Hamiltonian mechanics emerged in 1833 as a reformulation of
Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...
. Introduced by
Sir William Rowan Hamilton
Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Irela ...
, Hamiltonian mechanics replaces (generalized) velocities
used in Lagrangian mechanics with (generalized) ''momenta''. Both theories provide interpretations of
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
and describe the same physical phenomena.
Hamiltonian mechanics has a close relationship with geometry (notably,
symplectic geometry and
Poisson structures) and serves as a
link between classical and
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
.
Overview
Phase space coordinates (p,q) and Hamiltonian H
Let
be a
mechanical system with the
configuration space and the smooth Lagrangian
Select a standard coordinate system
on
The quantities
are called ''momenta''. (Also ''generalized momenta'', ''conjugate momenta'', and ''canonical momenta''). For a time instant
the
Legendre transformation
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ...
of
is defined as the map
which is assumed to have a smooth inverse
For a system with
degrees of freedom, the Lagrangian mechanics defines the ''energy function''
The inverse of the Legendre transform of
turns
into a function
known as the . The Hamiltonian satisfies
which implies that
where the velocities
are found from the (
-dimensional) equation
which, by assumption, is uniquely solvable for
The (
-dimensional) pair
is called ''phase space coordinates''. (Also ''canonical coordinates'').
From Euler-Lagrange equation to Hamilton's equations
In phase space coordinates
the (
-dimensional)
Euler-Lagrange equation
becomes ''Hamilton's equations'' in
dimensions
From stationary action principle to Hamilton's equations
Let
be the set of smooth paths
for which
and
The
action functional is defined via
where
and
(see above). A path
is a
stationary point
In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" in ...
of
(and hence is an equation of motion) if and only if the path
in phase space coordinates obeys the Hamilton's equations.
Basic physical interpretation
A simple interpretation of Hamiltonian mechanics comes from its application on a one-dimensional system consisting of one particle of mass . The value
of the Hamiltonian is the total energy of the system, i.e. the sum of
kinetic and
potential energy, traditionally denoted and , respectively. Here is the momentum and is the space coordinate. Then
is a function of alone, while is a function of alone (i.e., and are
scleronomic).
In this example, the time derivative of is the velocity, and so the first Hamilton equation means that the particle's velocity equals the derivative of its kinetic energy with respect to its momentum. The time derivative of the momentum equals the ''Newtonian force'', and so the second Hamilton equation means that the force equals the negative
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of potential energy.
Example
A spherical pendulum consists of a
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
''m'' moving without
friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of ...
on the surface of a
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
. The only
forces acting on the mass are the
reaction
Reaction may refer to a process or to a response to an action, event, or exposure:
Physics and chemistry
*Chemical reaction
*Nuclear reaction
* Reaction (physics), as defined by Newton's third law
*Chain reaction (disambiguation).
Biology and m ...
from the sphere and
gravity
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 interactions, approximately 1038 times weaker than the stro ...
.
Spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
are used to describe the position of the mass in terms of (''r'', ''θ'', ''φ''), where is fixed, .
The Lagrangian for this system is
Thus the Hamiltonian is
where
and
In terms of coordinates and momenta, the Hamiltonian reads
Hamilton's equations give the time evolution of coordinates and conjugate momenta in four first-order differential equations,
Momentum
, which corresponds to the vertical component of
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, is a constant of motion. That is a consequence of the rotational symmetry of the system around the vertical axis. Being absent from the Hamiltonian,
azimuth
An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north.
Mathematical ...
is a
cyclic coordinate
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lou ...
, which implies conservation of its conjugate momentum.
Deriving Hamilton's equations
Hamilton's equations can be derived by a calculation with the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
, generalized positions , and generalized velocities , where
. Here we work
off-shell
In physics, particularly in quantum field theory, configurations of a physical system that satisfy classical equations of motion are called "on the mass shell" or simply more often on shell; while those that do not are called "off the mass shell" ...
