BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Konstantin Druzhkov DTSTART:20240501T162000Z DTEND:20240501T180000Z DTSTAMP:20260423T022917Z UID:GDEq/111 DESCRIPTION:Title: I nternal Lagrangians and gauge systems\nby Konstantin Druzhkov as part of Geometry of differential equations seminar\n\n\nAbstract\nIn classical mechanics\, the Hamiltonian formalism is given in terms of instantaneous p hase spaces of mechanical systems. This explains why it can be interpreted as an encapsulation of the Lagrangian formalism into the intrinsic geomet ry of equations of motion. This observation can be generalized to the case of arbitrary variational equations. To do this\, we describe instantaneou s phase spaces using the intrinsic geometry of PDEs. The description is gi ven by the lifts of involutive codim-1 distributions from the base of a di fferential equation viewed as a bundle with a flat connection (Cartan dist ribution). Such lifts can be considered differential equations\, which one can regard as gauge systems. They encode instantaneous phase spaces. In a ddition\, each Lagrangian of a variational system generates a unique eleme nt of a certain cohomology of the system. We call such elements internal L agrangians. Internal Lagrangians can be varied within classes of paths in the instantaneous phase spaces. This fact yields a direct (non-covariant) reformulation of the Hamiltonian formalism in terms of the intrinsic geome try of PDEs. Finally\, the non-covariant internal variational principle gi ves rise to its covariant child.\n LOCATION:https://researchseminars.org/talk/GDEq/111/ END:VEVENT END:VCALENDAR