Coordinate transformation that preserves the form of Hamilton's equations
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (q, p) → (Q, P) that preserves the form of Hamilton's equations. This is sometimes known as form invariance. Although Hamilton's equations are preserved, it need not preserve the explicit form of the Hamiltonian itself. 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 q → Q do not affect the form of Lagrange's equations and, hence, do not affect the form of Hamilton's equations if the momentum is simultaneously changed by a Legendre transformation into
where
are the new co‑ordinates, grouped in canonical conjugate pairs of momenta
and corresponding positions
for
with
being the number of degrees of freedom in both co‑ordinate systems.
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).
Modern mathematical descriptions of canonical transformations are considered under the broader topic of symplectomorphism which covers the subject with advanced mathematical prerequisites such as cotangent bundles, exterior derivatives and symplectic manifolds.
Notation
Boldface variables such as q represent a list of N generalized coordinates that need not transform like a vector under rotation and similarly p represents the corresponding generalized momentum, e.g.,
A dot over a variable or list signifies the time derivative, e.g.,
and the equalities are read to be satisfied for all coordinates, for example:
The dot product notation between two lists of the same number of coordinates is a shorthand for the sum of the products of corresponding components, e.g.,
The dot product (also known as an "inner product") maps the two coordinate lists into one variable representing a single numerical value. The coordinates after transformation are similarly labelled with Q for transformed generalized coordinates and P for transformed generalized momentum.
Restricted canonical transformations are coordinate transformations where transformed coordinates Q and P do not have explicit time dependence, ie.
and
. The functional form of Hamilton's equations is
In general, a transformation (q, p) → (Q, P) does not preserve the form of Hamilton's equations but in the absence of time dependence in transformation, some simplifications are possible. Following the formal definition for a canonical transformation, it can be shown that for this type of transformation, the new Hamiltonian (sometimes called the Kamiltonian[1]) can be expressed as:
where it differs by a partial time derivative of a function known as generator, which reduces to being only a function of time for restricted canonical transformations.
In addition to leaving the form of the Hamiltonian unchanged, it is also permits the use of the unchanged Hamiltonian in the Hamilton's equations of motion due to the above form as:
Although canonical transformations refers to a more general set of transformations of phase space corresponding with less permissive transformations of the Hamiltonian, it provides simpler conditions to obtain results that can be further generalized. All of the following conditions, with the exception of bilinear invariance condition, can be generalized for canonical transformations, including time dependance.
Indirect conditions
Since restricted transformations have no explicit time dependence (by definition), the time derivative of a new generalized coordinate Qm is
![{\displaystyle {\begin{aligned}{\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} }}\\&=\lbrace Q_{m},H\rbrace \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3798150a68c3109affdaec92f501c4551409e293)
where {⋅, ⋅} is the Poisson bracket.
Similarly for the identity for the conjugate momentum, Pm using the form of the Kamiltonian it follows that:
Due to the form of the Hamiltonian equations of motion,
if the transformation is canonical, the two derived results must be equal, resulting in the equations:
The analogous argument for the generalized momenta Pm leads to two other sets of equations:
These are the indirect conditions to check whether a given transformation is canonical.
Symplectic condition
Sometimes the Hamiltonian relations are represented as:
Where
and
. Similarly, let
.
From the relation of partial derivatives, converting the
relation in terms of partial derivatives with new variables gives
where
. Similarly for
,
Due to form of the Hamiltonian equations for
,
where
can be used due to the form of Kamiltonian. Equating the two equations gives the symplectic condition as:[2]
The left hand side of the above is called the Poisson matrix of
, denoted as
. Similarly, a Lagrange matrix of
can be constructed as
.[3] It can be shown that the symplectic condition is also equivalent to
by using the
property. The set of all matrices
which satisfy symplectic conditions form a symplectic group. The symplectic conditions are equivalent with indirect conditions as they both lead to the equation
, which is used in both of the derivations.
Invariance of Poisson Bracket
The Poisson bracket which is defined as:
can be represented in matrix form as:
Hence using partial derivative relations and symplectic condition gives:[4]
The symplectic condition can also be recovered by taking
and
which shows that
. Thus these conditions are equivalent to symplectic conditions. Furthermore, it can be seen that
, which is also the result of explicitly calculating the matrix element by expanding it.[3]
Invariance of Lagrange Bracket
The Lagrange bracket which is defined as:
can be represented in matrix form as:
Using similar derivation, gives:
The symplectic condition can also be recovered by taking
and
which shows that
. Thus these conditions are equivalent to symplectic conditions. Furthermore, it can be seen that
, which is also the result of explicitly calculating the matrix element by expanding it.[3]
Bilinear invariance conditions
These set of conditions only apply to restricted canonical transformations or canonical transformations that are independent of time variable.
Consider arbitrary variations of two kinds, in a single pair of generalized coordinate and the corresponding momentum:[5]
The area of the infinitesimal parallelogram is given by:
It follows from the
symplectic condition that the infinitesimal area is conserved under canonical transformation:
Note that the new coordinates need not be completely oriented in one coordinate momentum plane.