, meaning
are independent coordinates in phase space, not constrained to follow any equations of motion (in particular,
is not a derivative of
). The
total differential
In calculus, the differential represents the principal part of the change in a function ''y'' = ''f''(''x'') with respect to changes in the independent variable. The differential ''dy'' is defined by
:dy = f'(x)\,dx,
where f'(x) is the ...
of the Lagrangian is:
The generalized momentum coordinates were defined as
, so we may rewrite the equation as:
After rearranging, one obtains:
The term in parentheses on the left-hand side is just the Hamiltonian
defined previously, therefore:
One may also calculate the total differential of the Hamiltonian
with respect to coordinates
instead of
, yielding:
One may now equate these two expressions for
, one in terms of
, the other in terms of
:
Since these calculations are off-shell, one can equate the respective coefficients of
on the two sides:
On-shell, one substitutes parametric functions
which define a trajectory in phase space with velocities
, obeying
Lagrange's equations
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lo ...
:
Rearranging and writing in terms of the on-shell
gives:
Thus Lagrange's equations are equivalent to Hamilton's equations:
In the case of time-independent
and
, i.e.
, Hamilton's equations consist of first-order
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s, while Lagrange's equations consist of second-order equations. Hamilton's equations usually do not reduce the difficulty of finding explicit solutions, but important theoretical results can be derived from them, because coordinates and momenta are independent variables with nearly symmetric roles.
Hamilton's equations have another advantage over Lagrange's equations: if a system has a symmetry, so that some coordinate
does not occur in the Hamiltonian (i.e. a ''cyclic coordinate''), the corresponding momentum coordinate
is conserved along each trajectory, and that coordinate can be reduced to a constant in the other equations of the set. This effectively reduces the problem from coordinates to coordinates: this is the basis of
symplectic reduction in geometry. In the Lagrangian framework, the conservation of momentum also follows immediately, however all the generalized velocities
still occur in the Lagrangian, and a system of equations in coordinates still has to be solved.
The Lagrangian and Hamiltonian approaches provide the groundwork for deeper results in classical mechanics, and suggest analogous formulations in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
: the
path integral formulation
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional i ...
and the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
.
Properties of the Hamiltonian
*The value of the Hamiltonian
is the total energy of the system if and only if the energy function
has the same property. (See definition of
*
when
form a solution of Hamilton's equations. Indeed,
and everything but the final term cancels out.
*
does not change under ''point transformations'', i.e. smooth changes
of space coordinates. (Follows from the invariance of the energy function
under point transformations. The invariance of
can be established directly).
*
(See Deriving Hamilton's equations).
*
(Compare Hamilton's and Euler-Lagrange equations or see Deriving Hamilton's equations).
*
if and only if
A coordinate for which the last equation holds is called ''cyclic'' (or ''ignorable''). Every cyclic coordinate
reduces the number of degrees of freedom by
causes the corresponding momentum
to be conserved, and makes Hamilton's equations
easier to solve.
Hamiltonian of a charged particle in an electromagnetic field
A sufficient illustration of Hamiltonian mechanics is given by the Hamiltonian of a charged particle in an
electromagnetic field. In
Cartesian coordinates the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
of a non-relativistic classical particle in an electromagnetic field is (in
SI Units):
where is the
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
of the particle, is the
electric scalar potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
, and the are the components of the
magnetic vector potential that may all explicitly depend on
and
.
This Lagrangian, combined with
Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
, produces the
Lorentz force law
and is called
minimal coupling In analytical mechanics and quantum field theory, minimal coupling refers to a coupling between fields which involves only the charge distribution and not higher multipole moments of the charge distribution. This minimal coupling is in contrast to, ...
.
Note that the values of scalar potential and vector potential would change during a
gauge transformation
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
, and the Lagrangian itself will pick up extra terms as well; But the extra terms in Lagrangian add up to a total time derivative of a scalar function, and therefore won't change the Euler–Lagrange equation.
The
canonical momenta
In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example,
hat x,\hat p_ ...
are given by:
Note that canonical momenta are not
gauge invariant
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
, and are not physically measurable. However, the
kinetic momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and ...
:
is gauge invariant and physically measurable.
The Hamiltonian, as the
Legendre transformation
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ...
of the Lagrangian, is therefore:
This equation is used frequently in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
.
Under
gauge transformation
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
:
where is any scalar function of space and time, the aforementioned Lagrangian, canonical momenta, and Hamiltonian transform like:
which still produces the same Hamilton's equation:
In quantum mechanics, the
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
will also undergo a
local
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
* Local government, a form of public administration, usually the lowest tier of administrat ...
