Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. It arose from Lagrangian mechanics, another re-formulation of classical mechanics, introduced by Joseph Louis Lagrange in 1788. It can however be formulated without recourse to Lagrangian mechanics, using symplectic spaces. See the section on its mathematical formulation for this. Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... William Rowan Hamilton Sir William Rowan Hamilton (August 4, 1805 – September 2, 1865) was an Irish mathematician, physicist, and astronomer who made important contributions to the development of optics, dynamics, and algebra. ... Lagrangian mechanics is a re-formulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. ... Joseph-Louis Lagrange, comte de lEmpire (January 25, 1736 – April 10, 1813; b. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ...

As a reformulation of Lagrangian mechanics

Starting with Lagrangian mechanics, the equations of motion are based on generalized coordinates In physics, equations of motion are equations that describe the behavior of a system (e. ... Generalized coordinates include any nonstandard coordinate system applied to the analysis of a physical system, especially in the study of Lagrangian dynamics. ...

and matching generalized velocities

We write the Lagrangian as A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ...

with the subscripted variables understood to represent all N variables of that type. Hamiltonian mechanics aims to replace the generalized velocity variables with generalized momentum variables, also known as conjugate momenta. By doing so, it is possible to handle certain systems, such as aspects of quantum mechanics that would otherwise be even more complicated.

For each generalized velocity, there is one corresponding conjugate momentum, defined as: In mathematics and physics, the canonical coordinates are a special set of coordinates on the cotangent bundle of a manifold. ...

In Cartesian coordinates, the generalized momenta are precisely the physical linear momenta. In circular polar coordinates, the generalized momentum corresponding to the angular velocity is the physical angular momentum. For an arbitrary choice of generalized coordinates, it may not be possible to obtain an intuitive interpretation of the conjugate momenta. Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... In classical mechanics, momentum (pl. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... Gyroscope. ...

One thing which is not too obvious in this coordinate dependent formulation is that different generalized coordinates are really nothing more than different coordinatizations of the same symplectic manifold. In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ...

The Hamiltonian is the Legendre transform of the Lagrangian: Diagram illustrating the Legendre transformation of the function f(x) . The function is shown in red, and the tangent line at x0  is shown in blue. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ...

If the transformation equations defining the generalized coordinates are independent of t, it can be shown that H is equal to the total energy E = T + V.

Each side in the definition of H produces a differential:

Substituting the previous definition of the conjugate momenta into this equation and matching coefficients, we obtain the equations of motion of Hamiltonian mechanics, known as the canonical equations of Hamilton: In physics and mathematics, Hamiltons equations is the set of differential equations that arise in Hamiltonian mechanics, but also in many other related and sometimes apparently not related areas of science. ...

Hamilton's equations are first-order differential equations, and thus easier to solve than Lagrange's equations, which are second-order. However, the steps leading to the equations of motion are more onerous than in Lagrangian mechanics - beginning with the generalized coordinates and the Lagrangian, we must calculate the Hamiltonian, express each generalized velocity in terms of the conjugate momenta, and replace the generalized velocities in the Hamiltonian with the conjugate momenta. All in all, there is little labor saved from solving a problem with Hamiltonian mechanics rather than Lagrangian mechanics. Ultimately, it will produce the same solution as Lagrangian mechanics and Newton's laws of motion. An illustration of a differential equation. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

The principal appeal of the Hamiltonian approach is that it provides the groundwork for deeper results in the theory of classical mechanics.

Geometry of Hamiltonian systems

A Hamiltonian system may be understood as a fiber bundle E over time R, with the fibers E_t, t ∈ R being the position space. The Lagrangian is thus a function on the jet bundle J over E; taking the fiberwise Legendre transform of the Lagrangian produces a function on the dual bundle over time whose fiber at t is the cotangent space T^*E_t, which comes equipped with a natural symplectic form, and this latter function is the Hamiltonian. In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... A pocket watch, a device used to keep time There are two distinct views on the meaning of time. ... In mathematics, a level set of a real-valued function f of n variables is a set of the form { (x1,...,xn) | f(x1,...,xn) = c } where c is a constant. ... In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. ... In mathematics, two differentiable functions f and g are said to be Legendre transforms of each other if their first derivatives are inverse functions of each other: f and g are then said to be related by a Legendre transformation. ... In differential geometry, one can attach to every point p of a differentiable manifold a vector space called the cotangent space at p. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ...

