Hamiltonian mechanics

From Wikipedia, the free encyclopedia

Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics. It uses a different mathematical formalism, providing a more abstract understanding of the theory. Historically, it was an important reformulation of classical mechanics, which later 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 q_i, p_i is indexed to the frame of reference of the system.

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

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

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 in the Hamilton's equations. This approach is equivalent to the one used in Lagrangian mechanics. In fact, as is shown below, 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 represents 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

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

Note that 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 ${\displaystyle {\dot {q}}^{i}}$ and time,

1. The momenta are calculated by differentiating the Lagrangian with respect to the (generalized) velocities:

   ${\displaystyle p_{i}(q^{i},{\dot {q}}^{i},t)={\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}}$

2. The velocities ${\displaystyle {\dot {q}}^{i}}$ are expressed in terms of the momenta p_i 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:
   ${\displaystyle {\mathcal {H}}=\sum _{i}{\dot {q}}^{i}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}-{\mathcal {L}}=\sum _{i}{\dot {q}}^{i}p_{i}-{\mathcal {L}}}$
   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 q_i, and generalized velocities q̇_i:

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

The generalized momenta were defined as

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

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

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

This can be rewritten as

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

which after rearranging leads to

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

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

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

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:

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

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

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

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

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

On-shell, Lagrange's equations indicate that

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

A rearrangement of this yields

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

Thus Hamilton's equations hold on-shell:

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

As a reformulation of Lagrangian mechanics

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

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

and matching generalized velocities

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

We write the Lagrangian as

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

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, defined as:

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

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.

One thing which is not too obvious in this coordinate dependent formulation is that different generalized coordinates are really nothing more than different coordinate patches on the same symplectic manifold (see Mathematical formalism, below).

The Hamiltonian is the Legendre transform of the Lagrangian:

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

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:

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

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

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

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 don't 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, of course the result that the corresponding momentum is conserved still follows immediately, but all the generalized velocities still occur in the Lagrangian - we still have to solve a system of equations in n coordinates.

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 E_t, t ∈ R, 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 T^∗E_t, which comes equipped with a natural symplectic form, and this latter function is the Hamiltonian.

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.

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 (a special type of symplectic 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

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

If we have a probability distribution, ρ, then (since the phase space velocity (ṗ_i, q̇_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

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

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

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

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. 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.

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.

Charged particle in an electromagnetic field

A good illustration of Hamiltonian mechanics is given by the Hamiltonian of a charged particle in an electromagnetic field. In Cartesian coordinates (i.e. q_i = x_i), the Lagrangian of a non-relativistic classical particle in an electromagnetic field is (in SI Units):

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

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

The generalized momenta are given by:

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

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

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

If we substitute the definition of the momenta, and the definitions of the velocities in terms of the momenta, into the definition of the Hamiltonian given above, and then simplify and rearrange, we get:

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

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:

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

Thus the particle's canonical (total) momentum is

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

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

Solving for the velocity, we get

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

So the Hamiltonian is

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

From this we get the force equation (equivalent to the Euler–Lagrange equation)

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

from which one can derive

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

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

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

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 = eφ.

William Rowan Hamilton

From Wikipedia, the free encyclopedia

Sir William Rowan Hamilton
Sir William Rowan Hamilton (1805–1865)
Born	4 August 1805 Dublin, Ireland
Died	2 September 1865 (aged 60) Dublin, Ireland
Residence	Ireland
Nationality	Irish
Alma mater	Trinity College, Dublin
Known for	Hamilton's principle Hamiltonian mechanics Hamiltonians Hamilton–Jacobi equation Quaternions Biquaternions Hamiltonian path Icosian calculus Nabla symbol Versor Coining the word 'tensor' Hamiltonian vector field Icosian game Universal algebra Hodograph Hamiltonian group Cayley–Hamilton theorem
Spouse(s)	Helen Maria Bayly
Children	William Edwin Hamilton, Archibald Henry Hamilton, Helen Eliza Amelia O'Regan Hamilton
Awards	Royal Medal (1835)
Scientific career
Fields	Mathematics, astronomy, physics
Institutions	Trinity College, Dublin
Academic advisors	John Brinkley
Influences	Zerah Colburn John T. Graves
Influenced	Peter Guthrie Tait