U(1)
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
group transformation during the Gauge Transformation, which implies that all physical results must be invariant under local U(1) transformations.
Relativistic charged particle in an electromagnetic field
The
relativistic Lagrangian for a particle (
rest mass
The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, i ...
and
charge
Charge or charged may refer to:
Arts, entertainment, and media Films
* '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* ''Charge'' (David Ford album)
* ''Charge'' (Machel Montano album)
* ''Charge!!'', an album by The Aqu ...
) is given by:
Thus the particle's canonical momentum is
that is, the sum of the kinetic momentum and the potential momentum.
Solving for the velocity, we get
So the Hamiltonian is
This results in the force equation (equivalent to the
Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
)
from which one can derive
The above derivation makes use of the
vector calculus identity
The following are important identities involving derivatives and integrals in vector calculus.
Operator notation
Gradient
For a function f(x, y, z) in three-dimensional Cartesian coordinate variables, the gradient is the vector field:
\o ...
:
An equivalent expression for the Hamiltonian as function of the relativistic (kinetic) momentum,
, is
This has the advantage that kinetic momentum
can be measured experimentally whereas canonical momentum
cannot. Notice that the Hamiltonian (
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 ...
) can be viewed as the sum of the
relativistic energy (kinetic+rest),
, plus the
potential energy,
.
From symplectic geometry to Hamilton's equations
Geometry of Hamiltonian systems
The Hamiltonian can induce 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 ...
on a
smooth even-dimensional manifold in several equivalent ways, the best known being the following:
As a
closed nondegenerate
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case.
T ...
symplectic 2-form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
ω. According to the
Darboux's theorem
Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among ...
, in a small neighbourhood around any point on there exist suitable local coordinates
(''
canonical
The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical examp ...
'' or ''symplectic'' coordinates) in which the
symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping that is
; Bilinear: Linear in each argument ...
becomes:
The form
induces a
natural isomorphism
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
of the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
with the
cotangent space
In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
:
This is done by mapping a vector
to the 1-form
where
for all
Due to the
bilinearity
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
Definition
Vector spaces
Let V, W ...
and non-degeneracy of
and the fact that
the mapping
is indeed a
linear isomorphism
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
. This isomorphism is ''natural'' in that it does not change with change of coordinates on
Repeating over all
we end up with an isomorphism
between the infinite-dimensional space of smooth vector fields and that of smooth 1-forms. For every
and
(In algebraic terms, one would say that the
-modules
and
are isomorphic). If
then, for every fixed
and
is known as a
Hamiltonian vector field In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is ...
. The respective differential equation on
is called . Here
and
is the (time-dependent) value of the vector field
at
A Hamiltonian system may be understood as a
fiber bundle over
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, ...
, with the
fiber
Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorpora ...
being the position space at time . The Lagrangian is thus a function on the
jet bundle
In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. ...
over ; taking the fiberwise
Legendre transform
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ...
of the Lagrangian produces a function on the dual bundle over time whose fiber at is the
cotangent space
In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
, which comes equipped with a natural
symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping that is
; Bilinear: Linear in each argument ...
, and this latter function is the Hamiltonian. The correspondence between Lagrangian and Hamiltonian mechanics is achieved with the
tautological one-form
In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T^Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus p ...
.
Any
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
real-valued function on a
symplectic manifold can be used to define a
Hamiltonian system
A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can ...
. The function is known as "the Hamiltonian" or "the energy function." The symplectic manifold is then called the
phase space. The Hamiltonian induces a special
vector field on the symplectic manifold, known as the
Hamiltonian vector field In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is ...
.
The Hamiltonian vector field induces a
Hamiltonian flow In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field ...
on the manifold. This is a one-parameter family of transformations of the manifold (the parameter of the curves is commonly called "the time"); in other words, an
isotopy of
symplectomorphism
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the sy ...
s, starting with the identity. By
Liouville's theorem, each symplectomorphism preserves the
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of th ...
on the
phase space. The collection of symplectomorphisms induced by the Hamiltonian flow is commonly called "the Hamiltonian mechanics" of the Hamiltonian system.
The symplectic structure induces a
Poisson bracket. The Poisson bracket gives the space of functions on the manifold the structure of a
Lie algebra.
