Phase space coordinates (p,q) and Hamiltonian H

Let ${\displaystyle (M,{\mathcal {L}})}$ be a mechanical system with the configuration space ${\displaystyle M}$ and the smooth Lagrangian ${\displaystyle {\mathcal {L}}.}$ Select a standard coordinate system ${\displaystyle ({\boldsymbol {q}},{\boldsymbol {\dot {q}}})}$ on ${\displaystyle M.}$ The quantities ${\displaystyle \textstyle p_{i}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)~{\stackrel {\text{def}}{=}}~{\partial {\mathcal {L}}}/{\partial {\dot {q}}^{i}}}$ are called momenta. (Also generalized momenta, conjugate momenta, and canonical momenta). For a time instant ${\displaystyle t,}$ the Legendre transformation of ${\displaystyle {\mathcal {L}}}$ is defined as the map ${\displaystyle ({\boldsymbol {q}},{\boldsymbol {\dot {q}}})\to \left({\boldsymbol {p}},{\boldsymbol {q}}\right)}$ which is assumed to have a smooth inverse ${\displaystyle ({\boldsymbol {p}},{\boldsymbol {q}})\to ({\boldsymbol {q}},{\boldsymbol {\dot {q}}}).}$ For a system with ${\displaystyle n}$ degrees of freedom, the Lagrangian mechanics defines the energy function

$${\displaystyle E_{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)\,{\stackrel {\text{def}}{=}}\,\sum _{i=1}^{n}{\dot {q}}^{i}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}-{\mathcal {L}}.}$$

The Legendre transform of ${\displaystyle {\mathcal {L}}}$ turns ${\displaystyle E_{\mathcal {L}}}$ into a function ${\displaystyle {\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}},t)}$ known as the Hamiltonian. The Hamiltonian satisfies

$${\displaystyle {\mathcal {H}}\left({\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {\dot {q}}}}},{\boldsymbol {q}},t\right)=E_{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)}$$

which implies that

$${\displaystyle {\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}},t)=\sum _{i=1}^{n}p_{i}{\dot {q}}^{i}-{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t),}$$

where the velocities ${\displaystyle {\boldsymbol {\dot {q}}}=({\dot {q}}^{1},\ldots ,{\dot {q}}^{n})}$ are found from the (${\displaystyle n}$-dimensional) equation ${\displaystyle \textstyle {\boldsymbol {p}}={\partial {\mathcal {L}}}/{\partial {\boldsymbol {\dot {q}}}}}$ which, by assumption, is uniquely solvable for ${\displaystyle {\boldsymbol {\dot {q}}}.}$ The (${\displaystyle 2n}$-dimensional) pair ${\displaystyle ({\boldsymbol {p}},{\boldsymbol {q}})}$ is called phase space coordinates. (Also canonical coordinates).

From Euler–Lagrange equation to Hamilton's equations

In phase space coordinates ${\displaystyle ({\boldsymbol {p}},{\boldsymbol {q}}),}$ the (${\displaystyle n}$-dimensional) Euler–Lagrange equation

$${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {q}}}}-{\frac {d}{dt}}{\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {\dot {q}}}}}=0}$$

becomes Hamilton's equations in ${\displaystyle 2n}$ dimensions

From stationary action principle to Hamilton's equations

Let ${\displaystyle {\mathcal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})}$ be the set of smooth paths ${\displaystyle {\boldsymbol {q}}:[a,b]\to M}$ for which ${\displaystyle {\boldsymbol {q}}(a)={\boldsymbol {x}}_{a}}$ and ${\displaystyle {\boldsymbol {q}}(b)={\boldsymbol {x}}_{b}.}$ The action functional ${\displaystyle {\mathcal {S}}:{\mathcal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})\to \mathbb {R} }$ is defined via

