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 symplectomorphisms and hence obey Liouville's theorem. This was soon generalized to flows generated by a Hamiltonian over a Poisson manifold. In 1973, Yoichiro Nambu suggested a generalization involving Nambu–Poisson manifolds with more than one Hamiltonian. [1]
Specifically, consider a differential manifold M, for some integer N ≥ 2; one has a smooth N-linear map from N copies of C∞ (M) to itself, such that it is completely antisymmetric: the Nambu bracket,
which acts as a derivation
whence the Filippov Identities (FI) [2] (evocative of the Jacobi identities, but unlike them, not antisymmetrized in all arguments, for N ≥ 2 ):
so that {f1, ..., fN−1, •} acts as a generalized derivation over the N-fold product {. ,..., .}.
There are N− 1 Hamiltonians, H1, ..., HN−1, generating an incompressible flow,
The generalized phase-space velocity is divergenceless, enabling Liouville's theorem. The case N = 2 reduces to a Poisson manifold, and conventional Hamiltonian mechanics.
For larger even N, the N−1 Hamiltonians identify with the maximal number of independent invariants of motion (cf. Conserved quantity) characterizing a superintegrable system that evolves in N-dimensional phase space. Such systems are also describable by conventional Hamiltonian dynamics; but their description in the framework of Nambu mechanics is substantially more elegant and intuitive, as all invariants enjoy the same geometrical status as the Hamiltonian: the trajectory in phase space is the intersection of the N − 1 hypersurfaces specified by these invariants. Thus, the flow is perpendicular to all N − 1 gradients of these Hamiltonians, whence parallel to the generalized cross product specified by the respective Nambu bracket.
Nambu mechanics can be extended to fluid dynamics, where the resulting Nambu brackets are non-canonical and the Hamiltonians are identified with the Casimir of the system, such as enstrophy or helicity. [3] [4]
Quantizing Nambu dynamics leads to intriguing structures [5] that coincide with conventional quantization ones when superintegrable systems are involved—as they must.
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, , equipped with a closed nondegenerate differential 2-form , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena.
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. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself as one of the new canonical momentum coordinates.
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the 18th century and onward, after Newtonian mechanics. Since Newtonian mechanics considers vector quantities of motion, particularly accelerations, momenta, forces, of the constituents of the system, an alternative name for the mechanics governed by Newton's laws and Euler's laws is vectorial mechanics.
In theoretical physics, the Batalin–Vilkovisky (BV) formalism was developed as a method for determining the ghost structure for Lagrangian gauge theories, such as gravity and supergravity, whose corresponding Hamiltonian formulation has constraints not related to a Lie algebra. The BV formalism, based on an action that contains both fields and "antifields", can be thought of as a vast generalization of the original BRST formalism for pure Yang–Mills theory to an arbitrary Lagrangian gauge theory. Other names for the Batalin–Vilkovisky formalism are field-antifield formalism, Lagrangian BRST formalism, or BV–BRST formalism. It should not be confused with the Batalin–Fradkin–Vilkovisky (BFV) formalism, which is the Hamiltonian counterpart.
In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities. For example,
In classical mechanics, the Laplace–Runge–Lenz (LRL) vector is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit; equivalently, the LRL vector is said to be conserved. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.
In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory, to the greatest extent possible.
A first class constraint is a dynamical quantity in a constrained Hamiltonian system whose Poisson bracket with all the other constraints vanishes on the constraint surface in phase space. To calculate the first class constraint, one assumes that there are no second class constraints, or that they have been calculated previously, and their Dirac brackets generated.
In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics.
In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in.
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, such that its behaviour has far fewer degrees of freedom than the dimensionality of its phase space; that is, its evolution is restricted to a submanifold within its phase space.
Hamiltonian fluid mechanics is the application of Hamiltonian methods to fluid mechanics. Note that this formalism only applies to nondissipative fluids.
In physics, canonical quantum gravity is an attempt to quantize the canonical formulation of general relativity. It is a Hamiltonian formulation of Einstein's general theory of relativity. The basic theory was outlined by Bryce DeWitt in a seminal 1967 paper, and based on earlier work by Peter G. Bergmann using the so-called canonical quantization techniques for constrained Hamiltonian systems invented by Paul Dirac. Dirac's approach allows the quantization of systems that include gauge symmetries using Hamiltonian techniques in a fixed gauge choice. Newer approaches based in part on the work of DeWitt and Dirac include the Hartle–Hawking state, Regge calculus, the Wheeler–DeWitt equation and loop quantum gravity.
In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequencies of oscillatory or rotational motion without solving the equations of motion. Action-angle coordinates are chiefly used when the Hamilton–Jacobi equations are completely separable. Action-angle variables define an invariant torus, so called because holding the action constant defines the surface of a torus, while the angle variables parameterize the coordinates on the torus.
In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a -dimensional symplectic manifold for which the following conditions hold:
In dynamical systems theory, the Liouville–Arnold theorem states that if, in a Hamiltonian dynamical system with n degrees of freedom, there are also n independent, Poisson commuting first integrals of motion, and the energy level set is compact, then there exists a canonical transformation to action-angle coordinates in which the transformed Hamiltonian is dependent only upon the action coordinates and the angle coordinates evolve linearly in time. Thus the equations of motion for the system can be solved in quadratures if the level simultaneous set conditions can be separated. The theorem is named after Joseph Liouville and Vladimir Arnold.
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.
Hamiltonian truncation is a numerical method used to study quantum field theories (QFTs) in spacetime dimensions. Hamiltonian truncation is an adaptation of the Rayleigh–Ritz method from quantum mechanics. It is closely related to the exact diagonalization method used to treat spin systems in condensed matter physics. The method is typically used to study QFTs on spacetimes of the form , specifically to compute the spectrum of the Hamiltonian along . A key feature of Hamiltonian truncation is that an explicit ultraviolet cutoff is introduced, akin to the lattice spacing a in lattice Monte Carlo methods. Since Hamiltonian truncation is a nonperturbative method, it can be used to study strong-coupling phenomena like spontaneous symmetry breaking.