If and are smooth functions on then the smooth function is properly defined; it is called a ''Poisson bracket'' of functions and and is denoted . The Poisson bracket has the following properties:
# bilinearity
# antisymmetry
#
Leibniz rule:
#
Jacobi identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
:
# non-degeneracy: if the point on is not critical for then a smooth function exists such that
.
Given a function
if there is a
probability distribution , then (since the phase space velocity
has zero divergence and probability is conserved) its convective derivative can be shown to be zero and so
This is called
Liouville's theorem. Every
smooth function over the
symplectic manifold generates a one-parameter family of
symplectomorphism
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the sy ...
s and if , then is conserved and the symplectomorphisms are
symmetry transformation
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
s.
A Hamiltonian may have multiple conserved quantities . If the symplectic manifold has dimension and there are functionally independent conserved quantities which are in involution (i.e., ), then the Hamiltonian is
Liouville integrable. The
Liouville–Arnold theorem says that, locally, any Liouville integrable Hamiltonian can be transformed via a symplectomorphism into a new Hamiltonian with the conserved quantities as coordinates; the new coordinates are called ''action-angle coordinates''. The transformed Hamiltonian depends only on the , and hence the equations of motion have the simple form
for some function .
There is an entire field focusing on small deviations from integrable systems governed by the
KAM theorem.
The integrability of Hamiltonian vector fields is an open question. In general, Hamiltonian systems are
chaotic
Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kid ...
; concepts of measure, completeness, integrability and stability are poorly defined.
Riemannian manifolds
An important special case consists of those Hamiltonians that are
quadratic forms, that is, Hamiltonians that can be written as
where is a smoothly varying
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on the
fibers , the
cotangent space
In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
to the point in the
configuration space, sometimes called a cometric. This Hamiltonian consists entirely of the
kinetic term
In physics, a kinetic term is the part of the Lagrangian that is bilinear in the fields (and for nonlinear sigma models, they are not even bilinear), and usually contains two derivatives with respect to time (or space); in the case of fermions, ...
.
If one considers a
Riemannian manifold or a
pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
, the
Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
induces a linear isomorphism between the tangent and cotangent bundles. (See
Musical isomorphism). Using this isomorphism, one can define a cometric. (In coordinates, the matrix defining the cometric is the inverse of the matrix defining the metric.) The solutions to the
Hamilton–Jacobi equation
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mecha ...
s for this Hamiltonian are then the same as the
geodesics on the manifold. In particular, the
Hamiltonian flow In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field ...
in this case is the same thing as the
geodesic flow
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
. The existence of such solutions, and the completeness of the set of solutions, are discussed in detail in the article on
geodesics. See also
Geodesics as Hamiltonian flows In mathematics, the geodesic equations are second-order non-linear differential equations, and are commonly presented in the form of Euler–Lagrange equations of motion. However, they can also be presented as a set of coupled first-order equatio ...
.
Sub-Riemannian manifolds
When the cometric is degenerate, then it is not invertible. In this case, one does not have a Riemannian manifold, as one does not have a metric. However, the Hamiltonian still exists. In the case where the cometric is degenerate at every point of the configuration space manifold , so that the
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
of the cometric is less than the dimension of the manifold , one has a
sub-Riemannian manifold.
The Hamiltonian in this case is known as a sub-Riemannian Hamiltonian. Every such Hamiltonian uniquely determines the cometric, and vice versa. This implies that every
sub-Riemannian manifold is uniquely determined by its sub-Riemannian Hamiltonian, and that the converse is true: every sub-Riemannian manifold has a unique sub-Riemannian Hamiltonian. The existence of sub-Riemannian geodesics is given by the
Chow–Rashevskii theorem
In sub-Riemannian geometry, the Chow–Rashevskii theorem (also known as Chow's theorem) asserts that any two points of a connected sub-Riemannian manifold, endowed with a bracket generating distribution, are connected by a horizontal path in the ...
.
The continuous, real-valued
Heisenberg group
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
::\begin
1 & a & c\\
0 & 1 & b\\
0 & 0 & 1\\
\end
under the operation of matrix multiplication. Elements ...
provides a simple example of a sub-Riemannian manifold. For the Heisenberg group, the Hamiltonian is given by
is not involved in the Hamiltonian.
