Hamiltonian mechanics emerged in 1833 as a reformulation of
Lagrangian mechanics. Introduced by
Sir William Rowan Hamilton,^{
[1]} Hamiltonian mechanics replaces (generalized) velocities ${\dot {q}}^{i}$ used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of
classical mechanics and describe the same physical phenomena.
Let $(M,{\mathcal {L}})$ be a
mechanical system with the
configuration space$M$ and the smooth Lagrangian ${\mathcal {L}}.$ Select a standard coordinate system $({\boldsymbol {q}},{\boldsymbol {\dot {q}}})$ on $M.$ The quantities $\textstyle p_{i}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)~{\stackrel {\text{def}}{=}}~{\partial {\mathcal {L}}}/{\partial {\dot {q}}^{i}}$ are called momenta. (Also generalized momenta, conjugate momenta, and canonical momenta). For a time instant $t,$ the
Legendre transformation of ${\mathcal {L}}$ is defined as the map $({\boldsymbol {q}},{\boldsymbol {\dot {q}}})\to \left({\boldsymbol {p}},{\boldsymbol {q}}\right)$ which is assumed to have a smooth inverse $({\boldsymbol {p}},{\boldsymbol {q}})\to ({\boldsymbol {q}},{\boldsymbol {\dot {q}}}).$ For a system with $n$ degrees of freedom, the Lagrangian mechanics defines the energy function
The Legendre transform of ${\mathcal {L}}$ turns $E_{\mathcal {L}}$ into a function ${\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}},t)$ known as the Hamiltonian. The Hamiltonian satisfies
where the velocities ${\boldsymbol {\dot {q}}}=({\dot {q}}^{1},\ldots ,{\dot {q}}^{n})$ are found from the ($n$-dimensional) equation $\textstyle {\boldsymbol {p}}={\partial {\mathcal {L}}}/{\partial {\boldsymbol {\dot {q}}}}$ which, by assumption, is uniquely solvable for ${\boldsymbol {\dot {q}}}.$ The ($2n$-dimensional) pair $({\boldsymbol {p}},{\boldsymbol {q}})$ is called phase space coordinates. (Also canonical coordinates).
From Euler-Lagrange equation to Hamilton's equations
In phase space coordinates $({\boldsymbol {p}},{\boldsymbol {q}}),$ the ($n$-dimensional)
Euler-Lagrange equation
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 ${\boldsymbol {q}}:[a,b]\to M$ for which ${\boldsymbol {q}}(a)={\boldsymbol {x}}_{a}$ and ${\boldsymbol {q}}(b)={\boldsymbol {x}}_{b}.$ The
action functional${\mathcal {S}}:{\mathcal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})\to \mathbb {R}$ is defined via
where ${\boldsymbol {q}}={\boldsymbol {q}}(t),$ and ${\boldsymbol {p}}=\partial {\mathcal {L}}/\partial {\boldsymbol {\dot {q}}}$ (see above). A path ${\boldsymbol {q}}\in {\mathcal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})$ is a
stationary point of ${\mathcal {S}}$ (and hence is an equation of motion) if and only if the path $({\boldsymbol {p}}(t),{\boldsymbol {q}}(t))$ 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 $H(p,q)$ 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
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.
A spherical pendulum consists of a
massm moving without
friction on the surface of a
sphere. The only
forces acting on the mass are the
reaction from the sphere and
gravity.
Spherical coordinates are used to describe the position of the mass in terms of (r, θ, φ), where r is fixed, r = l.
