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 nondegenerate symplectic 2-form ω. According to
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 or
symplectic coordinates) in which the symplectic form becomes: \omega = \sum_{i=1}^n dp_i \wedge dq_i \, . The form \omega induces a
natural isomorphism of the
tangent space with the
cotangent space: . This is done by mapping a vector \xi \in T_x M to the 1-form , where \omega_\xi (\eta) = \omega(\eta, \xi) for all . Due to the
bilinearity and non-degeneracy of , and the fact that , the mapping \xi \to \omega_\xi is indeed a
linear isomorphism. This isomorphism is
natural in that it does not change with change of coordinates on M. Repeating over all , we end up with an isomorphism J^{-1} : \text{Vect}(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 {{tmath|1= \xi,\eta \in \text{Vect}(M) }}, J^{-1}(f\xi + g\eta) = fJ^{-1}(\xi) + gJ^{-1}(\eta). (In algebraic terms, one would say that the C^\infty(M,\Reals)-modules \text{Vect}(M) and \Omega^1(M) are isomorphic). If , then, for every fixed , , and {{tmath|1= J(dH) \in \text{Vect}(M) }}. J(dH) is known as a
Hamiltonian vector field. The respective differential equation on M \dot{x} = 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 . A Hamiltonian system may be understood as a
fiber bundle over
time , with the
fiber being the position space at time . The Lagrangian is thus a function on the
jet bundle over ; taking the fiberwise
Legendre transform 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 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. 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 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: \{F_1 \cdot F_2, G\} = F_1\{F_2, G\} + F_2\{F_1, G\} •
Jacobi identity: \{\{H,F\}, G\} + \{\{F, G\}, H\} + \{\{G, H\}, F\} \equiv 0 • non-degeneracy: if the point on is not critical for then a smooth function exists such that {{tmath|1= \{F, G\}(x) \neq 0 }}. Given a function \frac{\mathrm{d}}{\mathrm{d}t} f = \frac{\partial }{\partial t} f + \left\{f,\mathcal{H}\right\}, if there is a
probability distribution , then (since the phase space velocity (\dot{p}_i, \dot{q}_i) has zero divergence and probability is conserved) its convective derivative can be shown to be zero and so \frac{\partial}{\partial t} \rho = - \left\{\rho ,\mathcal{H}\right\} This is called
Liouville's theorem. Every
smooth function over the
symplectic manifold generates a one-parameter family of
symplectomorphisms and if {{math|1={
G,
H} = 0}}, 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{G}_i = 0 \quad , \quad \dot{\varphi}_i = F_i(G) 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; 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 \mathcal{H}(q,p) = \tfrac{1}{2} \langle p, p\rangle_q where is a smoothly varying
inner product on the
fibers , the
cotangent space to the point in the
configuration space, sometimes called a cometric. This Hamiltonian consists entirely of the kinetic term. If one considers a
Riemannian manifold or a
pseudo-Riemannian manifold, the
Riemannian metric 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 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. See also
Geodesics as Hamiltonian flows.
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 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. The continuous, real-valued
Heisenberg group provides a simple example of a sub-Riemannian manifold. For the Heisenberg group, the Hamiltonian is given by \mathcal{H}\left(x,y,z,p_x,p_y,p_z\right) = \tfrac{1}{2}\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 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 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, but not for
quantum mechanics, 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 over and to the algebra of
Moyal brackets. Specifically, the more general form of the Hamilton's equation reads \frac{\mathrm{d}f}{\mathrm{d}t} = \left\{f, \mathcal{H}\right\} + \frac{\partial f}{\partial t} , 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. 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
Phase space formulation and
Wigner–Weyl transform). This more algebraic approach not only permits ultimately extending
probability distributions in
phase space to
Wigner quasi-probability distributions, but, at the mere Poisson bracket classical setting, also provides more power in helping analyze the relevant
conserved quantities in a system. == See also ==