Hamiltonian System
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A Hamiltonian system is a
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
governed by Hamilton's equations. 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 rel ...
, 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 bound non-stellar objects in or out of orbit around a star or star system. Generally speaking, systems with one or more planets constitute a planetary system, although such systems may also consis ...
or an
electron The electron (, or in nuclear reactions) 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 partic ...
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 classica ...
. 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) ''momen ...
and
dynamical systems theory Dynamical systems theory is an area of mathematics used to describe the behavior of complex systems, complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theo ...
.


Overview

Informally, a Hamiltonian system is a mathematical formalism developed by Hamilton 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 o ...
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 ro ...
to the general problem,
Poincaré Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science * Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré * Luci ...
showed for the first time that it exhibits deterministic chaos. Formally, a Hamiltonian system is a dynamical system characterised by the scalar function H(\boldsymbol,\boldsymbol,t), also known as the Hamiltonian. The state of the system, \boldsymbol, is described by the generalized coordinates \boldsymbol and \boldsymbol, corresponding to generalized momentum and position respectively. Both \boldsymbol and \boldsymbol are real-valued vectors with the same dimension ''N''. Thus, the state is completely described by the 2''N''-dimensional vector :\boldsymbol = (\boldsymbol,\boldsymbol) and the evolution equations are given by Hamilton's equations: :\begin & \frac = -\frac, \\ pt& \frac = +\frac. \end The trajectory \boldsymbol(t) 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 o ...
defined by Hamilton's equations and the initial condition \boldsymbol(t = 0) = \boldsymbol_0\in\mathbb^.


Time-independent Hamiltonian systems

If the Hamiltonian is not explicitly time-dependent, i.e. if H(\boldsymbol,\boldsymbol,t) = H(\boldsymbol,\boldsymbol), then the Hamiltonian does not vary with time at all: and thus the Hamiltonian is a
constant of motion In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a ''mathematical'' constraint, the natural consequence of the equations of motion, rather than ...
, 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 hea ...
of the system: H = E. Examples of such systems are the undamped pendulum, the
harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...
, and dynamical billiards.


Example

An example of a time-independent Hamiltonian system is the harmonic oscillator. Consider the system defined by the coordinates \boldsymbol = m\dot and \boldsymbol = x. Then the Hamiltonian is given by : H = \frac + \frac. 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. Writing : \nabla_ H(\boldsymbol) = \begin \frac \\ \frac \\ \end the evolution equation of the dynamical system can be written as :\frac = M_N \nabla_ H(\boldsymbol) where :M_N = \begin 0 & I_N \\ -I_N & 0 \\ \end 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 ...
. 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. :\begin \frac\oint_ d\boldsymbol &= \oint_\frac\cdot d\hat_ \\ &= \oint_ \left(M_N \nabla_ H(\boldsymbol)\right) \cdot d\hat_ \\ &= \int_\nabla_\cdot \left(M_N \nabla_ H(\boldsymbol)\right) \, dV \\ &= \int_\sum_^N\sum_^N\left(\frac - \frac\right) \, dV \\ &= 0 \end where the third equality comes from the divergence theorem.


Examples

* Dynamical billiards *
Planetary system A planetary system is a set of gravitationally bound non-stellar objects in or out of orbit around a star or star system. Generally speaking, systems with one or more planets constitute a planetary system, although such systems may also consis ...
s, more specifically, the n-body problem. * Canonical general relativity


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 frequencies of oscillatory or rotational motion without solving ...
* 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 i ...
*
Symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called s ...
* Kolmogorov–Arnold–Moser theorem


References


Further reading

* Almeida, A. M. (1992).'' Hamiltonian systems: Chaos and quantization''. Cambridge monographs on mathematical physics. Cambridge (u.a.:
Cambridge Univ. Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University 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.


External links

* {{DEFAULTSORT:Hamiltonian System Hamiltonian mechanics