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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 Ireland ...
, Hamiltonian mechanics replaces (generalized) velocities \dot q^i 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 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 ...
and
Poisson structure In differential geometry, a Poisson structure on a smooth manifold M is a Lie bracket \ (called a Poisson bracket in this special case) on the algebra (M) of smooth functions on M , subject to the Leibniz rule : \ = \h + g \ . Equivalently ...
s) 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 chemistry, ...
.


Overview


Phase space coordinates (p,q) and Hamiltonian H

Let (M, \mathcal L) be a
mechanical system A machine is a physical system using power to apply forces and control movement to perform an action. The term is commonly applied to artificial devices, such as those employing engines or motors, but also to natural biological macromolecul ...
with the configuration space M and the smooth Lagrangian \mathcal L. Select a standard coordinate system (\boldsymbol,\boldsymbol) on M. The quantities \textstyle p_i(\boldsymbol,\boldsymbol,t) ~\stackrel~ / are called ''momenta''. (Also ''generalized momenta'', ''conjugate momenta'', and ''canonical momenta''). For a time instant t, the Legendre transformation of \mathcal is defined as the map (\boldsymbol, \boldsymbol) \to \left(\boldsymbol,\boldsymbol\right) which is assumed to have a smooth inverse (\boldsymbol,\boldsymbol) \to (\boldsymbol,\boldsymbol). For a system with n degrees of freedom, the Lagrangian mechanics defines the ''energy function'' E_(\boldsymbol,\boldsymbol,t)\, \stackrel\, \sum^n_ \dot q^i \frac - \mathcal L. The inverse of the Legendre transform of \mathcal turns E_ into a function \mathcal H(\boldsymbol,\boldsymbol,t) known as the . The Hamiltonian satisfies \mathcal H\left(\frac,\boldsymbol,t\right) = E_(\boldsymbol,\boldsymbol,t) which implies that \mathcal H(\boldsymbol,\boldsymbol,t) = \sum^n_ p_i\dot q^i - \mathcal L(\boldsymbol,\boldsymbol,t), where the velocities \boldsymbol = (\dot q^1,\ldots, \dot q^n) are found from the (n-dimensional) equation \textstyle \boldsymbol = / which, by assumption, is uniquely solvable for \boldsymbol. The (2n-dimensional) pair (\boldsymbol,\boldsymbol) is called ''phase space coordinates''. (Also ''canonical coordinates'').


From Euler-Lagrange equation to Hamilton's equations

In phase space coordinates (\boldsymbol,\boldsymbol), the (n-dimensional) Euler-Lagrange equation \frac - \frac\frac = 0 becomes ''Hamilton's equations'' in 2n dimensions


From stationary action principle to Hamilton's equations

Let \mathcal P(a,b,\boldsymbol x_a,\boldsymbol x_b) be the set of smooth paths \boldsymbol q: ,b\to M for which \boldsymbol q(a) = \boldsymbol x_a and \boldsymbol q(b) = \boldsymbol x_. The action functional \mathcal S : \mathcal P(a,b,\boldsymbol x_a,\boldsymbol x_b) \to \Reals is defined via \mathcal S boldsymbol q= \int_a^b \mathcal L(t,\boldsymbol q(t),\dot(t))\, dt = \int_a^b \left(\sum^n_ p_i\dot q^i - \mathcal H(\boldsymbol,\boldsymbol,t) \right)\, dt, where \boldsymbol = \boldsymbol(t), and \boldsymbol = \partial \mathcal L/\partial \boldsymbol (see above). A path \boldsymbol q \in \mathcal P(a,b,\boldsymbol x_a,\boldsymbol x_b) 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 \mathcal S (and hence is an equation of motion) if and only if the path (\boldsymbol(t),\boldsymbol(t)) 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 H(p,q) of the Hamiltonian is the total energy of the system, i.e. the sum of
kinetic Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to its motion Art and ent ...
and
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potenti ...
, traditionally denoted and , respectively. Here is the momentum and is the space coordinate. Then \mathcal = T + V \quad , \quad T = \frac \quad , \quad V = V(q) is a function of alone, while is a function of alone (i.e., and are
scleronomic A mechanical system is scleronomous if the equations of constraints do not contain the time as an explicit variable and the equation of constraints can be described by generalized coordinates. Such constraints are called scleronomic constraints. ...
). 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 ele ...
''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 the c ...
. The only
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
s acting on the mass are the reaction 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 are used to describe the position of the mass in terms of (''r'', ''θ'', ''φ''), where is fixed, . The Lagrangian for this system is L = \frac ml^2\left( \dot^2+\sin^2\theta\ \dot^2 \right) + mgl\cos\theta. Thus the Hamiltonian is H = P_\theta\dot \theta + P_\phi\dot \phi - L where P_\theta = \frac = ml^2\dot \theta and P_\phi=\frac = ml^2\sin^2 \!\theta \, \dot \phi . In terms of coordinates and momenta, the Hamiltonian reads H = \underbrace_+\underbrace_ = \frac + \frac - mgl\cos\theta Hamilton's equations give the time evolution of coordinates and conjugate momenta in four first-order differential equations, \begin \dot &=\\ \dot &=\\ \dot &=\cos\theta-mgl\sin\theta \\ \dot &=0. \end Momentum P_\phi, 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 ...
L_z = l\sin\theta \times ml\sin\theta\,\dot\phi, 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 ...
\phi is a cyclic coordinate, which implies conservation of its conjugate momentum.


