Hamiltonian mechanics

Hamiltonian mechanics is a re-formulation of classical mechanics that was discovered in 1833 by Irish mathematician William Rowan Hamilton. It arose from Lagrangian mechanics, another re-formulation of classical mechanics, introduced by Joseph Louis Lagrange in 1788. It can however be formulated without recourse to Lagrangian mechanics, using symplectic spaces. See the section on its mathematical formulation for this. The Hamiltonian method differs from the Lagrange one in that instead of expressing second-order differential constraints on an n-dimensional coordinate space, it expresses first-order constraints on a 2n-dimensional phase space[1].

As with Lagrangian mechanics, Hamilton's equations provide a new and equivalent way of looking at classical mechanics. Generally, these equations do not provide a more convenient way of solving a particular problem. Rather, they provide deeper insights into both the general structure of classical mechanics and its connection to quantum mechanics as understood through Hamiltonian mechanics, as well as its connection to other areas of science.

Simplified overview of uses

For a closed system the sum of the kinetic and potential energy in the system is represented by a set of differential equations known as the Hamilton equations for that system. Hamiltonians can be used to describe such simple systems as a bouncing ball, a pendulum or an oscillating spring in which energy changes from kinetic to potential and back again over time. Hamiltonians can also be employed to model the energy of other more complex dynamic systems such as planetary orbits and in quantum mechanics. ^[1]

Failed to parse (Missing texvc executable; please see math/README to configure.): \dot p = -\frac{\partial \mathcal{H}}{\partial q}

Failed to parse (Missing texvc executable; please see math/README to configure.): \dot q =~~\frac{\partial \mathcal{H}}{\partial p}

In the above equations, the dot denotes the ordinary derivative with respect to time of the functions p = p(t) (called generalized momenta) and q = q(t) (called generalized coordinates), taking values in some vector space, and Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H}

= Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H}(p,q,t)

is the so-called Hamiltonian, or (scalar valued) Hamiltonian function. Thus, a little bit more explicitly, one should write

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm d}{\mathrm dt}p(t) = -\frac{\partial}{\partial q}\mathcal{H}(p(t),q(t),t)

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm d}{\mathrm dt}q(t) =~~\frac{\partial}{\partial p}\mathcal{H}(p(t),q(t),t)

and specify the domain of values in which the parameter t ("time") varies.

For a quite detailed derivation of these equations from Lagrangian mechanics, see below.

Basic physical interpretation, mnemotechnics

The simplest interpretation of the equations is as follows: The Hamiltonian Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H} represents the energy of the physical system, which is the sum of kinetic and potential energy, traditionally denoted T & V respectively:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H} = T + V , \quad T = \frac{p^2}{2m} , \quad V = V(q) = V(x).

Using Hamilton's equations

1. First write out L = T – V. Express T and V as though you were going to use Lagrange's equation.
2. Calculate the momenta by differentiating the Lagrangian with respect to velocity.
3. Express the velocities in terms of the momenta by inverting the expressions in step (2).
4. Calculate the Hamiltonian using the usual definition, Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H} = \sum_i p_i {\dot q_i} - \mathcal{L}

. Substitute for the velocities using the results in step (3).

1. Apply Hamilton's equations.

Notes

Hamilton's equations are appealing in view of their beautiful simplicity and (slightly broken) symmetry. They have been analyzed under almost every imaginable angle of view, from basic physics up to symplectic geometry. A lot is known about solutions of these equations, yet the exact general case solution of the equations of motion cannot be given explicitly for a system of more than two massive point particles. The finding of conserved quantities plays an important role in the search for solutions or information about their nature. In models with an infinite number of degrees of freedom, this is of course even more complicated. An interesting and promising area of research is the study of integrable systems, where an infinite number of independent conserved quantities can be constructed.

Deriving Hamilton's equations

We can derive Hamilton's equations by looking at how the Lagrangian changes as you change the time and the positions and velocities of particles.

Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Now the generalized momenta were defined as Failed to parse (Missing texvc executable; please see math/README to configure.): p_i = \frac{\partial \mathcal{L}}{\partial {\dot q_i}}

and Lagrange's equations tell us that

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial \mathcal{L}}{\partial {\dot q_i}} - \frac{\partial \mathcal{L}}{\partial q_i} = F_i

where Failed to parse (Missing texvc executable; please see math/README to configure.): F_i

is the generalized force.  We can rearrange this to get

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial \mathcal{L}}{\partial q_i} = {\dot p}_i - F_i

and substitute the result into the variation of the Lagrangian

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d}\mathcal{L} = \sum_i \left[ \left( {\dot p}_i - F_i \right) \mathrm{d} q_i + p_i \mathrm{d} {\dot q_i} \right] + \frac{\partial \mathcal{L}}{\partial t}\mathrm{d}t

We can rewrite this as

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d} \mathcal{L} = \sum_i \left [ \left ( {\dot p}_i - F_i \right ) \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

and rearrange again to get

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d} \left ( \sum_i p_i {\dot q_i} - \mathcal{L} \right ) = \sum_i \left [ \left ( F_i-{\dot p}_i \right ) \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 we have defined before, so we find that;

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d} \mathcal{H} = \sum_i \left [ \left ( F_i-{\dot p}_i \right ) \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

where the second equality holds because of the definition of the partial derivatives. Associating terms from both sides of the equation above yields Hamilton's equations:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial \mathcal{H}}{\partial q_j} = F_j - \dot{p}_j, \qquad \frac{\partial \mathcal{H}}{\partial p_j} = \dot{q}_j, \qquad \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

Failed to parse (Missing texvc executable; please see math/README to configure.): \left\{\, q_j | j=1, \ldots,N \,\right\}

and matching generalized velocities

Failed to parse (Missing texvc executable; please see math/README to configure.): \left\{\, \dot{q}_j | j=1, \ldots ,N \,\right\}

We write the Lagrangian as

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{L}(q_j, \dot{q}_j, t)

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): p_j = {\partial \mathcal{L} \over \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 coordinatizations of the same symplectic manifold.

The Hamiltonian is the Legendre transform of the Lagrangian:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H}\left(q_j,p_j,t\right) = \sum_i \dot{q}_i p_i - \mathcal{L}(q_j,\dot{q}_j,t)

If the transformation equations defining the generalized coordinates are independent of t, and the Lagrangian is a sum of products of functions (in the generalised 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H} produces a differential:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{matrix} \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{matrix}

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial \mathcal{H}}{\partial q_j} = - \dot{p}_j, \qquad \frac{\partial \mathcal{H}}{\partial p_j} = \dot{q}_j, \qquad \frac{\partial \mathcal{H}}{\partial t } = - {\partial \mathcal{L} \over \partial t}

Hamilton's equations are first-order differential equations, and thus easier to solve than Lagrange's equations, which are second-order. However, the steps leading to the equations of motion are more onerous than in Lagrangian mechanics - beginning with the generalized coordinates and the Lagrangian, we must calculate the Hamiltonian, express each generalized velocity in terms of the conjugate momenta, and replace the generalized velocities in the Hamiltonian with the conjugate momenta. All in all, there is little labor saved from solving a problem with Hamiltonian mechanics rather than Lagrangian mechanics. Ultimately, it will produce the same solution as Lagrangian mechanics and Newton's laws of motion.

The principal appeal of the Hamiltonian approach is that it provides the groundwork for deeper results in the theory of classical 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

The Hamilton's equations above work perfectly for classical mechanics, but not for the quantum mechanics, since the differential equations assume that we can find out the position and momentum of the particle simultaneously at any point in time. The equations can be further generalized to apply to quantum mechanics as well as to classical mechanics through the use of the Poisson algebra over p and q. In this case, the more general form of the Hamilton's equation reads

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d}f}{\mathrm{d}t} = \{f, \mathcal{H}\} + \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 Poisson bracket without resorting to differential equations, see Lie algebra, as Poisson bracket is just a different name for the Lie bracket in a Poisson algebra.

In fact, this more algebraic approach not only allows us to use probability distributions and wavefunctions for q and p, but also provides more power in the classical setting, in particular by helping to find the conserved quantities.

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 symplectic vector field.

The symplectic vector field, also called the Hamiltonian vector field, induces a Hamiltonian flow on the manifold. The integral curves of the vector field are a one-parameter family of transformations of the manifold; the parameter of the curves is commonly called the time. The time evolution is given by symplectomorphisms. 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 Hamiltonian vector field also induces a special operation, the Poisson bracket. The Poisson bracket acts on functions on the symplectic manifold, thus giving the space of functions on the manifold the structure of a Lie algebra.

In particular, given a function f

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d}}{\mathrm{d}t} f=\frac{\partial }{\partial t} f + \{\,f,\mathcal{H}\,\}.