Momentum $P_{\phi }$, which corresponds to the vertical component of
angular momentum$L_{z}=l\sin \theta \times ml\sin \theta \,{\dot {\phi }}$, 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$\phi$ 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 q^{i}, and generalized velocities q̇^{i}, where $i=1,\ldots ,n$.^{
[3]} Here we work
off-shell, meaning $q^{i},{\dot {q}}^{i},t$ are independent coordinates in phase space, not constrained to follow any equations of motion (in particular, ${\dot {q}}^{i}$ is not a derivative of $q^{i}$). The
total differential of the Lagrangian is:
The term in parentheses on the left-hand side is just the Hamiltonian ${\textstyle {\mathcal {H}}=\sum p_{i}{\dot {q}}^{i}-{\mathcal {L}}}$ defined previously, therefore:
One may also calculate the total differential of the Hamiltonian ${\mathcal {H}}$ with respect to coordinates $q^{i},p_{i},t$ instead of $q^{i},{\dot {q}}^{i},t$, yielding:
Since these calculations are off-shell, one can equate the respective coefficients of $\mathrm {d} q^{i},\mathrm {d} p_{i},\mathrm {d} t$ on the two sides:
On-shell, one substitutes parametric functions $q^{i}=q^{i}(t)$ which define a trajectory in phase space with velocities ${\textstyle {\dot {q}}^{i}={\tfrac {d}{dt}}q^{i}(t)}$, obeying
Lagrange's equations:
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 2n first-order
differential equations, while Lagrange's equations consist of n 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 n coordinates to (n − 1) coordinates: this is the basis of
symplectic reduction 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 n coordinates still has to be solved.^{
[4]}
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 ${\mathcal {H}}).$
${\frac {d{\mathcal {H}}}{dt}}={\frac {\partial {\mathcal {H}}}{\partial t}}$ when $\mathbf {p} (t),\mathbf {q} (t)$ form a solution of Hamilton's equations. Indeed, ${\textstyle {\frac {d{\mathcal {H}}}{dt}}={\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {p}}}}\cdot {\dot {\boldsymbol {p}}}+{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {q}}}}\cdot {\dot {\boldsymbol {q}}}+{\frac {\partial {\mathcal {H}}}{\partial t}},}$ and everything but the final term cancels out.
${\mathcal {H}}$ does not change under point transformations, i.e. smooth changes ${\boldsymbol {q}}\leftrightarrow {\boldsymbol {q'}}$ of space coordinates. (Follows from the invariance of the energy function $E_{\mathcal {L}}$ under point transformations. The invariance of $E_{\mathcal {L}}$ can be established directly).
$-{\frac {\partial {\mathcal {H}}}{\partial q^{i}}}={\dot {p}}_{i}={\frac {\partial {\mathcal {L}}}{\partial q^{i}}}.$ (Compare Hamilton's and Euler-Lagrange equations or see Deriving Hamilton's equations).
${\frac {\partial {\mathcal {H}}}{\partial q^{i}}}=0$ if and only if ${\frac {\partial {\mathcal {L}}}{\partial q^{i}}}=0.$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. In
Cartesian coordinates the
Lagrangian of a non-relativistic classical particle in an electromagnetic field is (in
SI Units):
In quantum mechanics, the
wave function will also undergo a
localU(1) group transformation^{
[5]} 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
An equivalent expression for the Hamiltonian as function of the relativistic (kinetic) momentum, $\mathbf {P} =\gamma m{\dot {\mathbf {x} }}(t)=\mathbf {p} -q\mathbf {A}$, is
This has the advantage that kinetic momentum $\mathbf {P}$ can be measured experimentally whereas canonical momentum $\mathbf {p}$ cannot. Notice that the Hamiltonian (
total energy) can be viewed as the sum of the
relativistic energy (kinetic+rest), $E=\gamma mc^{2}$, plus the
potential energy, $V=q\varphi$.
The form $\omega$ induces a
natural isomorphism of the
tangent space with the
cotangent space: $T_{x}M\cong T_{x}^{*}M.$ This is done by mapping a vector $\xi \in T_{x}M$ to the 1-form $\omega _{\xi }\in T_{x}^{*}M,$ 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. 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 $J^{-1}:{\text{Vect}}(M)\to \Omega ^{1}(M)$ between the infinite-dimensional space of smooth vector fields and that of smooth 1-forms. For every $f,g\in C^{\infty }(M,\mathbb {R} )$ and $\xi ,\eta \in {\text{Vect}}(M),$
(In algebraic terms, one would say that the $C^{\infty }(M,\mathbb {R} )$-modules ${\text{Vect}}(M)$ and $\Omega ^{1}(M)$ are isomorphic). If $H\in C^{\infty }(M\times \mathbb {R} _{t},\mathbb {R} ),$ then, for every fixed $t\in \mathbb {R} _{t},$$dH\in \Omega ^{1}(M),$ and $J(dH)\in {\text{Vect}}(M).$$J(dH)$ is known as a
Hamiltonian vector field. The respective differential equation on $M$
${\dot {x}}=J(dH)(x)$
is called Hamilton's equation. Here $x=x(t)$ and $J(dH)(x)\in T_{x}M$ is the (time-dependent) value of the vector field $J(dH)$ at $x\in M.$
A Hamiltonian system may be understood as a
fiber bundleE over
timeR, with the
fiberE_{t} being the position space at time t ∈ R. The Lagrangian is thus a function on the
jet bundleJ 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 spaceT^{∗}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.
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 ω^{2}(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:
if there is a
probability distributionρ, 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
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
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
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
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