100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

420 R. Gambini and J. PullinTo clarify ideas, let us consider an example. The model consists of aparameterized free particle in a two dimensional space-time under the influenceof a linear potential. The discrete Lagrangian is given by,L n ≡ L(q a n, π a n, N n , q a n+1, π a n+1, N n+1 ) (9)= π a n (qa n+1 − qa n ) − N n[π 0 n + 1 2 (π1 n )2 + αq 1 n ].We have chosen a first order formulation for the particle (otherwise thereare no constraints and the example is trivial). However, this Lagrangian isof the type we considered in this paper, one simply needs to consider allvariables, q a , π a , N as configuration variables. The system is clearly singularsince the π ′ s and N only appear at level n (or in the continuum Lagrangian,their time derivatives are absent). When considered as a Type I generatingfunction, the above Lagrangian leads to the equationsp a π, n+1 =p a q, n+1 =∂L n∂π a n+1∂L n∂q a n+1= 0, (10)= π a n, (11)andp N, n+1 =∂L n∂N n+1= 0, (12)p a π, n = −∂L n∂π a np a q, n = − ∂L n∂q a n= −(q a n+1 − qa n ) + π1 n N nδ a 1 + N nδ a 0 , (13)= π a n + δ a 1αN n , (14)p N, n = − ∂L n∂N n= π 0 n + 1 2 (π1 n )2 + αq 1 n . (15)The constraints (10,12,14,15) can be imposed strongly to eliminate the π’sand the N’s and obtain an explicit evolution scheme for the q’s and thep q ’s,qn 0 = q0 n+1 − C n+1αp 1 , (16)q, n+1qn 1 = qn+1 1 − C n+1α , (17)p 0 q, n = p0 q, n+1 , (18)p 1 q, n = p1 q, n+1 + C n+1p 1 , (19)q, n+1