Sir William Rowan Hamilton MRIA (4 August 1805 – 2 September 1865) was an Irish mathematician. While still an undergraduate he was appointed Andrews professor of Astronomy and Royal Astronomer of Ireland, and lived at Dunsink Observatory. He made important contributions to optics, classical mechanics and algebra. Although Hamilton was not a physicist–he regarded himself as a pure mathematician–his work was of major importance to physics, particularly his reformulation of Newtonian mechanics, now called Hamiltonian mechanics. This work has proven central to the modern study of classical field theories such as electromagnetism, and to the development of quantum mechanics. In pure mathematics, he is best known as the inventor of quaternions.

Hamilton is said to have shown immense talent at a very early age. Astronomer Bishop Dr. John Brinkley remarked of the 18-year-old Hamilton, "This young man, I do not say will be, but is, the first mathematician of his age."

Life

William Rowan Hamilton's scientific career included the study of geometrical optics, classical mechanics, adaptation of dynamic methods in optical systems, applying quaternion and vector methods to problems in mechanics and in geometry, development of theories of conjugate algebraic couple functions (in which complex numbers are constructed as ordered pairs of real numbers), solvability of polynomial equations and general quintic polynomial solvable by radicals, the analysis on Fluctuating Functions (and the ideas from Fourier analysis), linear operators on quaternions and proving a result for linear operators on the space of quaternions (which is a special case of the general theorem which today is known as the Cayley–Hamilton theorem). Hamilton also invented "icosian calculus", which he used to investigate closed edge paths on a dodecahedron that visit each vertex exactly once.

Early life

Hamilton was the fourth of nine children born to Sarah Hutton (1780–1817) and Archibald Hamilton (1778–1819), who lived in Dublin at 29 Dominick Street, later renumbered to 36. Hamilton's father, who was from Dublin, worked as a solicitor. By the age of three, Hamilton had been sent to live with his uncle James Hamilton, a graduate of Trinity College who ran a school in Talbots Castle in Trim, Co. Meath.

His uncle soon discovered that Hamilton had a remarkable ability to learn languages, and from a young age, had displayed an uncanny ability to acquire them (although this is disputed by some historians, who claim he had only a very basic understanding of them). At the age of seven, he had already made very considerable progress in Hebrew, and before he was thirteen he had acquired, under the care of his uncle (a linguist), almost as many languages as he had years of age. These included the classical and modern European languages, and Persian, Arabic, Hindustani, Sanskrit, and even Marathi and Malay. He retained much of his knowledge of languages to the end of his life, often reading Persian and Arabic in his spare time, although he had long since stopped studying languages, and used them just for relaxation.

In September 1813, the American calculating prodigy Zerah Colburn was being exhibited in Dublin. Colburn was 9, a year older than Hamilton. The two were pitted against each other in a mental arithmetic contest with Colburn emerging the clear victor. In reaction to his defeat, Hamilton dedicated less time to studying languages and more time to studying mathematics.

Hamilton was part of a small but well-regarded school of mathematicians associated with Trinity College in Dublin, which he entered at age 18. The college awarded him two Optimes, or off-the-chart grades. He studied both classics and mathematics, and was appointed Professor of Astronomy just prior to his graduation (BA, 1827, he was awarded MA in 1837). He then took up residence at Dunsink Observatory where he spent the rest of his life.

