# Hamiltonian mechanics

(Redirected from Hamilton's equations)

Hamiltonian mechanics is an equivalent but more abstract reformulation of classical mechanic theory. Historically, it contributed to the formulation of statistical mechanics and quantum mechanics.

Hamiltonian mechanics was first formulated by William Rowan Hamilton in 1833, starting from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788.

## Overview

In Hamiltonian mechanics, a classical physical system is described by a set of canonical coordinates r = (q, p), where each component of the coordinate qi, pi is indexed to the frame of reference of the system.

The time evolution of the system is uniquely defined by Hamilton's equations:

${\frac {\mathrm {d} {\boldsymbol {p}}}{\mathrm {d} t}}=-{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {q}}}}\quad ,\quad {\frac {\mathrm {d} {\boldsymbol {q}}}{\mathrm {d} t}}=+{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {p}}}}$

where H = H(q, p, t) is the Hamiltonian, which often corresponds to the total energy of the system. For a closed system, it is the sum of the kinetic and potential energy in the system.

In Newtonian mechanics, the time evolution is obtained by computing the total force being exerted on each particle of the system, and from Newton's second law, the time evolutions of both position and velocity are computed. In contrast, in Hamiltonian mechanics, the time evolution is obtained by computing the Hamiltonian of the system in the generalized coordinates and inserting it into Hamilton's equations. This approach is equivalent to the one used in Lagrangian mechanics. The Hamiltonian is the Legendre transform of the Lagrangian when holding q and t fixed and defining p as the dual variable, and thus both approaches give the same equations for the same generalized momentum. The main motivation to use Hamiltonian mechanics instead of Lagrangian mechanics comes from the symplectic structure of Hamiltonian systems.

While Hamiltonian mechanics can be used to describe simple systems such as a bouncing ball, a pendulum or an oscillating spring in which energy changes from kinetic to potential and back again over time, its strength is shown in more complex dynamic systems, such as planetary orbits in celestial mechanics. The more degrees of freedom the system has, the more complicated its time evolution is and, in most cases, it becomes chaotic.

### Basic physical interpretation

A simple interpretation of Hamiltonian mechanics comes from its application on a one-dimensional system consisting of one particle of mass m. The Hamiltonian can represent the total energy of the system, which is the sum of kinetic and potential energy, traditionally denoted T and V, respectively. Here q is the space coordinate and p is the momentum mv. Then

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

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 the momentum p equals the Newtonian force, and so the first Hamilton equation means that the force equals the negative gradient of potential energy. The time derivative of q is the velocity, and so the second Hamilton equation means that the particle's velocity equals the derivative of its kinetic energy with respect to its momentum.

### Calculating a Hamiltonian from a Lagrangian

Given a Lagrangian in terms of the generalized coordinates qi and generalized velocities ${\dot {q}}^{i}$  and time,

1. The momenta are calculated by differentiating the Lagrangian with respect to the (generalized) velocities:
$p_{i}(q^{i},{\dot {q}}^{i},t)={\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}$
2. The velocities ${\dot {q}}^{i}$  are expressed in terms of the momenta pi by inverting the expressions in the previous step.
3. The Hamiltonian is calculated using the usual definition of H as the Legendre transformation of L:
${\mathcal {H}}=\sum _{i}{\dot {q}}^{i}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}-{\mathcal {L}}=\sum _{i}{\dot {q}}^{i}p_{i}-{\mathcal {L}}$
Then the velocities are substituted for through the above results.

## Deriving Hamilton's equations

Hamilton's equations can be derived by looking at how the total differential of the Lagrangian depends on time, generalized positions qi, and generalized velocities i:

$\mathrm {d} {\mathcal {L}}=\sum _{i}\left({\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}\mathrm {d} {\dot {q}}^{i}\right)+{\frac {\partial {\mathcal {L}}}{\partial t}}\mathrm {d} t$

The generalized momenta were defined as

$p_{i}={\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}$

If this is substituted into the total differential of the Lagrangian, one gets

$\mathrm {d} {\mathcal {L}}=\sum _{i}\left({\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\mathrm {d} q^{i}+p_{i}\mathrm {d} {\dot {q}}^{i}\right)+{\frac {\partial {\mathcal {L}}}{\partial t}}\mathrm {d} t$

This can be rewritten as

$\mathrm {d} {\mathcal {L}}=\sum _{i}\left({\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\mathrm {d} q^{i}+\mathrm {d} \left(p_{i}{\dot {q}}^{i}\right)-{\dot {q}}^{i}\mathrm {d} p_{i}\right)+{\frac {\partial {\mathcal {L}}}{\partial t}}\mathrm {d} t\,.$

$\mathrm {d} \left(\sum _{i}p_{i}{\dot {q}}^{i}-{\mathcal {L}}\right)=\sum _{i}\left(-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\dot {q}}^{i}\mathrm {d} p_{i}\right)-{\frac {\partial {\mathcal {L}}}{\partial t}}\mathrm {d} t$

