Hamiltonian mechanics

Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena.

Sir William Rowan Hamilton

Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics.

OverviewEdit

Phase space coordinates (p,q) and Hamiltonian HEdit

Let   be a mechanical system with the configuration space   and the smooth Lagrangian   Select a standard coordinate system   on   The quantities   are called momenta. (Also generalized momenta, conjugate momenta, and canonical momenta). For a time instant   the Legendre transformation is defined as the map   which we will assume to have a smooth inverse   For a system with   degrees of freedom, the Lagrangian mechanics defines the energy function

 

The inverse of the Legendre transform turns   into a function   known as Hamiltonian. Formally,

 

which implies that

 

where the velocities   are found from the ( -dimensional) equation   which, by assumption, is uniquely solvable for   The ( -dimensional) pair   is called phase space coordinates. (Also canonical coordinates).

Terminology remark. Some sources define Legendre transformation as a time-dependent functional

 

where, as before, the function   satisfies   Under the latter definition, the Hamiltonian   is the Legendre transformation of the Lagrangian  

From Euler-Lagrange equation to Hamilton's equationsEdit

In phase space coordinates   the ( -dimensional) Euler-Lagrange equation

 

becomes Hamilton's equations in   dimensions

 

From stationary action principle to Hamilton's equationsEdit

Let   be the set of smooth paths   for which   and   The action functional   is defined via

 

where   and   (see above). A path   is a stationary point of   (and hence is an equation of motion) if and only if the path   in phase space coordinates obeys the Hamilton's equations.

Basic physical interpretationEdit

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

 

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

In this example, the time derivative of 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.

ExampleEdit

A spherical pendulum consists of a mass m moving without friction on the surface of a sphere. The only forces acting on the mass are the reaction from the sphere and gravity. Spherical coordinates are used to describe the position of the mass in terms of (r, θ, φ), where r is fixed, r=l.

 
Spherical pendulum: angles and velocities.

The Lagrangian for this system is[1]

 

Thus the Hamiltonian is

 

where

 

and

 

In terms of coordinates and momenta, the Hamiltonian reads

 

Hamilton's equations give the time evolution of coordinates and conjugate momenta in four first-order differential equations,

 
 
 
 .

Momentum  , which corresponds to the vertical component of angular momentum  , is a constant of motion. That is a consequence of the rotational symmetry of the system around the vertical axis. Being absent from the Hamiltonian, azimuth   is a cyclic coordinate, which implies conservation of its conjugate momentum.

Deriving Hamilton's equationsEdit

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:[2]

 

The generalized momenta were defined as

 

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

 

This can be rewritten as

 

which after rearranging leads to

 

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

 

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:

 

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

 

Since this calculation was done off-shell (i.e. without taking into account the equations of motion) one can associate corresponding terms from both sides of this equation to yield:

 

On-shell, Lagrange's equations indicate that

 

A rearrangement of this yields

 

Thus Hamilton's equations are

 

Hamilton's equations consist of 2n first-order differential equations, while Lagrange's equations consist of n second-order equations. Hamilton's equations usually do not reduce the difficulty of finding explicit solutions, but they still offer some advantages: 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. This effectively 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.[3] The Lagrangian and Hamiltonian approaches provide the groundwork for deeper results in the theory of classical mechanics, and for formulations of quantum mechanics.

Properties of the Hamiltonian HEdit

  • The value of the Hamiltonian   is the total energy of the system if and only if the energy function   has the same property. (See definition of  
  •   on the solutions of Hamilton's equations.
Indeed,   and everything but the final term cancels out.
  •   does not change under point transformations, i.e. smooth changes   of space coordinates. (Follows from the invariance of the energy function   under point transformations. The invariance of   can be established directly).
  •   (See Deriving Hamilton's equations).
  •   (Compare Hamilton's and Euler-Lagrange equations or see Deriving Hamilton's equations).
  •   if and only if  
The coordinate for which this is true is called cyclic (or ignorable). Every cyclic coordinate   reduces the number of degrees of freedom by   causes the corresponding momentum   to be conserved, and makes the Hamilton's equations easier to solve.

Hamiltonian of a charged particle in an electromagnetic fieldEdit

A sufficient illustration of Hamiltonian mechanics is given by the Hamiltonian of a charged particle in an electromagnetic field. In Cartesian coordinates the Lagrangian of a non-relativistic classical particle in an electromagnetic field is (in SI Units):

 

where q is the electric charge of the particle, φ is the electric scalar potential, and the Ai are the components of the magnetic vector potential that may all explicitly depend on   and  .

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

 

and is called minimal coupling.

Note that the values of scalar potential and vector potential would change during a gauge transformation,[4] and the Lagrangian itself will pick up extra terms as well; But the extra terms in Lagrangian add up to a total time derivative of a scalar function, and therefore won't change the Euler–Lagrange equation.

The canonical momenta are given by:

 

Note that canonical momenta are not gauge invariant, and are not physically measurable. However, the kinetic momentum:

 

is gauge invariant and physically measurable.

The Hamiltonian, as the Legendre transformation of the Lagrangian, is therefore:

 

This equation is used frequently in quantum mechanics.

