian L is defined, in terms of the kinetic energy T andthe potential energy V (with respect to some reference situation), by L = T − V .Here V is a function of the q i only, not of the ˙q i . Applying the EL equation to Lwe obtain Lagrange’s equations,∂L= d ( ) ∂L, i =1, 2,...,n.∂q i dt ∂˙q i◮Using Hamilton’s principle derive the wave equation for small transverse oscillations of ataut string.In this example we are in fact considering a generalisation of (22.21) to a case involvingone isolated independent coordinate t, togetherwithacontinuum in which the q i becomethe continuous variable x. The expressions for T and V therefore become integrals over xrather than sums over the label i.If ρ and τ are the local density and tension of the string, both of which may depend onx, then, referring to figure 22.9, the kinetic and potential energies of the string are givenbyand (22.21) becomes∫ lT =0L = 1 2ρ2∫ t1( ) 2 ∫ ∂ylτdx, V =∂t0 2t 0dt∫ l0[ρ( ) 2 ∂y− τ∂t789( ) 2 ∂ydx∂x( ) ] 2 ∂ydx.∂x