The term on the left-hand side is just the Hamiltonian that was defined before, therefore

$\mathrm {d} {\mathcal {H}}=\sum _{i}\left(-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\dot {q}}^{i}\mathrm {d} p_{i}\right)-{\frac {\partial {\mathcal {L}}}{\partial t}}\mathrm {d} t$

It is also possible to calculate the total differential of the Hamiltonian H with respect to time directly, similar to what was carried on with the Lagrangian L above, yielding:

$\mathrm {d} {\mathcal {H}}=\sum _{i}\left({\frac {\partial {\mathcal {H}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\frac {\partial {\mathcal {H}}}{\partial p_{i}}}\mathrm {d} p_{i}\right)+{\frac {\partial {\mathcal {H}}}{\partial t}}\mathrm {d} t$

It follows from the previous two independent equations that their right-hand sides are equal with each other. The result is

$\sum _{i}\left(-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\dot {q}}^{i}\mathrm {d} p_{i}\right)-{\frac {\partial {\mathcal {L}}}{\partial t}}\mathrm {d} t=\sum _{i}\left({\frac {\partial {\mathcal {H}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\frac {\partial {\mathcal {H}}}{\partial p_{i}}}\mathrm {d} p_{i}\right)+{\frac {\partial {\mathcal {H}}}{\partial t}}\mathrm {d} t$

Since this calculation was done off-shell, one can associate corresponding terms from both sides of this equation to yield:

${\frac {\partial {\mathcal {H}}}{\partial q^{i}}}=-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial p_{i}}}={\dot {q}}^{i}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial t}}=-{\partial {\mathcal {L}} \over \partial t}$

On-shell, Lagrange's equations indicate that

${\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}=0$

A rearrangement of this yields

${\frac {\partial {\mathcal {L}}}{\partial q^{i}}}={\dot {p}}_{i}$

Thus Hamilton's equations hold on-shell:

${\frac {\partial {\mathcal {H}}}{\partial q^{j}}}=-{\dot {p}}_{j}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial p_{j}}}={\dot {q}}^{j}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial t}}=-{\partial {\mathcal {L}} \over \partial t}$

## As a reformulation of Lagrangian mechanics

Starting with Lagrangian mechanics, the equations of motion are based on generalized coordinates

$\left.\left\{q^{j}\right|j=1,\ldots ,N\right\}$

and matching generalized velocities

$\left.\left\{{\dot {q}}^{j}\right|j=1,\ldots ,N\right\}$

The Lagrangian is written as

${\mathcal {L}}\left(q^{j},{\dot {q}}^{j},t\right)$

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, which is defined as:

$p_{j}={\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{j}}}$

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.

Something that may not be obvious in this coordinate dependent formulation is that different generalized coordinates are nothing more than different coordinate patches on the same symplectic manifold (see Mathematical formalism, below).

The Hamiltonian is the Legendre transform of the Lagrangian:

${\mathcal {H}}\left(q^{j},p_{j},t\right)=\left(\sum _{i}{\dot {q}}^{i}p_{i}\right)-{\mathcal {L}}\left(q^{j},{\dot {q}}^{j},t\right).$

If the transformation equations defining the generalized coordinates are independent of t, and the Lagrangian is a sum of products of functions (in the generalized coordinates) which are homogeneous of order 0, 1 or 2, then 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:

{\begin{aligned}\mathrm {d} {\mathcal {H}}&=\sum _{i}\left[\left({\partial {\mathcal {H}} \over \partial q^{i}}\right)\mathrm {d} q^{i}+\left({\partial {\mathcal {H}} \over \partial p_{i}}\right)\mathrm {d} p_{i}\right]+\left({\partial {\mathcal {H}} \over \partial t}\right)\mathrm {d} t\qquad \qquad \quad \quad \\\\&=\sum _{i}\left[{\dot {q}}^{i}\,\mathrm {d} p_{i}+p_{i}\,\mathrm {d} {\dot {q}}^{i}-\left({\partial {\mathcal {L}} \over \partial q^{i}}\right)\mathrm {d} q^{i}-\left({\partial {\mathcal {L}} \over \partial {\dot {q}}^{i}}\right)\mathrm {d} {\dot {q}}^{i}\right]-\left({\partial {\mathcal {L}} \over \partial t}\right)\mathrm {d} t.\end{aligned}}

Substituting the previous definition of the conjugate momenta into this equation and matching coefficients, the equations of motion of Hamiltonian mechanics, known as the canonical equations of Hamilton, are obtained:

${\frac {\partial {\mathcal {H}}}{\partial q^{j}}}=-{\dot {p}}_{j}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial p_{j}}}={\dot {q}}^{j}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial t}}=-{\frac {\partial {\mathcal {L}}}{\partial t}}.$

Hamilton's equations consist of 2n first-order differential equations, while Lagrange's equations consist of n second-order equations. However, Hamilton's equations usually do not reduce the difficulty of finding explicit solutions. They still offer some advantages, since important theoretical results can be derived 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, such that a coordinate does not occur in the Hamiltonian, the corresponding momentum is conserved, and that coordinate can be ignored in the other equations of the set. Effectively, this reduces the problem from n coordinates to (n − 1) coordinates. In the Lagrangian framework, the result that the corresponding momentum is conserved still follows immediately, but all the generalized velocities still occur in the Lagrangian. A system of equations in n coordinates still has to be solved.