Personal life

While attending Trinity College, Hamilton proposed to his friend's sister, who rejected him. Hamilton, being a sensitive young man, became sick and depressed, and almost committed suicide. He was rejected again in 1831 by Aubrey De Vere (1814-1902). Luckily, Hamilton found a woman who would accept his proposal. She was Helen Marie Bayly, a country preacher's daughter, and they married in 1833. Hamilton had three children with Bayly: William Edwin Hamilton (born 1834), Archibald Henry (born 1835), and Helen Elizabeth (born 1840). Hamilton's married life turned out to be difficult and unhappy as Bayly proved to be pious, shy, timid, and chronically ill.

Optics and mechanics

Hamilton made important contributions to optics and to classical mechanics. His first discovery was in an early paper that he communicated in 1823 to Dr. Brinkley, who presented it under the title of "Caustics" in 1824 to the Royal Irish Academy. It was referred as usual to a committee. While their report acknowledged its novelty and value, they recommended further development and simplification before publication. Between 1825 and 1828 the paper grew to an immense size, mostly by the additional details that the committee had suggested. But it also became more intelligible, and the features of the new method were now easily seen. Until this period Hamilton himself seems not to have fully understood either the nature or importance of optics, as later he intended to apply his method to dynamics.

In 1827, Hamilton presented a theory of a single function, now known as Hamilton's principal function, that brings together mechanics, optics, and mathematics, and which helped to establish the wave theory of light. He proposed it when he first predicted its existence in the third supplement to his "Systems of Rays", read in 1832. The Royal Irish Academy paper was finally entitled "Theory of Systems of Rays" (23 April 1827), and the first part was printed in 1828 in the Transactions of the Royal Irish Academy. The more important contents of the second and third parts appeared in the three voluminous supplements (to the first part) which were published in the same Transactions, and in the two papers "On a General Method in Dynamics", which appeared in the Philosophical Transactions in 1834 and 1835. In these papers, Hamilton developed his great principle of "Varying Action". The most remarkable result of this work is the prediction that a single ray of light entering a biaxial crystal at a certain angle would emerge as a hollow cone of rays. This discovery is still known by its original name, "conical refraction".

The step from optics to dynamics in the application of the method of "Varying Action" was made in 1827, and communicated to the Royal Society, in whose Philosophical Transactions for 1834 and 1835 there are two papers on the subject, which, like the "Systems of Rays", display a mastery over symbols and a flow of mathematical language almost unequaled. The common thread running through all this work is Hamilton's principle of "Varying Action". Although it is based on the calculus of variations and may be said to belong to the general class of problems included under the principle of least action which had been studied earlier by Pierre Louis Maupertuis, Euler, Joseph Louis Lagrange, and others, Hamilton's analysis revealed much deeper mathematical structure than had been previously understood, in particular the symmetry between momentum and position. Paradoxically, the credit for discovering the quantity now called the Lagrangian and Lagrange's equations belongs to Hamilton. Hamilton's advances enlarged greatly the class of mechanical problems that could be solved, and they represent perhaps the greatest addition which dynamics had received since the work of Isaac Newton and Lagrange. Many scientists, including Liouville, Jacobi, Darboux, Poincaré, Kolmogorov, and Arnold, have extended Hamilton's work, thereby expanding our knowledge of mechanics and differential equations.

While Hamilton's reformulation of classical mechanics is based on the same physical principles as the mechanics of Newton and Lagrange, it provides a powerful new technique for working with the equations of motion. More importantly, both the Lagrangian and Hamiltonian approaches which were initially developed to describe the motion of discrete systems, have proven critical to the study of continuous classical systems in physics, and even quantum mechanical systems. In this way, the techniques find use in electromagnetism, quantum mechanics, quantum relativity theory, and quantum field theory.

