Can one write down a Hamiltonian in the absence of a Lagrangian? How can I define the Hamiltonian independent of the Lagrangian? For instance, let's assume that i have a set of field equations that cannot be integrated to an action. Is there any prescription to construct the Hamiltonian of a such system starting from the field equations?
 A: The field equations must be conservative in a fairly precise sense in order that this can be done in a physically appropriate sense. 
Then there are several Hamiltonian approaches to field theory: the De Donder-Weyl formalism and the multisymplectic formalism. Although both formalisms can accommodate Lagrangians, the can also be understood without any Lagrangian, in a purely Hamiltonian form. Both formalisms can be made fully covariant.
This holds for classical fields. How to quantize a theory in one of these formalisms is very poorly understood.
A: Comments to the question (v2):


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*First of all, let us stress that OP is correct, that a given set of equations of motion does not necessarily have a variational/action principle, cf. e.g. this Phys.SE post and links therein.

*On one hand, if there exists a Lagrangian formulation, then one may in principle obtain a Hamiltonian formulation via a (possible singular) Legendre transformation. Traditionally this is done via the Dirac-Bergmann recipe/cookbook, see e.g. Refs. 1-2. 

*On the other hand, if we have a (possible constrained) Hamiltonian formulation, of the type discussed in Refs. 1 and 2, then it is possible to give a Hamiltonian action formulation, which in itself can be interpreted as a Lagrangian formulation, e.g. after integration out momentum variables along the lines indicated in my Phys.SE answer here. 

*In other words, Lagrangian and Hamiltonian formulations traditionally go hand in hands. Thus it is unclear what precisely OP is looking for.
References:


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*P.A.M. Dirac, Lectures on QM, (1964).

*M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, 1994.