$${\displaystyle {\mathcal {S}}[{\boldsymbol {q}}]=\int _{a}^{b}{\mathcal {L}}(t,{\boldsymbol {q}}(t),{\dot {\boldsymbol {q}}}(t))\,dt=\int _{a}^{b}\left(\sum _{i=1}^{n}p_{i}{\dot {q}}^{i}-{\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}},t)\right)\,dt,}$$

where ${\displaystyle {\boldsymbol {q}}={\boldsymbol {q}}(t),}$ and ${\displaystyle {\boldsymbol {p}}=\partial {\mathcal {L}}/\partial {\boldsymbol {\dot {q}}}}$ (see above). A path ${\displaystyle {\boldsymbol {q}}\in {\mathcal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})}$ is a stationary point of ${\displaystyle {\mathcal {S}}}$ (and hence is an equation of motion) if and only if the path ${\displaystyle ({\boldsymbol {p}}(t),{\boldsymbol {q}}(t))}$ 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 nonrelativistic particle of mass m. The value ${\displaystyle H(p,q)}$ of the Hamiltonian is the total energy of the system, in this case the sum of kinetic and potential energy, traditionally denoted T and V, respectively. Here p is the momentum mv and q is the space coordinate. Then

$${\displaystyle {\mathcal {H}}=T+V,\qquad T={\frac {p^{2}}{2m}},\qquad V=V(q)}$$

T is a function of p alone, while V is a function of q alone (i.e., T and V are scleronomic).

In this example, the time derivative of q 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 p equals the Newtonian force, and so the second Hamilton equation means that the force equals the negative gradient of potential energy.

Relativistic charged particle in an electromagnetic field

The relativistic Lagrangian for a particle (rest mass ${\displaystyle m}$ and charge ${\displaystyle q}$) is given by:

$${\displaystyle {\mathcal {L}}(t)=-mc^{2}{\sqrt {1-{\frac {{{\dot {\mathbf {x} }}(t)}^{2}}{c^{2}}}}}+q{\dot {\mathbf {x} }}(t)\cdot \mathbf {A} \left(\mathbf {x} (t),t\right)-q\varphi \left(\mathbf {x} (t),t\right)}$$

Thus the particle's canonical momentum is

$${\displaystyle \mathbf {p} (t)={\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {x} }}}}={\frac {m{\dot {\mathbf {x} }}}{\sqrt {1-{\frac {{\dot {\mathbf {x} }}^{2}}{c^{2}}}}}}+q\mathbf {A} }$$

that is, the sum of the kinetic momentum and the potential momentum.

Solving for the velocity, we get

$${\displaystyle {\dot {\mathbf {x} }}(t)={\frac {\mathbf {p} -q\mathbf {A} }{\sqrt {m^{2}+{\frac {1}{c^{2}}}{\left(\mathbf {p} -q\mathbf {A} \right)}^{2}}}}}$$

So the Hamiltonian is

$${\displaystyle {\mathcal {H}}(t)={\dot {\mathbf {x} }}\cdot \mathbf {p} -{\mathcal {L}}=c{\sqrt {m^{2}c^{2}+{\left(\mathbf {p} -q\mathbf {A} \right)}^{2}}}+q\varphi }$$

This results in the force equation (equivalent to the Euler–Lagrange equation)

$${\displaystyle {\dot {\mathbf {p} }}=-{\frac {\partial {\mathcal {H}}}{\partial \mathbf {x} }}=q{\dot {\mathbf {x} }}\cdot ({\boldsymbol {\nabla }}\mathbf {A} )-q{\boldsymbol {\nabla }}\varphi =q{\boldsymbol {\nabla }}({\dot {\mathbf {x} }}\cdot \mathbf {A} )-q{\boldsymbol {\nabla }}\varphi }$$