The Lagrangian and Hamiltonian approaches provide the groundwork for deeper results in the theory of classical mechanics, and for formulations of quantum mechanics.

## Geometry of Hamiltonian systems

A Hamiltonian system may be understood as a fiber bundle E over time R, with the fibers Et, tR, 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 TEt, 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.

## 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

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

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.

## 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 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.

Given a function f

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

if there is a probability distribution, ρ, then (since the phase space velocity (i, 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 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 Gi. If the symplectic manifold has dimension 2n and there are n functionally independent conserved quantities Gi which are in involution (i.e., {Gi, Gj} = 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 Gi as coordinates; the new coordinates are called action-angle coordinates. The transformed Hamiltonian depends only on the Gi, and hence the equations of motion have the simple form

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

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 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 ⟨ , ⟩q is a smoothly varying inner product on the fibers T
q
Q
, 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

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

pz 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.

## 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 (i.e. qi = xi) and Lorenz gauge, the Lagrangian of a non-relativistic classical particle in an electromagnetic field is (in SI Units):

${\mathcal {L}}=\sum _{i}{\tfrac {1}{2}}m{\dot {x}}_{i}^{2}+\sum _{i}e{\dot {x}}_{i}A_{i}-e\varphi$

where e is the electric charge of the particle (not necessarily the elementary charge), φ is the electric scalar potential, and the Ai are the components of the magnetic vector potential (these may be modified through a gauge transformation).

This Lagrangian, combined with Euler–Lagrange equation, produces the Lorentz force law

$m{\ddot {\mathbf {x} }}=q\mathbf {E} +q{\dot {\mathbf {x} }}\times \mathbf {B} \,,$

and is called minimal coupling.

The generalized momenta are given by:

$p_{i}={\frac {\partial {\mathcal {L}}}{\partial {\dot {x}}_{i}}}=m{\dot {x}}_{i}+eA_{i}$

Rearranging, the velocities are expressed in terms of the momenta:

${\dot {x}}_{i}={\frac {p_{i}-eA_{i}}{m}}$

If the definition of the momenta is substituted, as well as the definitions of the velocities in terms of the momenta, into the definition of the Hamiltonian given above, and this is then simplified and rearranged, it is rendered as:

${\mathcal {H}}=\left\{\sum _{i}{\dot {x}}_{i}p_{i}\right\}-{\mathcal {L}}=\sum _{i}{\frac {\left(p_{i}-eA_{i}\right)^{2}}{2m}}+e\varphi$

This equation is used frequently in quantum mechanics.

## Relativistic charged particle in an electromagnetic field

The Lagrangian for a relativistic charged particle is given by:

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

Thus the particle's canonical (total) momentum is

$\mathbf {P} (t)={\frac {\partial {\mathcal {L}}(t)}{\partial {\dot {\mathbf {x} }}(t)}}={\frac {m{\dot {\mathbf {x} }}(t)}{\sqrt {1-{\frac {{{\dot {\mathbf {x} }}(t)}^{2}}{c^{2}}}}}}+e\mathbf {A} \left(\mathbf {x} (t),t\right)$

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

Solving for the velocity, we get

${\dot {\mathbf {x} }}(t)={\frac {\mathbf {P} (t)-e\mathbf {A} (\mathbf {x} (t),t)}{\sqrt {m^{2}+{\frac {1}{c^{2}}}{\left(\mathbf {P} (t)-e\mathbf {A} \left(\mathbf {x} (t),t\right)\right)}^{2}}}}$

So the Hamiltonian is

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

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

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

from which one can derive

${\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {m{\dot {\mathbf {x} }}}{\sqrt {1-{\frac {{\dot {\mathbf {x} }}^{2}}{c^{2}}}}}}\right)=e\mathbf {E} +e{\dot {\mathbf {x} }}\times \mathbf {B}$

An equivalent expression for the Hamiltonian as function of the relativistic (kinetic) momentum, p = γm(t), is

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

This has the advantage that p can be measured experimentally whereas P cannot. Notice that the Hamiltonian (total energy) can be viewed as the sum of the relativistic energy (kinetic+rest), E = γmc2, plus the potential energy, V = .