Mathematical studies

Hamilton's mathematical studies seem to have been undertaken and carried to their full development without any assistance whatsoever, and the result is that his writings do not belong to any particular "school". Not only was Hamilton an expert as an arithmetic calculator, but he seems to have occasionally had fun in working out the result of some calculation to an enormous number of decimal places. At the age of eight Hamilton engaged Zerah Colburn, the American "calculating boy", who was then being exhibited as a curiosity in Dublin. Two years later, aged ten, Hamilton stumbled across a Latin copy of Euclid, which he eagerly devoured; and at twelve he studied Newton's Arithmetica Universalis. This was his introduction to modern analysis. Hamilton soon began to read the Principia, and at sixteen Hamilton had mastered a great part of it, as well as some more modern works on analytical geometry and the differential calculus.

Around this time Hamilton was also preparing to enter Trinity College, Dublin, and therefore had to devote some time to classics. In mid-1822 he began a systematic study of Laplace's Mécanique Céleste.

From that time Hamilton appears to have devoted himself almost wholly to mathematics, though he always kept himself well acquainted with the progress of science both in Britain and abroad. Hamilton found an important defect in one of Laplace's demonstrations, and he was induced by a friend to write out his remarks, so that they could be shown to Dr. John Brinkley, then the first Royal Astronomer of Ireland, and an accomplished mathematician. Brinkley seems to have immediately perceived Hamilton's talents, and to have encouraged him in the kindest way.

Hamilton's career at College was perhaps unexampled. Amongst a number of extraordinary competitors, he was first in every subject and at every examination. He achieved the rare distinction of obtaining an optime both for Greek and for physics. Hamilton might have attained many more such honours (he was expected to win both the gold medals at the degree examination), if his career as a student had not been cut short by an unprecedented event. This was Hamilton's appointment to the Andrews Professorship of Astronomy in the University of Dublin, vacated by Dr. Brinkley in 1827. The chair was not exactly offered to him, as has been sometimes asserted, but the electors, having met and talked over the subject, authorised Hamilton's personal friend (also an elector) to urge Hamilton to become a candidate, a step which Hamilton's modesty had prevented him from taking. Thus, when barely 22, Hamilton was established at the Dunsink Observatory, near Dublin.

Hamilton was not especially suited for the post, because although he had a profound acquaintance with theoretical astronomy, he had paid little attention to the regular work of the practical astronomer. Hamilton's time was better employed in original investigations than it would have been spent in observations made even with the best of instruments. Hamilton was intended by the university authorities who elected him to the professorship of astronomy to spend his time as he best could for the advancement of science, without being tied down to any particular branch. If Hamilton had devoted himself to practical astronomy, the University of Dublin would assuredly have furnished him with instruments and an adequate staff of assistants.

He was twice awarded the Cunningham Medal of the Royal Irish Academy. The first award, in 1834, was for his work on conical refraction, for which he also received the Royal Medal of the Royal Society the following year. He was to win it again in 1848.

In 1835, being secretary to the meeting of the British Association which was held that year in Dublin, he was knighted by the lord-lieutenant. Other honours rapidly succeeded, among which his election in 1837 to the president's chair in the Royal Irish Academy, and the rare distinction of being made a corresponding member of the Saint Petersburg Academy of Sciences. Later, in 1864, the newly established United States National Academy of Sciences elected its first Foreign Associates, and decided to put Hamilton's name on top of their list.

Quaternions

Quaternion Plaque on Broom Bridge

The other great contribution Hamilton made to mathematical science was his discovery of quaternions in 1843. However, in 1840, Benjamin Olinde Rodrigues had already reached a result that amounted to their discovery in all but name.

Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a 2-dimensional plane) to higher spatial dimensions. He failed to find a useful 3-dimensional system (in modern terminology, he failed to find a real, three-dimensional skew-field), but in working with four dimensions he created quaternions. According to Hamilton, on 16 October he was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation

i² = j² = k² = ijk = −1

suddenly occurred to him; Hamilton then promptly carved this equation using his penknife into the side of the nearby Broom Bridge (which Hamilton called Brougham Bridge). This event marks the discovery of the quaternion group.

