Canonical transformation


Canonical transformation

In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (mathbf{q}, mathbf{p}, t) ightarrow (mathbf{Q}, mathbf{P}, t) that preserves the form of Hamilton's equations, although it might not preserve the Hamiltonian itself. This is sometimes known as form invariance. Canonical transformations are useful in their own right, and also form the basis for the Hamilton–Jacobi equations (a useful method for calculating conserved quantities) and Liouville's theorem (itself the basis for classical statistical mechanics).

Since Lagrangian mechanics is based on generalized coordinates, transformations of the coordinates mathbf{q} ightarrow mathbf{Q} do not affect the form of Lagrange's equations and, hence, do not affect the form of Hamilton's equations if we simultaneously change the momentum by a Legendre transform into:P_i=frac{partial L}{partial dot{Q}_i}.

Therefore, coordinate transformations (also called point transformations) are a "type" of canonical transformation. However, the class of canonical transformations is much broader, since the old generalized coordinates, momenta and even time may be combined to form the new generalized coordinates and momenta. Canonical transformations that do not include the time explicitly are called restricted canonical transformations (many textbooks consider only this type).

For clarity, we restrict the presentation here to calculus and classical mechanics. Readers familiar with more advanced mathematics such as cotangent bundles, exterior derivatives and symplectic manifolds should read the related symplectomorphism article. (Canonical transformations are a special case of a symplectomorphism.) However, a brief introduction to the modern mathematical description is included at the end of this article.

Notation

Boldface variables such as mathbf{q} represent a list of N generalized coordinates, e.g.,

:mathbf{q} equiv (q_{1}, q_{2}, ldots, q_{N-1}, q_{N})

that need not transform like a vector under rotation. As usual, the dot signifies the time derivative, e.g., dot{mathbf{q equiv frac{dmathbf{q{dt}. The dot product is defined here as the sum of the products of corresponding components, e.g.,

:mathbf{p} cdot mathbf{q} equiv sum_{k=1}^{N} p_{k} q_{k}

Direct Approach

The functional form of Hamilton's equations is

:dot{mathbf{p = -frac{partial H}{partial mathbf{q:dot{mathbf{q =~~frac{partial H}{partial mathbf{p

By definition, the transformed coordinates have analogous dynamics

:dot{mathbf{P = -frac{partial K}{partial mathbf{Q:dot{mathbf{Q =~~frac{partial K}{partial mathbf{P

where K(mathbf{Q}, mathbf{P}) is a new Hamiltonian that must be determined.

Unfortunately, a generic transformation (mathbf{q}, mathbf{p}, t) ightarrow (mathbf{Q}, mathbf{P}, t) does not preserve the form of Hamilton's equations. We may "check" whether a given restricted transformation between (mathbf{q}, mathbf{p}) and (mathbf{Q}, mathbf{P}) is canonical as follows. Since the transformation has no explicit time dependence (by assumption), the time derivative of a new generalized coordinate Q_{m} is

:dot{Q}_{m} = frac{partial Q_{m{partial mathbf{q cdot dot{mathbf{q + frac{partial Q_{m{partial mathbf{p cdot dot{mathbf{p = frac{partial Q_{m{partial mathbf{q cdot frac{partial H}{partial mathbf{p - frac{partial Q_{m{partial mathbf{p cdot frac{partial H}{partial mathbf{q

We also have the identity for the conjugate momentum P_{m}

:frac{partial H}{partial P_{m = frac{partial H}{partial mathbf{q cdot frac{partial mathbf{q{partial P_{m + frac{partial H}{partial mathbf{p cdot frac{partial mathbf{p{partial P_{m

If the transformation is canonical, these two must be equal, resulting in the equations

:left( frac{partial Q_{m{partial p_{n ight)_{mathbf{q}, mathbf{p = -left( frac{partial q_{n{partial P_{m ight)_{mathbf{Q}, mathbf{P

:left( frac{partial Q_{m{partial q_{n ight)_{mathbf{q}, mathbf{p = left( frac{partial p_{n{partial P_{m ight)_{mathbf{Q}, mathbf{P

The analogous argument for the generalized momenta P_{m} leads to two other sets of equations

:left( frac{partial P_{m{partial p_{n ight)_{mathbf{q}, mathbf{p = left( frac{partial q_{n{partial Q_{m ight)_{mathbf{Q}, mathbf{P

:left( frac{partial P_{m{partial q_{n ight)_{mathbf{q}, mathbf{p = -left( frac{partial p_{n{partial Q_{m ight)_{mathbf{Q}, mathbf{P

These are the direct conditions to check whether a given transformation is canonical.

