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 *q*^{a} and
*p*_{a}, in terms
of which the symplectic structure has the form d*p*_{a}
∧ d*q*^{a}.

* __Equations of motion__:
In terms of (*X*, Ω, *H*),
they are given by the Poisson brackets d*f*/d*t* = {*f*,* H*}
= Ω^{ab} ∇_{a}* f*
∇_{b}* H*.

* __Canonical momenta__: The canonical momentum associated with a configuration
variable *q*^{a} is the coefficient in the boundary term in δ*S*
when *q*^{a} 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 *A*_{a}
for Ω_{ab}
= 2 ∇_{[a}* A*_{b]},
if \(\cal L\)_{l}* A*_{a}
= 0 for any generator *l*^{a} of
the Lie algebra, then, for each such generator, the Hamiltonian
is *H* = *A*_{a} *l*^{a};
However, if the group of canonical transformations is compact,
such an *A*_{a} can always be
found; Given any *A*_{a}, define
the "average" over
the group, *A*_{a}^{avg}:=
∫_{G} L_{g}* *A*_{a} d*g* /
∫_{G} d*g*,
where d*g* 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,

*p*_{a}:= ∂*L*/∂*q*^{·}^{a} , *H*(*p*,*q*):= *p*_{a} *q*^{·}^{a} – *L*(*q*,*q*^{·},*t*)
;

for a field theory, if *T*_{ab} is
obtained from the action in the usual way (> see energy-momentum)

*p*_{a}:= δ*L*/δ∂_{t}* q ^{a}* ,

@ __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* = *H*_{1} + *H*_{2},
with *H*_{i} 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}* p*_{a} *p*_{b}
in a conformally related
metric* g'*_{ab}:= (*E*–*V*) *g*_{ab};
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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