A plaque under the bridge was unveiled by the Taoiseach Éamon de Valera, himself a mathematician and student of quaternions, on 13 November 1958. Since 1989, the National University of Ireland, Maynooth has organised a pilgrimage called the Hamilton Walk, in which mathematicians take a walk from Dunsink Observatory to the bridge, where no trace of the carving remains, though a stone plaque does commemorate the discovery.

The quaternion involved abandoning commutativity, a radical step for the time. Not only this, but Hamilton had in a sense invented the cross and dot products of vector algebra. Hamilton also described a quaternion as an ordered four-element multiple of real numbers, and described the first element as the 'scalar' part, and the remaining three as the 'vector' part.

Hamilton introduced, as a method of analysis, both quaternions and biquaternions, the extension to eight dimensions by introduction of complex number coefficients. When his work was assembled in 1853, the book Lectures on Quaternions had "formed the subject of successive courses of lectures, delivered in 1848 and subsequent years, in the Halls of Trinity College, Dublin". Hamilton confidently declared that quaternions would be found to have a powerful influence as an instrument of research. When he died, Hamilton was working on a definitive statement of quaternion science. His son William Edwin Hamilton brought the Elements of Quaternions, a hefty volume of 762 pages, to publication in 1866. As copies ran short, a second edition was prepared by Charles Jasper Joly, when the book was split into two volumes, the first appearing 1899 and the second in 1901. The subject index and footnotes in this second edition improved the Elements accessibility.

One of the features of Hamilton's quaternion system was the differential operator del which could be used to express the gradient of a vector field or to express the curl. These operations were applied by Clerk Maxwell to the electrical and magnetic studies of Michael Faraday in Maxwell's Treatise on Electricity and Magnetism (1873). Though the del operator continues to be used, the real quaternions fall short as a representation of spacetime. On the other hand, the biquaternion algebra, in the hands of Arthur W. Conway and Ludwik Silberstein, provided representational tools for Minkowski space and the Lorentz group early in the twentieth century.

Today, the quaternions are used in computer graphics, control theory, signal processing, and orbital mechanics, mainly for representing rotations/orientations. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions, which are also used to telemeter their current attitude. The rationale is that combining quaternion transformations is more numerically stable than combining many matrix transformations. In control and modelling applications, quaternions do not have a computational singularity (undefined division by zero) that can occur for quarter-turn rotations (90 degrees) that are achievable by many Air, Sea and Space vehicles. In pure mathematics, quaternions show up significantly as one of the four finite-dimensional normed division algebras over the real numbers, with applications throughout algebra and geometry.

It is believed by some modern mathematicians that Hamilton's work on quaternions was satirized by Charles Lutwidge Dodgson in Alice in Wonderland. In particular, the Mad Hatter's tea party was meant to represent the folly of quaternions and the need to revert to Euclidean geometry.

Other originality

Hamilton originally matured his ideas before putting pen to paper. The discoveries, papers, and treatises previously mentioned might well have formed the whole work of a long and laborious life. But not to speak of his enormous collection of books, full to overflowing with new and original matter, which have been handed over to Trinity College, Dublin, the previous mentioned works barely form the greater portion of what Hamilton has published. Hamilton developed the variational principle, which was reformulated later by Carl Gustav Jacob Jacobi. He also introduced the icosian game or Hamilton's puzzle which can be solved using the concept of a Hamiltonian path.

Hamilton's extraordinary investigations connected with the solution of algebraic equations of the fifth degree, and his examination of the results arrived at by N. H. Abel, G. B. Jerrard, and others in their researches on this subject, form another contribution to science. There is next Hamilton's paper on fluctuating functions, a subject which, since the time of Joseph Fourier, has been of immense and ever increasing value in physical applications of mathematics. There is also the extremely ingenious invention of the hodograph. Of his extensive investigations into the solutions (especially by numerical approximation) of certain classes of physical differential equations, only a few items have been published, at intervals, in the Philosophical Magazine.