Liouville's theorem

The direct conditions allow us to prove Liouville's theorem, which states that the "volume" in phase space is conserved under canonical transformations, i.e.,

:int dmathbf{q} dmathbf{p} = int dmathbf{Q} dmathbf{P}

By calculus, the latter integral must equal the former times the Jacobian J

:int dmathbf{Q} dmathbf{P} = int J dmathbf{q} dmathbf{p}

where the Jacobian is the determinant of the matrix of partial derivatives, which we write as

:J equiv frac{partial (mathbf{Q}, mathbf{P})}{partial (mathbf{q}, mathbf{p})}

Exploiting the "division" property of Jacobians yields

:J equivfrac{partial (mathbf{Q}, mathbf{P})}{partial (mathbf{q}, mathbf{P})}left/frac{partial (mathbf{q}, mathbf{p})}{partial (mathbf{q}, mathbf{P})} ight.

Eliminating the repeated variables gives

:J equiv frac{partial (mathbf{Q})}{partial (mathbf{q})}left/frac{partial (mathbf{p})}{partial (mathbf{P})} ight.

Application of the direct conditions above yields J=1.

Generating function approach

To "guarantee" a valid transformation between (mathbf{q}, mathbf{p}, H) and (mathbf{Q}, mathbf{P}, K), we may resort to an indirect generating function approach. Both sets of variables must obey Hamilton's principle

:delta int_{t_{1^{t_{2 left [ mathbf{p} cdot dot{mathbf{q - H(mathbf{q}, mathbf{p}, t) ight] dt = 0

:delta int_{t_{1^{t_{2 left [ mathbf{P} cdot dot{mathbf{Q - K(mathbf{Q}, mathbf{P}, t) ight] dt = 0

To satisfy both variational integrals, we must have

:lambda left [ mathbf{p} cdot dot{mathbf{q - H(mathbf{q}, mathbf{p}, t) ight] = mathbf{P} cdot dot{mathbf{Q - K(mathbf{Q}, mathbf{P}, t) + frac{dG}{dt}

In general, the scaling factor lambda is set equal to one; canonical transformations for which lambda eq 1 are called extended canonical transformations.

Here G is a generating function of one old canonical coordinate (mathbf{q} or mathbf{p}), one new canonical coordinate (mathbf{Q} or mathbf{P}) and (possibly) the time t. Thus, there are four basic types of generating functions, depending on the choice of variables.As will be shown below, the generating function will define a transformation from old to new canonical coordinates, and any such transformation (mathbf{q}, mathbf{p}) ightarrow (mathbf{Q}, mathbf{P}) is guaranteed to be canonical.

Type 1 generating function

The type 1 generating function G_{1} depends only on the old and new generalized coordinates:G equiv G_{1}(mathbf{q}, mathbf{Q}, t)To derive the implicit transformation, we expand the defining equation above

:mathbf{p} cdot dot{mathbf{q - H(mathbf{q}, mathbf{p}, t) = mathbf{P} cdot dot{mathbf{Q - K(mathbf{Q}, mathbf{P}, t) + frac{partial G_{1{partial t} + frac{partial G_{1{partial mathbf{q cdot dot{mathbf{q + frac{partial G_{1{partial mathbf{Q cdot dot{mathbf{Q

Since the new and old coordinates are each independent, the following 2N+1 equations must hold

:mathbf{p} = ~~frac{partial G_{1{partial mathbf{q:mathbf{P} = -frac{partial G_{1{partial mathbf{Q:K = H + frac{partial G_{1{partial t}

These equations define the transformation (mathbf{q}, mathbf{p}) ightarrow (mathbf{Q}, mathbf{P}) as follows. The "first" set of N equations

:mathbf{p} = ~~frac{partial G_{1{partial mathbf{q

define relations between the new generalized coordinates mathbf{Q} and the old canonical coordinates (mathbf{q},mathbf{p}). Ideally, one can invert these relations to obtain formulae for each Q_{k} as a function of the old canonical coordinates. Substitution of these formulae for the mathbf{Q} coordinates into the "second" set of N equations

:mathbf{P} = -frac{partial G_{1{partial mathbf{Q

yields analogous formulae for the new generalized momenta mathbf{P} in terms of the old canonical coordinates (mathbf{q},mathbf{p}). We then invert both sets of formulae to obtain the "old" canonical coordinates (mathbf{q},mathbf{p}) as functions of the "new" canonical coordinates (mathbf{Q},mathbf{P}). Substitution of the inverted formulae into the final equation :K = H + frac{partial G_{1{partial t}yields a formula for K as a function of the new canonical coordinates (mathbf{Q},mathbf{P}).

In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let :G_{1} equiv mathbf{q} cdot mathbf{Q}This results in swapping the generalized coordinates for the momenta and vice versa :mathbf{p} = ~~frac{partial G_{1{partial mathbf{q = mathbf{Q}:mathbf{P} = -frac{partial G_{1{partial mathbf{Q = -mathbf{q}and K=H. This example illustrates how independent the coordinates and momenta are in the Hamiltonian formulation; they're equivalent variables.

