Hamiltonian Dynamics  

In General > s.a. hamiltonian systems [including boundaries]; Momentum; phase space.
* Motivation: An elegant, geometrical way of expressing the dynamical content of a physical theory (usually the system must be non-dissipative); It is convenient for the study of symmetries and conservation laws, and necessary for the covariant quantization method.
* Idea: Choose a phase space or Hamiltonian manifold (with a symplectic structure and preferred Hamiltonian function H, whose Hamiltonian vector field gives time evolution); Usually, one starts with a space manifold, and a configuration space of states defined on it, and uses as phase space the cotangent bundle over configuration space, with natural canonically conjugate coordinates qa and pa, in terms of which the symplectic structure has the form dpa ∧ dqa.
* Equations of motion: In terms of (X, Ω, H), they are given by the Poisson brackets df/dt = {f, H} = Ωaba fb H.
* Canonical momenta: The canonical momentum associated with a configuration variable qa is the coefficient in the boundary term in δS when qa is varied at the endpoint of a trajectory.

Mathematically > s.a. symplectic structures.
* Structure on phase space: A (pre)symplectic manifold with a preferred Hamiltonian function, i.e., a triple (X, Ω, H); The function H generates canonical transformations which correspond to time evolution.
* Generating functions: In general, it is easy to find the Hamiltonian vector field given the generating function, but not viceversa, except for some cases, like when the potential form for the symplectic structure is Lie-derived by all generators of the Lie algebra of canonical transformations; given the potential Aa for Ωab = 2 ∇[a Ab], if \(\cal L\)l Aa = 0 for any generator la of the Lie algebra, then, for each such generator, the Hamiltonian is H = Aa la; However, if the group of canonical transformations is compact, such an Aa can always be found; Given any Aa, define the "average" over the group, Aaavg:= ∫G Lg* Aa dg / ∫G dg, where dg is a left-invariant measure.
@ Geometric approach: Casetti et al PRP(00); Kocharyan in(93)ap/04; Miron a1203; Rajeev a1701 [generalized notion of curvature].

Approaches > s.a. gauge transformations [gauge-covariant canonical formalism]; lagrangian systems [relationship]; poisson structure.
* Possibilities: Canonical analysis of S; Noether theorem; Symplectic analysis of L.
* And Lagrangian formulation: The relationship is best understood in terms of the Weiss variational principle (> see lagrangian dynamics); For a mechanical system,

pa:= ∂L/∂q·a ,   H(p,q):= pa q·aL(q,q·,t) ;

for a field theory, if Tab is obtained from the action in the usual way (> see energy-momentum)

pa:= δL/δ∂t qa ,   H:= T00 .

@ Hamiltonian from Noether theorem: Francaviglia & Raiteri CQG(02)gq/01 [and general relativity with boundaries].
@ For field theories: Giachetta et al 97; Krupková JGP(02) [Lepagean form]; Danilenko TMP(13) [modified approach].

Related Concepts and Techniques > s.a. canonical quantum mechanics; conservation laws; constrained systems [and reduction]; Transport.
* Geometry: For 1 degree of freedom, the constant energy surfaces in phase space are elliptic manifolds; For 2 degrees of freedom, 2 out of the 8 possible geometries for 3-manifolds can occur as constant energy surfaces, not the hyperbolic one.
* Symplectic integration method: Used for H = H1 + H2, with Hi exactly integrable [@ in Berger et al CQG(97)gq/96].
* Maupertuis principle: The dynamics of a system with Hamiltonian H = \({1\over2}g_{ab}\,\dot q^a\dot q^b\) + V(q) and energy E can be mapped to geodesic motion H = g'ab pa pb in a conformally related metric g'ab:= (EV) gab; Used, e.g., for Bianchi models; > s.a. jacobi dynamics; variational principles.
@ Symmetries: Mukhanov & Wipf IJMPA(95)ht/94; Deriglazov & Evdokimov IJMPA(00)ht/99; Mignemi ht/00 [1D system]; Dorodnitsyn & Kozlov a0809 [and first integrals]; Kay PRA(09)-a0911 [rotational invariance]; Frolov a1407 [for constrained Hamiltonian systems]; > s.a. gauge transformations; noether's theorem.
@ Stability of equilibria / orbits: Ortega & Ratiu JGP(99), JGP(99); > s.a. classical systems.
@ Symplectic integration: Rangarajan PLA(01).
@ Maupertuis principle: in Arnold 89; Szydłowski et al JMP(96); Izquierdo et al mp/02-proc [and second-order variational calculus]; > s.a. chaos in general relativity; quantization.

References > s.a. classical mechanics; Perturbation Methods.
@ General: Lucey & Newman JMP(88); Bailey FP(04) [and history]; Jordan AJP(04)aug [quick tutorial]; Low a0903 [new derivation]; Galley PRL(13)-a1210 [formulation compatible with initial-value problems]; de Gosson a1501 [as a link between classical Hamiltonian flows and quantum propagators].
@ Books: Sudarshan & Mukunda 75; Abraham & Marsden 78; Calkin 96 [+ solutions 99]; Vilasi 01; Lowenstein 12 [III, r CP(12), PT(13)mar]; Curry 13; Hamill 13 [II]; Nolting 16; Cortés & Haupt book(17)-a1612 [lecture notes].
@ Non-uniqueness: Hojman JPA(91); Polyzou PRC(10)-a1001 [equivalent Hamiltonians].
@ Alternative Hamiltonian descriptions: Cawley PRD(80) [generalized]; Chruściński & Kijowski JGP(98) [gauge-invariant, charged particle]; Horikoshi & Kawamura PTEP-a1304 [hidden Nambu mechanics]; Horwitz et al a1511 [underlying geometrical manifold]; > s.a. magnetism [non-canonical]; statistical mechanics.
@ With respect to a timelike vector field with expansion: Roberts EPL(99)gq/98.
@ And multisymplectic formalism: Francaviglia et al mp/03-conf; Echeverría-Enríquez et al JMP(07)mp/05 [for field theories].
@ Other covariant: Lachièze-Rey a1602 [histories-based]; see hamiltonian systems [including generalizations]; modified symplectic structures.


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