Deriving Hamilton's equations

Hamilton's equations can be derived by a calculation with the Lagrangian \mathcal L, generalized positions , and generalized velocities , where i = 1,\ldots,n. Here we work off-shell, meaning q^i, \dot^i, t are independent coordinates in phase space, not constrained to follow any equations of motion (in particular, \dot^i is not a derivative of q^i). The total differential of the Lagrangian is: \mathrm \mathcal = \sum_i \left ( \frac \mathrm q^i + \frac \mathrm \dot^i \right ) + \frac \mathrmt \ . The generalized momentum coordinates were defined as p_i = \partial \mathcal/\partial \dot^i, so we may rewrite the equation as: \begin \mathrm \mathcal &=& \displaystyle\sum_i \left( \frac \mathrm q^i + p_i \mathrm \dot^i \right) + \frac\mathrmt \\ &=& \displaystyle \sum_i \left( \frac \mathrmq^i + \mathrm( p_i \dot^i) - \dot^i \mathrm p_i \right) + \frac\mathrmt\,. \end After rearranging, one obtains: \mathrm\! \left ( \sum_i p_i \dot^i - \mathcal \right ) = \sum_i \left( - \frac \mathrm q^i + \dot^i \mathrmp_i \right) - \frac\mathrmt\ . The term in parentheses on the left-hand side is just the Hamiltonian \mathcal H = \sum p_i \dot^i - \mathcal L defined previously, therefore: \mathrm \mathcal = \sum_i \left( - \frac \mathrm q^i + \dot^i \mathrm p_i \right) - \frac\mathrmt\ . One may also calculate the total differential of the Hamiltonian \mathcal H with respect to coordinates q^i, p_i, t instead of q^i, \dot^i, t, yielding: \mathrm \mathcal =\sum_i \left( \frac \mathrm q^i + \frac \mathrm p_i \right) + \frac\mathrmt\ . One may now equate these two expressions for d\mathcal H, one in terms of \mathcal L, the other in terms of \mathcal H: \sum_i \left( - \frac \mathrm q^i + \dot^i \mathrm p_i \right) - \frac\mathrmt \ =\ \sum_i \left( \frac \mathrm q^i + \frac \mathrm p_i \right) + \frac\mathrmt\ . Since these calculations are off-shell, one can equate the respective coefficients of \mathrmq^i, \mathrmp_i, \mathrmt on the two sides: \frac = - \frac \quad, \quad \frac = \dot^i \quad, \quad \frac = - \ . On-shell, one substitutes parametric functions q^i=q^i(t) which define a trajectory in phase space with velocities \dot q^i = \tfracq^i(t) , 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-Lou ...
: \frac \frac - \frac = 0\ . Rearranging and writing in terms of the on-shell p_i = p_i(t) gives: \frac = \dot_i\ . Thus Lagrange's equations are equivalent to Hamilton's equations: \frac =- \dot_i \quad , \quad \frac = \dot^i \quad , \quad \frac = - \frac\, . In the case of time-independent \mathcal H and \mathcal L, i.e. \partial\mathcal H/\partial t = -\partial\mathcal L/\partial t = 0, 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, ...
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 q_i does not occur in the Hamiltonian (i.e. a ''cyclic coordinate''), the corresponding momentum coordinate p_i 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 mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the actio ...
in geometry. In the Lagrangian framework, the conservation of momentum also follows immediately, however all the generalized velocities \dot q_i 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 chemistry, ...
: 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 \mathcal H is the total energy of the system if and only if the energy function E_ \mathcal L has the same property. (See definition of \mathcal H). *\frac = \frac when \mathbf p(t), \mathbf q(t) form a solution of Hamilton's equations. Indeed, \frac = \frac\cdot \dot\boldsymbol + \frac\cdot \dot\boldsymbol + \frac, and everything but the final term cancels out. * \mathcal H does not change under ''point transformations'', i.e. smooth changes \boldsymbol \leftrightarrow \boldsymbol of space coordinates. (Follows from the invariance of the energy function E_ under point transformations. The invariance of E_ can be established directly). *\frac = -\frac. (See Deriving Hamilton's equations). *-\frac = \dot p_i = \frac. (Compare Hamilton's and Euler-Lagrange equations or see Deriving Hamilton's equations). *\frac = 0 if and only if \frac=0.A coordinate for which the last equation holds is called ''cyclic'' (or ''ignorable''). Every cyclic coordinate q^i reduces the number of degrees of freedom by 1, causes the corresponding momentum p_i 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 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 ...