Type 2 generating function

The type 2 generating function G_{2} depends only on the old generalized coordinates and the new generalized momenta:G equiv -mathbf{Q} cdot mathbf{P} + G_{2}(mathbf{q}, mathbf{P}, t)where the -mathbf{Q} cdot mathbf{P} terms represent a Legendre transformation to change the right-hand side of the equation below. To derive the implicit transformation, we expand the defining equation above

:mathbf{p} cdot dot{mathbf{q - H(mathbf{q}, mathbf{p}, t) = -mathbf{Q} cdot dot{mathbf{P - K(mathbf{Q}, mathbf{P}, t) + frac{partial G_{2{partial t} + frac{partial G_{2{partial mathbf{q cdot dot{mathbf{q + frac{partial G_{2{partial mathbf{P cdot dot{mathbf{P

Since the old coordinates and new momenta are each independent, the following 2N+1 equations must hold

:mathbf{p} = frac{partial G_{2{partial mathbf{q:mathbf{Q} = frac{partial G_{2{partial mathbf{P:K = H + frac{partial G_{2{partial t}

These equations define the transformation (mathbf{q}, mathbf{p}) ightarrow (mathbf{Q}, mathbf{P}) as follows. The "first" set of N equations

:mathbf{p} = frac{partial G_{2{partial mathbf{q

define relations between the new generalized momenta mathbf{P} and the old canonical coordinates (mathbf{q},mathbf{p}). Ideally, one can invert these relations to obtain formulae for each P_{k} as a function of the old canonical coordinates. Substitution of these formulae for the mathbf{P} coordinates into the "second" set of N equations

:mathbf{Q} = frac{partial G_{2{partial mathbf{P

yields analogous formulae for the new generalized coordinates mathbf{Q} in terms of the old canonical coordinates (mathbf{q},mathbf{p}). We then invert both sets of formulae to obtain the "old" canonical coordinates (mathbf{q},mathbf{p}) as functions of the "new" canonical coordinates (mathbf{Q},mathbf{P}). Substitution of the inverted formulae into the final equation :K = H + frac{partial G_{2{partial t}yields a formula for K as a function of the new canonical coordinates (mathbf{Q},mathbf{P}).

In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let :G_{2} equiv mathbf{g}(mathbf{q}; t) cdot mathbf{P}where mathbf{g} is a set of N functions. This results in a point transformation of the generalized coordinates:mathbf{Q} = frac{partial G_{2{partial mathbf{P = mathbf{g}(mathbf{q}; t)

Type 3 generating function

The type 3 generating function G_{3} depends only on the old generalized momenta and the new generalized coordinates :G equiv mathbf{q} cdot mathbf{p} + G_{3}(mathbf{p}, mathbf{Q}, t)where the mathbf{q} cdot mathbf{p} terms represent a Legendre transformation to change the left-hand side of the equation below. To derive the implicit transformation, we expand the defining equation above

:-mathbf{q} cdot dot{mathbf{p - H(mathbf{q}, mathbf{p}, t) = mathbf{P} cdot dot{mathbf{Q - K(mathbf{Q}, mathbf{P}, t) + frac{partial G_{3{partial t} + frac{partial G_{3{partial mathbf{p cdot dot{mathbf{p + frac{partial G_{3{partial mathbf{Q cdot dot{mathbf{Q

Since the new and old coordinates are each independent, the following 2N+1 equations must hold

:mathbf{q} = -frac{partial G_{3{partial mathbf{p:mathbf{P} = -frac{partial G_{3{partial mathbf{Q:K = H + frac{partial G_{3{partial t}

These equations define the transformation (mathbf{q}, mathbf{p}) ightarrow (mathbf{Q}, mathbf{P}) as follows. The "first" set of N equations

:mathbf{q} = -frac{partial G_{3{partial mathbf{p

define relations between the new generalized coordinates mathbf{Q} and the old canonical coordinates (mathbf{q},mathbf{p}). Ideally, one can invert these relations to obtain formulae for each Q_{k} as a function of the old canonical coordinates. Substitution of these formulae for the mathbf{Q} coordinates into the "second" set of N equations

:mathbf{P} = -frac{partial G_{3{partial mathbf{Q

yields analogous formulae for the new generalized momenta mathbf{P} in terms of the old canonical coordinates (mathbf{q},mathbf{p}). We then invert both sets of formulae to obtain the "old" canonical coordinates (mathbf{q},mathbf{p}) as functions of the "new" canonical coordinates (mathbf{Q},mathbf{P}). Substitution of the inverted formulae into the final equation :K = H + frac{partial G_{3{partial t}yields a formula for K as a function of the new canonical coordinates (mathbf{Q},mathbf{P}).