Poisson algebras
Hamiltonian systems can be generalized in various ways. Instead of simply looking at the
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
of
smooth functions over a
symplectic manifold, Hamiltonian mechanics can be formulated on general
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
unital real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
Poisson algebra In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central ...
s. A
state
State may refer to:
Arts, entertainment, and media Literature
* ''State Magazine'', a monthly magazine published by the U.S. Department of State
* ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States
* ''Our S ...
is a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
on the Poisson algebra (equipped with some suitable
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
) such that for any element of the algebra, maps to a nonnegative real number.
A further generalization is given by
Nambu dynamics.
Generalization to quantum mechanics through Poisson bracket
Hamilton's equations above work well for
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, but not for
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
, since the differential equations discussed assume that one can specify the exact position and momentum of the particle simultaneously at any point in time. However, the equations can be further generalized to then be extended to apply to quantum mechanics as well as to classical mechanics, through the deformation of the
Poisson algebra In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central ...
over and to the algebra of
Moyal brackets.
Specifically, the more general form of the Hamilton's equation reads
where is some function of and , and is the Hamiltonian. To find out the rules for evaluating a
Poisson bracket without resorting to differential equations, see
Lie algebra; a Poisson bracket is the name for the Lie bracket in a
Poisson algebra In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central ...
. These Poisson brackets can then be extended to
Moyal brackets comporting to an inequivalent Lie algebra, as proven by
Hilbrand J. Groenewold, and thereby describe quantum mechanical diffusion in phase space (See the
phase space formulation
The phase-space formulation of quantum mechanics places the position ''and'' momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position ''or'' momentum representations (see also position and mome ...
and the
Wigner–Weyl transform
In quantum mechanics, the Wigner–Weyl transform or Weyl–Wigner transform (after Hermann Weyl and Eugene Wigner) is the invertible mapping between functions in the quantum phase space formulation and Hilbert space operators in the Schrödin ...
). This more algebraic approach not only permits ultimately extending
probability distributions in
phase space to
Wigner quasi-probability distribution
The Wigner quasiprobability distribution (also called the Wigner function or the Wigner–Ville distribution, after Eugene Wigner and :fr:Jean Ville, Jean-André Ville) is a quasiprobability distribution. It was introduced by Eugene Wigner in 193 ...
s, but, at the mere Poisson bracket classical setting, also provides more power in helping analyze the relevant
conserved quantities in a system.
See also
*
Canonical transformation
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canon ...
*
Classical field theory
*
Hamiltonian field theory
In theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory. It also has applications in quantum field theory ...
*
Covariant Hamiltonian field theory
In theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory. It also has applications in quantum field theory ...
*
Classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
*
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 '' ...
*
Hamiltonian system
A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can ...
*
Hamilton–Jacobi equation
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mecha ...
*
Hamilton–Jacobi–Einstein equation
In general relativity, the Hamilton–Jacobi–Einstein equation (HJEE) or Einstein–Hamilton–Jacobi equation (EHJE) is an equation in the Hamiltonian formulation of geometrodynamics in superspace, cast in the "geometrodynamics era" around t ...
*
Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...
*
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
*
Hamiltonian (quantum mechanics)
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltoni ...
*
Quantum Hamilton's equations
*
Quantum field theory
*
Hamiltonian optics Hamiltonian opticsH. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993, . and Lagrangian opticsVasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011, . are two formulations of geometrical ...
*
De Donder–Weyl theory
*
Geometric mechanics Geometric mechanics is a branch of mathematics applying particular geometric methods to many areas of mechanics, from mechanics of particles and rigid bodies to fluid mechanics to control theory.
Geometric mechanics applies principally to systems f ...
*
Routhian mechanics
*
Nambu mechanics
In mathematics, Nambu mechanics is a generalization of Hamiltonian mechanics involving multiple Hamiltonians. Recall that Hamiltonian mechanics is based upon the flows generated by a smooth Hamiltonian over a symplectic manifold. The flows are s ...
*
Hamiltonian fluid mechanics Hamiltonian fluid mechanics is the application of Hamiltonian methods to fluid mechanics. Note that this formalism only applies to nondissipative fluids.
Irrotational barotropic flow
Take the simple example of a barotropic, inviscid vorticity-f ...
*
Hamiltonian vector field In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is ...
References
Further reading
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External links
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