A Hamiltonian system is a
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
governed by
Hamilton's equations
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
. In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, this dynamical system describes the evolution of a
physical system
A physical system is a collection of physical objects.
In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the ...
such as a
planetary system
A planetary system is a set of gravitationally
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interacti ...
or an
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no kn ...
in an
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
. These systems can be studied in both
Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
and
dynamical systems theory
Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called '' ...
.
Overview
Informally, a Hamiltonian system is a mathematical formalism developed by
Hamilton Hamilton may refer to:
People
* Hamilton (name), a common British surname and occasional given name, usually of Scottish origin, including a list of persons with the surname
** The Duke of Hamilton, the premier peer of Scotland
** Lord Hamilt ...
to describe the
evolution equations of a physical system. The advantage of this description is that it gives important insights into the dynamics, even if the
initial value problem
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or ...
cannot be solved analytically. One example is the
planetary movement of three bodies: while there is no
closed-form solution
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th roo ...
to the general problem,
Poincaré showed for the first time that it exhibits
deterministic chaos
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have ...
.
Formally, a Hamiltonian system is a dynamical system characterised by the scalar function
, also known as the Hamiltonian.
The state of the system,
, is described by the
generalized coordinates
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
and
, corresponding to generalized momentum and position respectively. Both
and
are real-valued vectors with the same dimension ''N''. Thus, the state is completely described by the 2''N''-dimensional vector
:
and the evolution equations are given by
Hamilton's equations
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
:
:
The trajectory
is the solution of the
initial value problem
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or ...
defined by Hamilton's equations and the initial condition
.
Time-independent Hamiltonian systems
If the Hamiltonian is not explicitly time-dependent, i.e. if
, then the Hamiltonian does not vary with time at all:
[
and thus the Hamiltonian is a constant of motion, whose constant equals the total ]energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
of the system: . Examples of such systems are the undamped pendulum, the harmonic oscillator, and dynamical billiards
A dynamical billiard is a dynamical system in which a particle alternates between free motion (typically as a straight line) and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss of speed ...
.
Example
An example of a time-independent Hamiltonian system is the harmonic oscillator. Consider the system defined by the coordinates and . Then the Hamiltonian is given by
:
The Hamiltonian of this system does not depend on time and thus the energy of the system is conserved.
Symplectic structure
One important property of a Hamiltonian dynamical system is that it has a symplectic structure
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Ha ...
.[ Writing
:
the evolution equation of the dynamical system can be written as
:
where
:
and ''I''''N'' is the ''N''×''N'' ]identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
.
One important consequence of this property is that an infinitesimal phase-space volume is preserved.[ A corollary of this is Liouville's theorem, which states that on a Hamiltonian system, the phase-space volume of a closed surface is preserved under time evolution.][
:
where the third equality comes from the ]divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
.
Examples
*Dynamical billiards
A dynamical billiard is a dynamical system in which a particle alternates between free motion (typically as a straight line) and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss of speed ...
*Planetary system
A planetary system is a set of gravitationally
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interacti ...
s, more specifically, the n-body problem
In physics, the -body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally.Leimanis and Minorsky: Our interest is with Leimanis, who first discusses some histor ...
.
*Canonical general relativity
In physics, canonical quantum gravity is an attempt to quantize the canonical formulation of general relativity (or canonical gravity). It is a Hamiltonian formulation of Einstein's general theory of relativity. The basic theory was outlined by ...
See also
* Action-angle coordinates
In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequency, frequencies of oscillatory or rotational motion witho ...
* Liouville's theorem
* Integrable system
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
* Symplectic manifold
* Kolmogorov–Arnold–Moser theorem
The Kolmogorov–Arnold–Moser (KAM) theorem is a result in dynamical systems about the persistence of quasiperiodic motions under small perturbations. The theorem partly resolves the small-divisor problem that arises in the perturbation theory ...
References
Further reading
* Almeida, A. M. (1992).'' Hamiltonian systems: Chaos and quantization''. Cambridge monographs on mathematical physics. Cambridge (u.a.: Cambridge Univ. Press)
* Audin, M., (2008). ''Hamiltonian systems and their integrability''. Providence, R.I: American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
,
* Dickey, L. A. (2003). ''Soliton equations and Hamiltonian systems''. Advanced series in mathematical physics, v. 26. River Edge, NJ: World Scientific
World Scientific Publishing is an academic publisher of scientific, technical, and medical books and journals headquartered in Singapore. The company was founded in 1981. It publishes about 600 books annually, along with 135 journals in various ...
.
*Treschev, D., & Zubelevich, O. (2010). ''Introduction to the perturbation theory of Hamiltonian systems''. Heidelberg: Springer
Springer or springers may refer to:
Publishers
* Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag.
** Springer Nature, a multinationa ...
* Zaslavsky, G. M. (2007). ''The physics of chaos in Hamiltonian systems''. London: Imperial College Press
Imperial College Press (ICP) was formed in 1995 as a partnership between Imperial College of Science, Technology and Medicine in London and World Scientific publishing.
This publishing house was awarded the rights, by The Nobel Foundation, Swed ...
.
External links
*
{{DEFAULTSORT:Hamiltonian System
Hamiltonian mechanics