from which one can derive

$${\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {m{\dot {\mathbf {x} }}}{\sqrt {1-{\frac {{\dot {\mathbf {x} }}^{2}}{c^{2}}}}}}\right)&={\frac {\mathrm {d} }{\mathrm {d} t}}(\mathbf {p} -q\mathbf {A} )={\dot {\mathbf {p} }}-q{\frac {\partial \mathbf {A} }{\partial t}}-q({\dot {\mathbf {x} }}\cdot \nabla )\mathbf {A} \\&=q{\boldsymbol {\nabla }}({\dot {\mathbf {x} }}\cdot \mathbf {A} )-q{\boldsymbol {\nabla }}\varphi -q{\frac {\partial \mathbf {A} }{\partial t}}-q({\dot {\mathbf {x} }}\cdot \nabla )\mathbf {A} \\&=q\mathbf {E} +q{\dot {\mathbf {x} }}\times \mathbf {B} \end{aligned}}}$$

The above derivation makes use of the vector calculus identity:

$${\displaystyle {\tfrac {1}{2}}\nabla \left(\mathbf {A} \cdot \mathbf {A} \right)=\mathbf {A} \cdot \mathbf {J} _{\mathbf {A} }=\mathbf {A} \cdot (\nabla \mathbf {A} )=(\mathbf {A} \cdot \nabla )\mathbf {A} +\mathbf {A} \times (\nabla \times \mathbf {A} ).}$$

An equivalent expression for the Hamiltonian as function of the relativistic (kinetic) momentum, ${\displaystyle \mathbf {P} =\gamma m{\dot {\mathbf {x} }}(t)=\mathbf {p} -q\mathbf {A} }$, is

$${\displaystyle {\mathcal {H}}(t)={\dot {\mathbf {x} }}(t)\cdot \mathbf {P} (t)+{\frac {mc^{2}}{\gamma }}+q\varphi (\mathbf {x} (t),t)=\gamma mc^{2}+q\varphi (\mathbf {x} (t),t)=E+V}$$

This has the advantage that kinetic momentum ${\displaystyle \mathbf {P} }$ can be measured experimentally whereas canonical momentum ${\displaystyle \mathbf {p} }$ cannot. Notice that the Hamiltonian (total energy) can be viewed as the sum of the relativistic energy (kinetic+rest), ${\displaystyle E=\gamma mc^{2}}$, plus the potential energy, ${\displaystyle V=q\varphi }$.

Geometry of Hamiltonian systems

The Hamiltonian can induce a symplectic structure on a smooth even-dimensional manifold M²ⁿ in several equivalent ways, the best known being the following:^[6]

As a closed nondegenerate symplectic 2-form ω. According to the Darboux's theorem, in a small neighbourhood around any point on M there exist suitable local coordinates ${\displaystyle p_{1},\cdots ,p_{n},\ q_{1},\cdots ,q_{n}}$ (canonical or symplectic coordinates) in which the symplectic form becomes:

$${\displaystyle \omega =\sum _{i=1}^{n}dp_{i}\wedge dq_{i}\,.}$$

The form ${\displaystyle \omega }$ induces a natural isomorphism of the tangent space with the cotangent space: ${\displaystyle T_{x}M\cong T_{x}^{*}M.}$ This is done by mapping a vector ${\displaystyle \xi \in T_{x}M}$ to the 1-form ${\displaystyle \omega _{\xi }\in T_{x}^{*}M,}$ where ${\displaystyle \omega _{\xi }(\eta )=\omega (\eta ,\xi )}$ for all ${\displaystyle \eta \in T_{x}M.}$ Due to the bilinearity and non-degeneracy of ${\displaystyle \omega ,}$ and the fact that ${\displaystyle \dim T_{x}M=\dim T_{x}^{*}M,}$ the mapping ${\displaystyle \xi \to \omega _{\xi }}$ is indeed a linear isomorphism. This isomorphism is natural in that it does not change with change of coordinates on ${\displaystyle M.}$ Repeating over all ${\displaystyle x\in M,}$ we end up with an isomorphism ${\displaystyle J^{-1}:{\text{Vect}}(M)\to \Omega ^{1}(M)}$ between the infinite-dimensional space of smooth vector fields and that of smooth 1-forms. For every ${\displaystyle f,g\in C^{\infty }(M,\mathbb {R} )}$ and ${\displaystyle \xi ,\eta \in {\text{Vect}}(M),}$