Under gauge transformation:

 

where f(r,t) is any scalar function of space and time, the aforementioned Lagrangian, canonical momenta, and Hamiltonian transform like:

 

which still produces the same Hamilton's equation:

 

In quantum mechanics, the wave function will also undergo a local U(1) group transformation[5] during the Gauge Transformation, which implies that all physical results must be invariant under local U(1) transformations.

Relativistic charged particle in an electromagnetic fieldEdit

The relativistic Lagrangian for a particle (rest mass   and charge  ) is given by:

 

Thus the particle's canonical momentum is

 

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

Solving for the velocity, we get

 

So the Hamiltonian is

 

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

 

from which one can derive

 

The above derivation makes use of the vector calculus identity:

 

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

 

This has the advantage that kinetic momentum   can be measured experimentally whereas canonical momentum   cannot. Notice that the Hamiltonian (total energy) can be viewed as the sum of the relativistic energy (kinetic+rest),  , plus the potential energy,  .

From symplectic geometry to Hamilton's equationsEdit

Geometry of Hamiltonian systemsEdit

The Hamiltonian can induce a symplectic structure on a smooth even-dimensional manifold M2n in several different, but equivalent, ways the best known among which are the following:[6]

As a closed nondegenerate symplectic 2-form ω. According to the Darboux's theorem, in a small neighbourhood around any point on M in suitable local coordinates   exists the symplectic form

 

The local coordinates p, q are then called canonical or symplectic.

The form   allows to construct a natural isomorphism   of the tangent space   and the cotangent space   This is done by mapping a vector   to the 1-form   where   for an arbitrary   Due to the bilinearity and non-degeneracy of   and the fact that   the mapping   is indeed a linear isomorphism. This isomorphism is natural in that it does not change with change of coordinates on   Repeating for every   we end up with an isomorphism   between the infinite-dimensional space of smooth vector fields and that of smooth 1-forms. For every   and  

 

(In algebraic terms, one would say that the  -modules   and   are isomorphic). If   then, for every fixed     and     is known as a Hamiltonian vector field. The respective differential equation on  

 

is called Hamilton's equation. Here   and   is the (time-dependent) value of the vector field   at  

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.

Any smooth real-valued function H on a symplectic manifold can be used to define a Hamiltonian system. The function H is known as "the Hamiltonian" or "the energy function." The symplectic manifold is then called the phase space. The Hamiltonian induces a special vector field on the symplectic manifold, known as the Hamiltonian vector field.

The Hamiltonian vector field induces a Hamiltonian flow on the manifold. This is a one-parameter family of transformations of the manifold (the parameter of the curves is commonly called "the time"); in other words, an isotopy of symplectomorphisms, starting with the identity. By Liouville's theorem, each symplectomorphism preserves the volume form on the phase space. The collection of symplectomorphisms induced by the Hamiltonian flow is commonly called "the Hamiltonian mechanics" of the Hamiltonian system.

The symplectic structure induces a Poisson bracket. The Poisson bracket gives the space of functions on the manifold the structure of a Lie algebra.

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

  1. bilinearity
  2. antisymmetry
  3.   (Leibniz rule)
  4.   (Jacobi identity)
  5. non-degeneracy: if the point x on M is not critical for F then a smooth function G exists such that  .

Given a function f

 

if there is a probability distribution, ρ, then (since the phase space velocity   has zero divergence and probability is conserved) its convective derivative can be shown to be zero and so

 

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

 

for some function F.[7] There is an entire field focusing on small deviations from integrable systems governed by the KAM theorem.

The integrability of Hamiltonian vector fields is an open question. In general, Hamiltonian systems are chaotic; concepts of measure, completeness, integrability and stability are poorly defined.

Riemannian manifoldsEdit

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

 

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 manifoldsEdit

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

 

pz is not involved in the Hamiltonian.

Poisson algebrasEdit

Hamiltonian systems can be generalized in various ways. Instead of simply looking at the algebra of smooth functions over a symplectic manifold, Hamiltonian mechanics can be formulated on general commutative unital real Poisson algebras. A state is a continuous linear functional on the Poisson algebra (equipped with some suitable topology) such that for any element A of the algebra, A2 maps to a nonnegative real number.

A further generalization is given by Nambu dynamics.

Generalization to quantum mechanics through Poisson bracketEdit

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

 

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.

See alsoEdit

ReferencesEdit

  1. ^ Landau & Lifshitz 1976, pp. 33–34
  2. ^ This derivation is along the lines as given in Arnol'd 1989, pp. 65–66
  3. ^ Goldstein, Poole & Safko 2002, pp. 347–349
  4. ^ Srednicki, Mark (January 2007). Quantum Field Theory. Cambridge Core. doi:10.1017/cbo9780511813917. ISBN 9780511813917. Retrieved 2020-05-08.
  5. ^ Zinn-Justin, Jean; Guida, Riccardo (2008-12-04). "Gauge invariance". Scholarpedia. 3 (12): 8287. Bibcode:2008SchpJ...3.8287Z. doi:10.4249/scholarpedia.8287. ISSN 1941-6016.
  6. ^ Arnol'd, Kozlov & Neĩshtadt 1988, §3. Hamiltonian mechanics.
  7. ^ Arnol'd, Kozlov & Neĩshtadt 1988

Further readingEdit

External linksEdit