In practice, this procedure is easier than it sounds, because the generating function is usually simple.

Type 4 generating function

The type 4 generating function G_{4}(mathbf{p}, mathbf{P}, t) depends only on the old and new generalized momenta:G equiv mathbf{q} cdot mathbf{p} - mathbf{Q} cdot mathbf{P} + G_{4}(mathbf{p}, mathbf{P}, t)where the mathbf{q} cdot mathbf{p} - mathbf{Q} cdot mathbf{P} terms represent a Legendre transformation to change both sides of the equation below. To derive the implicit transformation, we expand the defining equation above

:-mathbf{q} cdot dot{mathbf{p - H(mathbf{q}, mathbf{p}, t) = -mathbf{Q} cdot dot{mathbf{P - K(mathbf{Q}, mathbf{P}, t) + frac{partial G_{4{partial t} + frac{partial G_{4{partial mathbf{p cdot dot{mathbf{p + frac{partial G_{4{partial mathbf{P cdot dot{mathbf{P

Since the new and old coordinates are each independent, the following 2N+1 equations must hold

:mathbf{q} = -frac{partial G_{4{partial mathbf{p:mathbf{Q} = ~~frac{partial G_{4{partial mathbf{P:K = H + frac{partial G_{4{partial t}

These equations define the transformation (mathbf{q}, mathbf{p}) ightarrow (mathbf{Q}, mathbf{P}) as follows. The "first" set of N equations

:mathbf{q} = -frac{partial G_{4{partial mathbf{p

define relations between the new generalized momenta mathbf{P} and the old canonical coordinates (mathbf{q},mathbf{p}). Ideally, one can invert these relations to obtain formulae for each P_{k} as a function of the old canonical coordinates. Substitution of these formulae for the mathbf{P} coordinates into the "second" set of N equations

:mathbf{Q} = frac{partial G_{4{partial mathbf{P

yields analogous formulae for the new generalized coordinates mathbf{Q} in terms of the old canonical coordinates (mathbf{q},mathbf{p}). We then invert both sets of formulae to obtain the "old" canonical coordinates (mathbf{q},mathbf{p}) as functions of the "new" canonical coordinates (mathbf{Q},mathbf{P}). Substitution of the inverted formulae into the final equation :K = H + frac{partial G_{4{partial t}yields a formula for K as a function of the new canonical coordinates (mathbf{Q},mathbf{P}).

In practice, this procedure is easier than it sounds, because the generating function is usually simple.

Motion as a canonical transformation

Motion itself (or, equivalently, a shift in the time origin) is a canonical transformation. If mathbf{Q}(t) equiv mathbf{q}(t+ au) and mathbf{P}(t) equiv mathbf{p}(t+ au), then Hamilton's principle is automatically satisfied

:delta int_{t_{1^{t_{2 left [ mathbf{P} cdot dot{mathbf{Q - K(mathbf{Q}, mathbf{P}, t) ight] dt = delta int_{t_{1}+ au}^{t_{2}+ au} left [ mathbf{p} cdot dot{mathbf{q - H(mathbf{q}, mathbf{p}, t+ au) ight] dt = 0

since a valid trajectory (mathbf{q}(t), mathbf{p}(t)) should always satisfy Hamilton's principle, regardless of the endpoints.

Modern mathematical description

In mathematical terms, canonical coordinates are any coordinates on the phase space (cotangent bundle) of the system that allow the canonical one-form to be written as

:sum_i p_i,dq^i

up to a total differential (exact form). The change of variable between one set of canonical coordinates and another is a canonical transformation. The index of the generalized coordinates mathbf{q} is written here as a "superscript" (q^{i}), not as a "subscript" as done above (q_{i}). The superscript conveys the contravariant transformation properties of the generalized coordinates, and does "not" mean that the coordinate is being raised to a power. Further details may be found at the symplectomorphism article.

History

The first major application of the canonical transformation was in 1846, by Charles Delaunay, in the study of the Earth-Moon-Sun system. This work resulted in the publication of a pair of large volumes as "Mémoires" by the French Academy of Sciences, in 1860 and 1867.

ee also

* Symplectomorphism
* Hamilton–Jacobi equation
* Liouville's theorem (Hamiltonian)
* Mathieu transformation
* List of canonical coordinate transformations

References

* Landau LD and Lifshitz EM (1976) "Mechanics", 3rd. ed., Pergamon Press. ISBN 0-08-021022-8 (hardcover) and ISBN 0-08-029141-4 (softcover).

* Goldstein H. (1980) "Classical Mechanics", 2nd. ed., Addison-Wesley. ISBN 0-201-02918-9


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