Mathematical formalism

Any smooth real-valued function H on a symplectic manifold can be used to define a Hamiltonian system. The function H 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 symplectic vector field. In mathematics, a smooth function is one that is infinitely (indefinitely) differentiable, i. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics and physics, the symplectic vector field, also known as the Hamiltonian vector field, is a vector field induced on a symplectic manifold by an energy function or Hamiltonian. ... For other senses of this term, see phase space (disambiguation). ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics and physics, the symplectic vector field, also known as the Hamiltonian vector field, is a vector field induced on a symplectic manifold by an energy function or Hamiltonian. ...

The symplectic vector field, also called the Hamiltonian vector field, induces a Hamiltonian flow on the manifold. The integral curves of the vector field are a one-parameter family of transformations of the manifold; the parameter of the curves is commonly called the time. The time evolution is given by symplectomorphisms. By Liouville's theorem, each symplectomorphism preserves the volume form on the phase space. The collection of symplectomorphisms induced by the Hamiltonian flow is commonly called the Hamiltonian mechanics of the Hamiltonian system. In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. ... In mathematics, a symplectomorphism (or Hamiltonian flow) is an isomorphism in the category of symplectic manifolds. ... In mathematical physics, Liouvilles theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. ... In mathematics, the volume form is a differential form that represents a unit volume of a Riemannian manifold or a pseudo-Riemannian manifold. ... For other senses of this term, see phase space (disambiguation). ...

The Hamiltonian vector field also induces a special operation, the Poisson bracket. The Poisson bracket acts on functions on the symplectic manifold, thus giving the space of functions on the manifold the structure of a Lie algebra. In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...

In particular, given a function f

If we have a probability distribution, ρ, then (since the phase space velocity () has zero divergence, and probability is conserved) its convective derivative can be shown to be zero and so In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...

This is called Liouville's theorem. Every smooth function G over the symplectic manifold generates a one-parameter family of symplectomorphisms and if { G, H } = 0, then G is conserved and the symplectomorphisms are symmetry transformations. In mathematical physics, Liouvilles theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. ... In mathematics, a smooth function is one that is infinitely (indefinitely) differentiable, i. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics, a symplectomorphism (or Hamiltonian flow) is an isomorphism in the category of symplectic manifolds. ... Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...

A Hamiltonian may have multiple conserved quantities G_i. If the symplectic manifold has dimension 2n and there are n functionally independent conserved quantities G_i which are in involution (i.e., { G_i, G_j } = 0), then the Hamiltonian is Liouville integrable. The Liouville–Arnol'd theorem says that locally, any Liouville integrable Hamiltonian can be transformed via a symplectomorphism in a new Hamiltonian with the conserved quantities G_i as coordinates; the new coordinates are called action-angle coordinates. The transformed Hamiltonian depends only on the G_i, and hence the equations of motion have the simple form

for some function F (Arnol'd et al., 1988). There is an entire field focusing on small deviations from integrable systems governed by the KAM theorem. The Kolmogorov-Arnold-Moser theorem is a theorem in non-linear dynamics that solves the small-divisor problem in classical perturbation theory. ...

The integrability of Hamiltonian vector fields is an open question. In general, Hamiltonian systems are chaotic; concepts of measure, completeness, integrability and stability are poorly defined. At this time, the study of dynamical systems is primarily qualitative, and not a quantitative science. A plot of the trajectory Lorenz system for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. ... In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ...

