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Variation

# Variation - Variational methods in which TIME is also...

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Variational methods in which TIME is also varied Shina Tan We traditionally fix the time variable of a path in the variations. In relativity, however, time is intimately related to space. Time and space are mixed if we go from one reference frame to a different frame. It is unnatural to variate spatial coordinates only but not time. Even in nonrelativistic physics, it is beneficial to variate both space and time. For instance, this more general variation offers a deeper insight into the energies of periodic motion (see Sec. 3.2). In the following we first consider general variations in Lagrangian dynamics, and then study those in Hamiltonian dynamics. In the latter case, the variables q j , p j , and t are all varied independently, yielding even greater flexibility. The general principles outlined in Secs. 1 and 2 are applicable to both nonrelativistic mechanics and relativity . 1 Lagrangian Dynamics Consider a Lagrangian system L = L ( q 1 , · · · , q s , ˙ q 1 , · · · , ˙ q s , t ) . (1) Let us call the ( s + 1)-dimensional space of all possible values of ( q 1 , · · · , q s , t ) the “QT space”. An arbitrary infinitesimal transformation in the QT space would transform any actual path ( q j , t ) , t 1 < t < t 2 to a neighboring path (called “varied path”) ( q j + δq j , t + δt ) , t 1 + δt 1 < t < t 2 + δt 2 which may or may not be a classically allowed path (namely, the varied path may or may not satisfy classical equations of motion as the actual path does). In particular, each of the two end points of the actual path is transformed to a neighboring point with slightly different values of q j and t . For any quantity X that depends on q j , ˙ q j and t , we define the differential of X as the infinitesimal change of X ALONG a path: dX X ( q j + dq j , ˙ q j + d ˙ q j , t + dt ) - X ( q j , ˙ q j , t ) . (2) 1

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We also define the variation of X as the infinitesimal change of X ACROSS two paths (the actual path and the varied one): δX X ( q j + δq j , ˙ q j + δ ˙ q j , t + δt ) - X ( q j , ˙ q j , t ) . (3) Because ( A - B ) - ( C - D ) = ( A - C ) - ( B - D ), the variation of the differential is equal to the differential of the variation: δdX = dδX. (4) We define an action S for any path: S [path] = Z Ldt, (5) where the integral is from one end point to the other end point. The variation of the action is defined as δS S [varied path] - S [actual path] . (6) Variation, like differential, obeys the product rule. So δS = Z ( δL ) dt + Lδdt . Since R Lδdt = R Ldδt , we can use integral by parts to treat the second term: Z Lδdt = Z Ldδt = Z d ( Lδt ) - ( dL ) δt = ( Lδt ) t 2 t 1 - Z ( dL ) δt. Thus δS = ( Lδt ) t 2 t 1 + Z ( δL ) dt - ( dL ) δt . 2
Since δL = j ( ∂L ∂q j δq j + ∂L ˙ q j δ ˙ q j ) + ∂L ∂t δt and dL = j ( ∂L ∂q j dq j + ∂L ˙ q j d ˙ q j ) + ∂L ∂t dt , we have ( δL ) dt - ( dL ) δt = X j ∂L ∂q j ( δq j ) dt - ( dq j ) δt + ∂L ˙ q j ( δ ˙ q j ) dt - ( d ˙ q j ) δt = X j ∂L ∂q j ( δq j ) dt - ( dq j ) δt + ∂L ˙ q j ( δ ˙ q j ) dt + ˙ q j δdt - ( d ˙ q j ) δt - ˙ q j dδt = X j ∂L ∂q j ( δq j ) dt - ( dq j ) δt + ∂L ˙ q j δ ( ˙ q j dt ) - d ( ˙ q j δt ) = X j ∂L ∂q j ( δq j ) dt - ( dq j ) δt + ∂L ˙ q j δ ( dq j ) - d ( ˙ q j δt ) = X j ∂L ∂q j ( δq j ) dt - ( dq j ) δt + ∂L

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