. In
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
the Lagrangian of a non-relativistic classical particle in an electromagnetic field is (in
SI Units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
): \mathcal = \sum_i \tfrac m \dot_i^2 + \sum_i q \dot_i A_i - q \varphi 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 res ...
of the particle, is the electric scalar potential, and the are the components of the
magnetic vector potential In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic ...
that may all explicitly depend on x_i and t. This Lagrangian, combined with Euler–Lagrange equation, produces the
Lorentz force In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
law m \ddot = q \mathbf + q \dot \times \mathbf \, , and is called minimal coupling. Note that the values of scalar potential and vector potential would change during a gauge transformation, 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 are given by: p_i = \frac = m \dot_i + q A_i Note that canonical momenta are not gauge invariant, and are not physically measurable. However, the kinetic momentum: P_i \equiv m\dot_i = p_i - q A_i is gauge invariant and physically measurable. The Hamiltonian, as the Legendre transformation of the Lagrangian, is therefore: \mathcal = \left\ - \mathcal = \sum_i \frac + q \varphi 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 chemistry, ...
. Under gauge transformation: \mathbf \rightarrow \mathbf+\nabla f \,, \quad \varphi \rightarrow \varphi-\dot f \,, where is any scalar function of space and time, the aforementioned Lagrangian, canonical momenta, and Hamiltonian transform like: L \rightarrow L'= L+q\frac \,, \quad \mathbf \rightarrow \mathbf = \mathbf+q\nabla f \,, \quad H \rightarrow H' = H-q\frac \,, which still produces the same Hamilton's equation: \begin \left.\frac\_&=\left.\frac\_(\dot x_ip'_i-L')=-\left.\frac\_ \\ &=-\left.\frac\_-q\left.\frac\_\frac \\ &= -\frac\left(\left.\frac\_+q\left.\frac\_\right)\\ &=-\dot p'_i \end 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 U(1) 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 ...
m and charge q) is given by: \mathcal(t) = - m c^2 \sqrt + q \dot(t) \cdot \mathbf \left(\mathbf(t),t\right) - q \varphi \left(\mathbf(t),t\right) Thus the particle's canonical momentum is \mathbf(t) = \frac = \frac + q \mathbf that is, the sum of the kinetic momentum and the potential momentum. Solving for the velocity, we get \dot(t) = \frac So the Hamiltonian is \mathcal(t) = \dot \cdot \mathbf - \mathcal = c \sqrt + q \varphi This results in the force equation (equivalent to the Euler–Lagrange equation) \dot = - \frac = q \dot\cdot(\boldsymbol \mathbf) - q \boldsymbol \varphi = q \boldsymbol(\dot \cdot\mathbf) - q \boldsymbol \varphi from which one can derive \begin \frac\mathrm\left(\frac \right) &=\frac\mathrm(\mathbf - q \mathbf)=\dot\mathbf-q\frac-q(\dot\mathbf\cdot\nabla)\mathbf \\ &=q \boldsymbol(\dot \cdot\mathbf) - q \boldsymbol \varphi -q\frac-q(\dot\mathbf\cdot\nabla)\mathbf \\ &= q \mathbf + q \dot \times \mathbf \end The above derivation makes use of the vector calculus identity: \tfrac \nabla \left( \mathbf \cdot \mathbf \right) = \mathbf \cdot \mathbf_\mathbf = \mathbf \cdot (\nabla \mathbf) = (\mathbf \cdot \nabla) \mathbf + \mathbf \times (\nabla \times \mathbf) . An equivalent expression for the Hamiltonian as function of the relativistic (kinetic) momentum, \mathbf = \gamma m \dot(t) = \mathbf - q \mathbf, is \mathcal(t) = \dot(t) \cdot \mathbf(t) +\frac + q \varphi (\mathbf(t),t)=\gamma mc^2+ q \varphi (\mathbf(t),t)=E+V This has the advantage that kinetic momentum \mathbf can be measured experimentally whereas canonical momentum \mathbf cannot. Notice that the Hamiltonian ( total energy) can be viewed as the sum of the relativistic energy (kinetic+rest), E = \gamma m c^2, plus the
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potenti ...
, V = q \varphi.