Riemannian manifolds

An important special case consists of those Hamiltonians that are quadratic forms, that is, Hamiltonians that can be written as In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

where is a cometric on the fiber , the cotangent space to the point q in the configuration space. This Hamiltonian consists entirely of the kinetic term. In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In differential geometry, one can attach to every point p of a differentiable manifold a vector space called the cotangent space at p. ... In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in most applications, it is a topological space or/and a vector space. ... In physics, a kinetic term is the part of the Lagrangian that is bilinear in the fields (this does not include the mass term!) (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, so that one has an invertible, non-degenerate metric, then the cometric is given simply as the inverse of the metric. The solutions to the Hamilton–Jacobi equations for this Hamiltonian are then the same as the geodesics on the manifold. In particular, the Hamiltonian flow in this case is the same thing as the geodesic flow. The existence of such solutions, and the completeness of the set of solutions, are discussed in detail in the article on geodesics. In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... In mathematics a metric or distance is a function which assigns a distance to elements of a set. ... In physics and mathematics, the Hamilton–Jacobi equation (HJE) is a reformulation of classical mechanics and, thus, equivalent to other formulations such as Newtons laws of motion, Lagrangian mechanics and Hamiltonian mechanics. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. Definition of geodesic depends on the type of curved space. If the space carries a natural metric then geodesics are defined to be (locally) the shortest path between points on the space. ... In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds. ... This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. Definition of geodesic depends on the type of curved space. If the space carries a natural metric then geodesics are defined to be (locally) the shortest path between points on the space. ...

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 q of the configuration space manifold Q, so that the rank of the cometric is less than the dimension of the manifold Q, one has a sub-Riemannian manifold. Rank means a wide variety of things in mathematics, including: Rank (linear algebra) Rank of a tensor Rank of an array Rank of an abelian group Rank (set theory) Rank-into-rank Rank of a greedoid This is a disambiguation page — a navigational aid which lists other pages that... In mathematics, a sub-Riemannian manifold is a certain type of generalization of a 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 mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. ...

The continuous, real-valued Heisenberg group provides a simple example of a sub-Riemannian manifold. For the Heisenberg group, the Hamiltonian is given by In mathematics, the Heisenberg group is a group of 3×3 upper triangular matrices of the form Elements a,b,c can be taken from some (arbitrary) commutative ring. ...

.

p_z is not involved in the Hamiltonian.

Poisson algebras

Hamiltonian systems can be generalized in various ways. Instead of simply looking at the algebra of smooth functions over a symplectic manifold, Hamiltonian mechanics can be formulated on general commutative unital real Poisson algebras. A state is a continuous linear functional on the Poisson algebra (equipped with some suitable topology) such that for any element A of the algebra, A² maps to a nonnegative real number. In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In mathematics, a smooth function is one that is infinitely (indefinitely) differentiable, i. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In mathematics, the real numbers may be described informally in several different ways. ... A Poisson algebra is an associative algebra together with a Lie bracket, satisfying Leibniz law. ... In functional analysis, a state on a C*-algebra is a positive linear functional of norm 1. ... In topology, a continuous function is generally defined as one for which preimages of open sets are open. ... In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

A further generalization is given by Nambu dynamics. Nambu dynamics is an interesting generalization of Hamiltonian mechanics. ...

References

• V.I. Arnol'd, Mathematical Methods of Classical Mechanics, Springer-Verlag (1989), [ISBN 0-387-96890-3]
• Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X
• V.I. Arnol'd, V.V. Kozlov and A.I. Neĩshtadt, "Mathematical aspects of classical and celestial mechanics." In: Encyclopaedia of Mathematical Sciences, Dynamical Systems III (vol. 3), Springer-Verlag, 1988.
• Rychlik, Marek, "Lagrangian and Hamiltonian mechanics -- A short introduction"
• Binney, James, "Classical Mechanics" (PostScript) lecture notes (PDF)
• Tong, David, Classical Dynamics (Cambridge lecture notes)

Vladimir I. Arnold (Moscow, December 2001). ... Ralph H. Abraham (born July 4, 1936) is an American mathematician. ... Jerrold E. Marsden. ... Vladimir I. Arnold (Moscow, December 2001). ... This article or section does not cite its references or sources. ... PDF redirects here. ...

See also

• Hamiltonian (quantum mechanics)