Infinitesimal Calculus of Variations

It is desirable that physical laws should beformulated infinitesimally, while it is well known thatthe calculus of variations, which has long beenconcerned with local or global horizons, gives aunifying viewpoint of various arenas of modern physics.The principal objective of this paper is toinfinitesimalize the calculus of variations by makinguse of the vanguard of modern differential geometry,namely, synthetic differential geometry, in whichnilpotent infinitesimals of various orders areabundantly and coherently available. Our treatment iscompletely coordinate-free, the decomposition of a stateinto its position and velocity components beingreplaced by the vertical-horizontal decompositionassociated with an appropriate connection. Within ournewly established infinitesimal calculus of variations, generalized conservation laws of momentum andenergy are demonstrated.