$${\displaystyle J^{-1}(f\xi +g\eta )=fJ^{-1}(\xi )+gJ^{-1}(\eta ).}$$

(In algebraic terms, one would say that the ${\displaystyle C^{\infty }(M,\mathbb {R} )}$-modules ${\displaystyle {\text{Vect}}(M)}$ and ${\displaystyle \Omega ^{1}(M)}$ are isomorphic). If ${\displaystyle H\in C^{\infty }(M\times \mathbb {R} _{t},\mathbb {R} ),}$ then, for every fixed ${\displaystyle t\in \mathbb {R} _{t},}$ ${\displaystyle dH\in \Omega ^{1}(M),}$ and ${\displaystyle J(dH)\in {\text{Vect}}(M).}$ ${\displaystyle J(dH)}$ is known as a Hamiltonian vector field. The respective differential equation on ${\displaystyle M}$

$${\displaystyle {\dot {x}}=J(dH)(x)}$$

is called Hamilton's equation. Here ${\displaystyle x=x(t)}$ and ${\displaystyle J(dH)(x)\in T_{x}M}$ is the (time-dependent) value of the vector field ${\displaystyle J(dH)}$ at ${\displaystyle x\in M.}$

A Hamiltonian system may be understood as a fiber bundle E over time R, with the fiber E_t being the position space at time t ∈ R. 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. The correspondence between Lagrangian and Hamiltonian mechanics is achieved with the tautological one-form.

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 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 F and G are smooth functions on M then the smooth function ω²(IdG, IdF) is properly defined; it is called a Poisson bracket of functions F and G and is denoted {F, G}. The Poisson bracket has the following properties:

1. bilinearity
2. antisymmetry
3. Leibniz rule: ${\displaystyle \{F_{1}\cdot F_{2},G\}=F_{1}\{F_{2},G\}+F_{2}\{F_{1},G\}}$
4. Jacobi identity: ${\displaystyle \{\{H,F\},G\}+\{\{F,G\},H\}+\{\{G,H\},F\}\equiv 0}$
5. non-degeneracy: if the point x on M is not critical for F then a smooth function G exists such that ${\displaystyle \{F,G\}(x)\neq 0}$.

Given a function f

$${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}f={\frac {\partial }{\partial t}}f+\left\{f,{\mathcal {H}}\right\},}$$

if there is a probability distribution ρ, then (since the phase space velocity ${\displaystyle ({\dot {p}}_{i},{\dot {q}}_{i})}$ has zero divergence and probability is conserved) its convective derivative can be shown to be zero and so

$${\displaystyle {\frac {\partial }{\partial t}}\rho =-\left\{\rho ,{\mathcal {H}}\right\}}$$

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.

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–Arnold theorem says that, locally, any Liouville integrable Hamiltonian can be transformed via a symplectomorphism into 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

$${\displaystyle {\dot {G}}_{i}=0\quad ,\quad {\dot {\varphi }}_{i}=F_{i}(G)}$$

for some function F.^[7] 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

$${\displaystyle {\mathcal {H}}(q,p)={\tfrac {1}{2}}\langle p,p\rangle _{q}}$$

where ⟨ , ⟩_q is a smoothly varying inner product on the fibers T^∗
_qQ, the cotangent space to the point q 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 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.

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

$${\displaystyle {\mathcal {H}}\left(x,y,z,p_{x},p_{y},p_{z}\right)={\tfrac {1}{2}}\left(p_{x}^{2}+p_{y}^{2}\right).}$$

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.

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 p and q to the algebra of Moyal brackets.

Specifically, the more general form of the Hamilton's equation reads

$${\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} t}}=\left\{f,{\mathcal {H}}\right\}+{\frac {\partial f}{\partial t}}}$$

where f is some function of p and q, and H 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 the phase space formulation and the 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.