Hamiltonian
Dynamics| 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, 
, H),
they are given by the Poisson brackets df/dt = {f, H}
= ab 
a f b H.
* Canonical momenta: The canonical momentum associated with a configuration
variable qa is the coefficient in the boundary term in 
S
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, a triple
(X, , H); 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 ab
= 2 [a Ab],
if 
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.
@ Mathematical: Casetti et al PRP(00) [geometric]; Kocharyan in(93)ap/04 [new
geometrical approach].
Approaches > s.a. 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·a – L(q,q·,t)
;
for a field theory, if Tab is obtained from the action in the usual way (> see energy-momentum)
pa:= L/t qa , H:= T00
.
@ Hamiltonian from Noether theorem: Francaviglia & Raiteri CQG(02)gq/01 [and
general relativity with boundaries].
@ For field theories: Krupková JGP(02)
[Lepagean form].
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 = (1/2) gab q·a 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:= (E–V) 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]; > 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; Szydlowski et al JMP(96);
Izquierdo
et al mp/02-in
[and second-order variational calculus]; > s.a.
chaos in general relativity; quantization.
References > s.a. Perturbation
Methods.
@ General: Sudarshan & Mukunda 75; Abraham & Marsden 78; Lucey & Newman
JMP(88);
Vilasi 00; Bailey FP(04)
[and history]; Jordan AJP(04)aug
[quick tutorial]; Low a0903 [new derivation].
@ Non-uniqueness: Hojman JPA(91).
@ Alternative Hamiltonian descriptions: Chruscinski & Kijowski JGP(98)
[gauge-invariant, charged particle]; > s.a. statistical
mechanics.
@ Wrt a timelike vector field with expansion: Roberts EPL(99)gq/98.
@ And multisymplectic formalism: Francaviglia et al mp/03-in;
Echeverría-Enríquez et al JMP(07)mp/05 [for
field theories].
> Other covariant: see hamiltonian
systems; modified symplectic
structures.
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