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\(t\)
= \(\{f,H\} = \Omega^{ab}\, \nabla_{\!a} f \nabla_{\!b} H\).

* __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 \(\Omega_{ab}\)
= 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; constrained systems
[and reduction].

* __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.

@ __General references__: de Gosson a1501 [as a link between classical Hamiltonian flows and quantum propagators];
Pavelka & Klika JCP-a1810 [self-regularization of Hamiltonian systems];
de Gosson a2002
[classical and semiclassical time evolutions of subsystems].

@ __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);
Babichev et al a1803 [and unbounded Hamiltonians];
> 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.

> __Other topics__: see conservation laws;
Machine Learning; Transport.

**References**
> s.a. classical mechanics / causality;
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];
Baumgarten a2001 [key role in physics].

@ __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];
Sanyal a1807
[the problem and a remedy, and higher-order gravity].

@ __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];
Hadad & Rosenblum a1905 [on non-Cauchy hypersurfaces];
Carcassi & Aidala SHPMP-a2004 [and conservation of information entropy];
> 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];
> s.a. hamiltonian systems [including generalizations];
modified symplectic structures.

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