If we have a probability distribution, ρ, then (since the phase space velocity (Failed to parse (Missing texvc executable; please see math/README to configure.): {\dot p_i} , {\dot q _i} ) has zero divergence, and probability is conserved) its convective derivative can be shown to be zero and so

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial}{\partial t} \rho = - \{\,\rho ,\mathcal{H}\,\}.

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–Arnol'd theorem says that locally, any Liouville integrable Hamiltonian can be transformed via a symplectomorphism in 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{G}_i = 0, \qquad \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. At this time, the study of dynamical systems is primarily qualitative, and not a quantitative science.

Riemannian manifolds

An important special case consists of those Hamiltonians that are quadratic forms, that is, Hamiltonians that can be written as

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H}(q,p)= \frac{1}{2} \langle p,p\rangle_q

where Failed to parse (Missing texvc executable; please see math/README to configure.): \langle\cdot,\cdot\rangle_q

is a cometric on the fiber Failed to parse (Missing texvc executable; please see math/README to configure.): T_q^*Q

, the cotangent space to the point q in the configuration space. This Hamiltonian consists entirely of the kinetic term.

If one considers a Riemannian manifold or a pseudo-Riemannian manifold, so that one has an invertible, non-degenerate metric, then the cometric is given simply as the inverse of 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H}(x,y,z,p_x,p_y,p_z)=\frac{1}{2}\left( p_x^2 + p_y^2 \right)

.

Failed to parse (Missing texvc executable; please see math/README to configure.): 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. Failed to parse (Missing texvc executable; please see math/README to configure.): q_i = x_i ), the Lagrangian of a non-relativistic classical particle in an electromagnetic field is (in SI Units):

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{L} = \sum_i \tfrac{1}{2} m \dot{x}_i^2 + \sum_i e \dot{x}_i A_i - e \phi,

where e is the electric charge of the particle (not necessarily the electron charge), Failed to parse (Missing texvc executable; please see math/README to configure.): \phi

is the  electric scalar potential, and the Failed to parse (Missing texvc executable; please see math/README to configure.): A_i
are the components of the magnetic vector potential (these may be modified through a  gauge tranformations).

The generalized momenta may be derived by:

Failed to parse (Missing texvc executable; please see math/README to configure.): p_j = \frac{\partial L}{ \partial \dot{x}_j} = m \dot{x}_j + e A_j.

Rearranging, we may express the velocities in terms of the momenta, as:

Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{x}_j = \frac{ p_j - e A_j }{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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H} = \sum_i \dot{x}_i p_i - \mathcal{L} = \sum_i \frac{ (p_i - e A_i)^2 } {2 m } + e \phi.

This equation is used frequently in quantum mechanics.

References

• V.I. Arnol'd, Mathematical Methods of Classical Mechanics, Springer-Verlag (1989), [ISBN 0-387-96890-3]
• Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X
• V.I. Arnol'd, V.V. Kozlov and A.I. Neĩshtadt, "Mathematical aspects of classical and celestial mechanics." In: Encyclopaedia of Mathematical Sciences, Dynamical Systems III (vol. 3), Springer-Verlag, 1988.
• A. M. Vinogradov , B. A. Kupershmidt "The structure of Hamiltonian mechanics" (djvu), London Math. Soc. Lect. Notes Ser., 60 (1981), Cambridge Univ. Press, London
• Binney, James, "Classical Mechanics" (PostScript) lecture notes (PDF)
• Tong, David, Classical Dynamics (Cambridge lecture notes)

See also

• Hamiltonian (quantum mechanics)
• Lagrangian mechanics
• Canonical transformation
• Classical mechanics
• Dynamical systems
• Quantum mechanics
• Maxwell's equations
• Field theory
• Hamilton–Jacobi equationar:ميكانيك هاملتوني

ca:Formulació hamiltoniana cs:Hamiltonovská formulace mechaniky de:Hamilton-Formalismus es:Mecánica hamiltoniana fa:مکانیک همیلتونی fr:Mécanique hamiltonienne gl:Mecánica hamiltoniana ko:해밀턴 역학 id:Mekanika Hamiltonian it:Meccanica hamiltoniana nl:Hamiltonformalisme ja:ハミルトン力学 no:Hamiltonmekanikk pt:Mecânica hamiltoniana ru:Гамильтонова механика fi:Hamiltonin mekaniikka uk:Механіка Гамільтона