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A canonical formalism based on the geometrical approach to the calculus of variations is given. The notion of multi-phase space is introduced which enables to define whole the canonical structure (physical quantities, Poisson bracket, canonical fields) without use of functional derivatives. All definitions are of pure geometrical (finite dimensional) character.The observable algebra (9 (physical quantities algebra) obtained here is much smaller then the algebra of all (sufficiently smooth) functional on the space of states, derived from the standard infinite-dimensional formulation. As it is known, the latter is much too large for purposes of quantization. As the examples prove, our algebra (9 could be an adequate start-point for quantization.For simplifying the language the notion of observable-valued distribution is introduced. Many concrete physical examples are given. E.g. it is shown that some problems connected with gauge in electrodynamics are automatically solved in this approach. The introduced language allows to obtain the Noether theorem in a most natural way.

It is shown that any theory of gravitation with a nonlinear Lagrangian depending on the Ricci tensor is equivalent to the Einstein theory of gravitation interacting with additional matter fields.Consider a metric theory of gravitation (possibly interacting with a bosonic matter field 4 A ) based on a nonlinear Lagrangian:where L,,, is a matter Lagrangian, R,, =R,,(g) is the Ricci curvature of a metric tensor g, y 1s the Levi-Civita connection of g, and ~= 8 . n G is the gravitational constant. The choice of the linear function F(g,,,R,, ) = R =R,,g@"(2)leads to the standard Einstein theory. Recently, e.g., alot of interest has been devoted to quadratic Lagrangians,'which will be discussed in the sequel as a special case of our problem. Now we consider a general nonlinear case (1). The Euler-Lagrange equations,form a system of fourth-order differential equations for the metric g unless F is linear. Equation (4) can be rewritten as where by D we denote the covariant derivative with respect to y. We introduce the auxiliary quantity A,"=( -detg ) ' / 2 a~/ a~, vwhich is at the moment merely an abbreviation for a tensor density built up for gpv and R,,:The specific form of relation (8) depends on the choice of the function F, i.e., on the choice of our Lagrangian (1) [e.g., for a quadratic Lagrangian (3) Xpv depends linearly on R,,; this example will be discussed in detail]. Relation (8) is a second-order differential equation for g,,. This enables us to rewrite Eq. (6) asThe above fourth-order differential equation for g,, is equivalent to the system (8) and (9) of second-order differential equations for independent quantities g,, andXpv. It is interesting to notice that the second-order differential operator acting on A, " is universal and does not depend on a special choice of a Lagrangian function (1). This leads to the following result, which we prove in

A general scheme of constructing a canonical structure (i.e. Poisson bracket, canonical fields) in classical field theories is proposed. The theory is manifestly independent of the particular choice of an initial space-like surface in space-time. The connection between dynamics and canonical structure is established. Applications to theories with a gauge and constraints are of special interest. Several physical examples are given.Recently one of us W. Szczyrba, using the general theory elaborated in the present paper has obtained a natural symplectic structure for a set on Einstein metrics in General Relativity. These results will be submitted for publication in Commun. math. Phys.The authors would like to thank Proffessors K. Maurin and I. Birula-Biaϊynicki for lively interest in their work and fruitful discussions.We thank also very much Professor D. Simms for his deep comments which were very valuable for us during the preparation of the manuscript.Our special thanks are due to Professor J. Ehlers for many profound remarks and improvements of the final version of this paper.