Besides all this, Hamilton was a voluminous correspondent. Often a single letter of Hamilton's occupied from fifty to a hundred or more closely written pages, all devoted to the minute consideration of every feature of some particular problem; for it was one of the peculiar characteristics of Hamilton's mind never to be satisfied with a general understanding of a question; Hamilton pursued the problem until he knew it in all its details. Hamilton was ever courteous and kind in answering applications for assistance in the study of his works, even when his compliance must have cost him much time. He was excessively precise and hard to please with reference to the final polish of his own works for publication; and it was probably for this reason that he published so little compared with the extent of his investigations.

Death and afterwards

Irish commemorative coin celebrating the 200th Anniversary of his birth.

Hamilton retained his faculties unimpaired to the very last, and steadily continued the task of finishing the Elements of Quaternions which had occupied the last six years of his life. He died on 2 September 1865, following a severe attack of gout precipitated by excessive drinking and overeating. He is buried in Mount Jerome Cemetery in Dublin. He had married Helen Bayly and had several children.

Hamilton is recognised as one of Ireland's leading scientists and, as Ireland becomes more aware of its scientific heritage, he is increasingly celebrated. The Hamilton Institute is an applied mathematics research institute at NUI Maynooth and the Royal Irish Academy holds an annual public Hamilton lecture at which Murray Gell-Mann, Frank Wilczek, Andrew Wiles, and Timothy Gowers have all spoken. The year 2005 was the 200th anniversary of Hamilton's birth and the Irish government designated that the Hamilton Year, celebrating Irish science. Trinity College Dublin marked the year by launching the Hamilton Mathematics Institute.

Two commemorative stamps were issued by Ireland in 1943 to mark the centenary of the announcement of quaternions. A 10 Euros commemorative silver Proof coin was issued by the Central Bank of Ireland in 2005 to commemorate 200 years since his birth.

The newest maintenance depot for the Dublin LUAS tram system has been named after him. It is located adjacent to the Broombridge stop on the Green Line.

Commemorations of Hamilton

• Hamilton's equations are a formulation of classical mechanics.
• Numerous other concepts and objects in mechanics, such as Hamilton's principle, Hamilton's principal function, the Hamilton–Jacobi equation, Cayley-Hamilton theorem are named after Hamilton.
• The Hamiltonian is the name of both a function (classical) and an operator (quantum) in physics, and, in a different sense, a term from graph theory.
• The RCSI Hamilton Society was founded in his name in 2004.
• The algebra of quaternions is usually denoted by H, or in blackboard bold by ${\displaystyle \mathbb {H} }$, in honour of Hamilton.

Quotations

• "Time is said to have only one dimension, and space to have three dimensions. ... The mathematical quaternion partakes of both these elements; in technical language it may be said to be 'time plus space', or 'space plus time': and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be."—William Rowan Hamilton (quoted in Robert Percival Graves' "Life of Sir William Rowan Hamilton" (3 volumes, 1882, 1885, 1889))
• "He used to carry on, long trains of algebraic and arithmetical calculations in his mind, during which he was unconscious of the earthly necessity of eating; we used to bring in a 'snack' and leave it in his study, but a brief nod of recognition of the intrusion of the chop or cutlet was often the only result, and his thoughts went on soaring upwards." – William Edwin Hamilton (his elder son)

Publications

• Hamilton, William Rowan (Royal Astronomer of Ireland), "Introductory Lecture on Astronomy". Dublin University Review and Quarterly Magazine Vol. I, Trinity College, January 1833.
• Hamilton, William Rowan, "Lectures on Quaternions". Royal Irish Academy, 1853.
• Hamilton (1866) Elements of Quaternions University of Dublin Press. Edited by William Edwin Hamilton, son of the deceased author.
• Hamilton (1899) Elements of Quaternions volume I, (1901) volume II. Edited by Charles Jasper Joly; published by Longmans, Green & Co..
• David R. Wilkins's collection of Hamilton's Mathematical Papers.