From symplectic geometry to Hamilton's equations


Geometry of Hamiltonian systems

The Hamiltonian can induce a symplectic structure on a smooth even-dimensional manifold in several equivalent ways, the best known being the following: As a
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
nondegenerate symplectic 2-form ω. According to the Darboux's theorem, in a small neighbourhood around any point on there exist suitable local coordinates p_1, \cdots, p_n, \ q_1, \cdots, q_n (''
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 becomes: \omega = \sum_^n dp_i \wedge dq_i \, . The form \omega 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: T_xM \cong T^*_xM. This is done by mapping a vector \xi \in T_x M to the 1-form \omega_\xi \in T^*_xM, where \omega_\xi (\eta) = \omega(\eta, \xi) for all \eta \in T_x M. Due to the bilinearity and non-degeneracy of \omega, and the fact that \dim T_x M = \dim T^*_x M, the mapping \xi \to \omega_\xi 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 M. Repeating over all x \in M, we end up with an isomorphism J^ : \text(M) \to \Omega^1(M) between the infinite-dimensional space of smooth vector fields and that of smooth 1-forms. For every f,g \in C^\infty(M,\Reals) and \xi,\eta \in \text(M), J^(f\xi + g\eta) = fJ^(\xi) + gJ^(\eta). (In algebraic terms, one would say that the C^\infty(M,\Reals)-modules \text(M) and \Omega^1(M) are isomorphic). If H \in C^\infty(M \times \R_t, \R), then, for every fixed t \in \R_t, dH \in \Omega^1(M), and J(dH) \in \text(M). J(dH) is known as a Hamiltonian vector field. The respective differential equation on M \dot = J(dH)(x) is called . Here x=x(t) and J(dH)(x) \in T_xM is the (time-dependent) value of the vector field J(dH) at x \in M. A Hamiltonian system may be understood as a
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
over
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
, 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 incorporate ...
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 , which comes equipped with a natural symplectic form, and this latter function is the Hamiltonian. The correspondence between Lagrangian and Hamiltonian mechanics is achieved with the tautological one-form. Any smooth real-valued function on a
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 sym ...
can be used to define a Hamiltonian system. The function is known as "the Hamiltonian" or "the energy function." The symplectic manifold is then called the
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
. The Hamiltonian induces a special vector field on the symplectic manifold, known as the Hamiltonian vector field. The Hamiltonian vector field induces a Hamiltonian flow 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 symplectomorphisms, 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 In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
. 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 In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. T ...
. The Poisson bracket gives the space of functions on the manifold the structure of a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
. 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 Leibniz's rule (named after Gottfried Wilhelm Leibniz) may refer to one of the following: * Product rule in differential calculus * General Leibniz rule, a generalization of the product rule * Leibniz integral rule * The alternating series test, al ...
: \ = F_1\ + F_2\ # Jacobi identity: \ + \ + \ \equiv 0 # non-degeneracy: if the point on is not critical for then a smooth function exists such that \(x) \neq 0. Given a function \frac f = \frac f + \left\, if there is a
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
, then (since the phase space velocity (\dot_i, \dot_i) has zero divergence and probability is conserved) its convective derivative can be shown to be zero and so \frac \rho = - \left\ This is called Liouville's theorem. Every
smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
over the
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 sym ...
generates a one-parameter family of symplectomorphisms and if , then is conserved and the symplectomorphisms are symmetry transformations. 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 \dot_i = 0 \quad , \quad \dot_i = F_i(G) for some function . There is an entire field focusing on small deviations from integrable systems governed by the
KAM theorem Kaam (Gurmukhi: ਕਾਮ ''Kāma'') in common usage, the term stands for 'excessive passion for sexual pleasure' and it is in this sense that it is considered to be an evil in Sikhism. In Sikhism it is believed that Kaam can be overcome ...
. 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 form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
s, that is, Hamiltonians that can be written as \mathcal(q,p) = \tfrac \langle p, p\rangle_q 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 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 incorporate ...
, the cotangent space 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 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 ...
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 induces a linear isomorphism between the tangent and cotangent bundles. (See
Musical isomorphism In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle \mathrmM and the cotangent bundle \mathrm^* M of a pseudo-Riemannian manifold induced b ...
). 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 equations for this Hamiltonian are then the same as the
geodesic 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 connecti ...
s on the manifold. In particular, the Hamiltonian flow 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
geodesic 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 connecti ...
s. 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 equation ...
.


