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Miloslav Znojil
Reviewer number:
Item's zbl-Number:
DE 017 478 486
Monterde, J.; Munoz Masque, J.:
Hamiltonian formalism in supermechanics
Int. J. Theor. Phys. 41, No. 3, 429 - 458 (2002)
70A05Axiomatics, foundations
Primary Classification:
58A50Supermanifolds and graded manifolds
Secondary Classification:
70HxxHamiltonian and Lagrangian mechanics
81S10Geometry and quantization, symplectic methods
graded symplectic structure; Poincare-Cartan form; the sheaf of the Berezinian Lagrangian densities; Batalin-Vilkovitsky structure; Koszul-Schouten bracket; graded first-order jet bundle; Euler-Lagrange equations; solving Hamiltonian equations; supersymmetric classical mechanics

A typical university course in classical mechanics (treated in
the spirit of the applied variational calculus) could contain,
say, the following chapters: ... 4. Action functional and
variational principle. 5. Euler - Lagrange equations. ... 7.
Poincare - Cartan forms. 8. Hamilton equations. ... 14 . Schouten
brackets. Precisely the same sections appear in the present
review, with the only re-qualification that the author speaks
about the classical mechanics in its supersymmetric
generalization. Originally intended to be a theory which involves
spin and distinguishes between the bosons and fermions (and
which, quite paradoxically, emerged first in its more complicated
quantum version), the subject became a full-fledged part of
mathematics after Berezin imagined the importance of the
underlying graded manifold structures and defined his integral in
a way which need not be cited equally well as one does not cite
Euler, Lagrange, Hamilton, Cartan and Poincare. In this context,
the authors develop the Hamilton - Cartan formalism on the space
of curves of a graded manifold. In this manner the even and odd
variables prove tractable, on precisely the same footing, in a
way presented in pleasing detail and emphasizing the existence of
the graded and symplectic structures on the solution manifolds.
The text is surprisingly compact (the authors often refer to
their older papers for more details) and pleasing the reader by a
number of useful insights associating, e.g., an equivalent
graded Lagrangian density to each Berezinian Lagrangian density
(with the link being given by the total derivative with respect
to the odd coordinate), offering the pattern of transition to the
Hamiltonian formalism in canonical coordinates, etc.
Remarks to the editors:

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