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 In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called ''horizontal ...
. 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 In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called ''horizontal ...
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. 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 \mathcal\left(x,y,z,p_x,p_y,p_z\right) = \tfrac\left( p_x^2 + p_y^2 \right). 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 function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
s over a
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 sym ...
, 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
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 i ...
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 g ...
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 chemistry, ...
, 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 i ...
over and to the algebra of
Moyal bracket In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product. The Moyal bracket was developed in about 1940 by José Enrique Moyal, but Moyal only succeeded in publishing his work in 1949 after a len ...
s. Specifically, the more general form of the Hamilton's equation reads \frac = \left\ + \frac where is some function of and , and is the Hamiltonian. To find out the rules for evaluating a
Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. T ...
without resorting to differential equations, see
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
; 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 i ...
. These Poisson brackets can then be extended to
Moyal bracket In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product. The Moyal bracket was developed in about 1940 by José Enrique Moyal, but Moyal only succeeded in publishing his work in 1949 after a len ...
s comporting to an inequivalent Lie algebra, as proven by
Hilbrand J. Groenewold Hilbrand Johannes "Hip" Groenewold (1910–1996) was a Dutch theoretical physicist who pioneered the largely operator-free formulation of quantum mechanics in phase space known as phase-space quantization. Biography Groenewold was born on 29 Jun ...
, and thereby describe quantum mechanical diffusion in phase space (See the phase space formulation 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 Schröd ...
). This more algebraic approach not only permits ultimately extending
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
s in
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
to
Wigner quasi-probability distribution The Wigner quasiprobability distribution (also called the Wigner function or the Wigner–Ville distribution, after Eugene Wigner and Jean-André Ville) is a quasiprobability distribution. It was introduced by Eugene Wigner in 1932 to study quan ...
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 *
Classical field theory A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum ...
*
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 * Hamilton–Jacobi equation * Hamilton–Jacobi–Einstein equation *
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 In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
* Hamiltonian optics * 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 ...
*
Routhian mechanics alt= In classical mechanics, Routh's procedure or Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the Routhian is the function which replaces both the ...
*
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 sy ...
* Hamiltonian fluid mechanics * Hamiltonian vector field


References


Further reading

* * * * * *


External links

* * * {{Authority control Classical mechanics Dynamical systems Mathematical physics