Hence, the condition is more generally stated as an invariance of the form
under canonical transformation, expanded as:
If the above is obeyed for any arbitrary variations, it would be only possible if the indirect conditions are met.[6][7] The form of the equation,
is also known as a symplectic product of the vectors
and
and the bilinear invariance condition can be stated as a local conservation of the symplectic product.[8]
Liouville's theorem
The indirect conditions allow us to prove Liouville's theorem, which states that the volume in phase space is conserved under canonical transformations, i.e.,
By calculus, the latter integral must equal the former times the determinant of Jacobian M
Where
Exploiting the "division" property of Jacobians yields
Eliminating the repeated variables gives
Application of the indirect conditions above yields
.[9]
Generating function approach
To guarantee a valid transformation between (q, p, H) and (Q, P, K), we may resort to a direct generating function approach. Both sets of variables must obey Hamilton's principle. That is the action integral over the Lagrangians
and
, obtained from the respective Hamiltonian via an "inverse" Legendre transformation, must be stationary in both cases (so that one can use the Euler–Lagrange equations to arrive at Hamiltonian equations of motion of the designated form; as it is shown for example here):
One way for both variational integral equalities to be satisfied is to have
Lagrangians are not unique: one can always multiply by a constant λ and add a total time derivative dG/dt and yield the same equations of motion (as discussed on Wikibooks). In general, the scaling factor λ is set equal to one; canonical transformations for which λ ≠ 1 are called extended canonical transformations. dG/dt is kept, otherwise the problem would be rendered trivial and there would be not much freedom for the new canonical variables to differ from the old ones.
Here G is a generating function of one old canonical coordinate (q or p), one new canonical coordinate (Q or P) and (possibly) the time t. Thus, there are four basic types of generating functions (although mixtures of these four types can exist), 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 (q, p) → (Q, P) is guaranteed to be canonical.
The various generating functions and its properties tabulated below is discussed in detail:
Properties of four basic Canonical Transformations[10] Generating Function | Generating Function Derivatives | Transformed Hamiltonian | Trivial Cases |
| | | | | | |
| | | | | |
| | | | | |
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Type 1 generating function
The type 1 generating function G1 depends only on the old and new generalized coordinates
To derive the implicit transformation, we expand the defining equation above
Since the new and old coordinates are each independent, the following 2N + 1 equations must hold
These equations define the transformation (q, p) → (Q, P) as follows: The first set of N equations
define relations between the new generalized coordinates Q and the old canonical coordinates (q, p). Ideally, one can invert these relations to obtain formulae for each Qk as a function of the old canonical coordinates. Substitution of these formulae for the Q coordinates into the second set of N equations
yields analogous formulae for the new generalized momenta P in terms of the old canonical coordinates (q, p). We then invert both sets of formulae to obtain the old canonical coordinates (q, p) as functions of the new canonical coordinates (Q, P). Substitution of the inverted formulae into the final equation
yields a formula for K as a function of the new canonical coordinates (Q, P).
In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let
This results in swapping the generalized coordinates for the momenta and vice versa
and K = H. This example illustrates how independent the coordinates and momenta are in the Hamiltonian formulation; they are equivalent variables.
Type 2 generating function
The type 2 generating function
depends only on the old generalized coordinates and the new generalized momenta
where the
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
Since the old coordinates and new momenta are each independent, the following 2N + 1 equations must hold
These equations define the transformation (q, p) → (Q, P) as follows: The first set of N equations
define relations between the new generalized momenta P and the old canonical coordinates (q, p). Ideally, one can invert these relations to obtain formulae for each Pk as a function of the old canonical coordinates. Substitution of these formulae for the P coordinates into the second set of N equations
yields analogous formulae for the new generalized coordinates Q in terms of the old canonical coordinates (q, p). We then invert both sets of formulae to obtain the old canonical coordinates (q, p) as functions of the new canonical coordinates (Q, P). Substitution of the inverted formulae into the final equation
yields a formula for K as a function of the new canonical coordinates (Q, P).
In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let
where g is a set of N functions. This results in a point transformation of the generalized coordinates
Type 3 generating function
The type 3 generating function
depends only on the old generalized momenta and the new generalized coordinates
where the
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
Since the new and old coordinates are each independent, the following 2N + 1 equations must hold
These equations define the transformation (q, p) → (Q, P) as follows: The first set of N equations
define relations between the new generalized coordinates Q and the old canonical coordinates (q, p). Ideally, one can invert these relations to obtain formulae for each Qk as a function of the old canonical coordinates. Substitution of these formulae for the Q coordinates into the second set of N equations
yields analogous formulae for the new generalized momenta P in terms of the old canonical coordinates (q, p). We then invert both sets of formulae to obtain the old canonical coordinates (q, p) as functions of the new canonical coordinates (Q, P). Substitution of the inverted formulae into the final equation
yields a formula for K as a function of the new canonical coordinates (Q, 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
depends only on the old and new generalized momenta
where the
terms represent a Legendre transformation to change both sides of the equation below. To derive the implicit transformation, we expand the defining equation above
Since the new and old coordinates are each independent, the following 2N + 1 equations must hold
These equations define the transformation (q, p) → (Q, P) as follows: The first set of N equations
define relations between the new generalized momenta P and the old canonical coordinates (q, p). Ideally, one can invert these relations to obtain formulae for each Pk as a function of the old canonical coordinates. Substitution of these formulae for the P coordinates into the second set of N equations
yields analogous formulae for the new generalized coordinates Q in terms of the old canonical coordinates (q, p). We then invert both sets of formulae to obtain the old canonical coordinates (q, p) as functions of the new canonical coordinates (Q, P). Substitution of the inverted formulae into the final equation
yields a formula for K as a function of the new canonical coordinates (Q, P).
Restrictions on generating functions
For example, using generating function of second kind:
and
, the first set of equations consisting of variables
,
and
has to be inverted to get
. This process is possible when the matrix defined by
is non-singular.[11]
Hence, restrictions are placed on generating functions to have the matrices:
,
,
and
, being non-singular.[12][13]
Limitations of generating functions
Since
is non-singular, it implies that
is also non-singular. Since the matrix
is inverse of
, the transformations of type 2 generating functions always have a non-singular
matrix. Similarly, it can be stated that type 1 and type 4 generating functions always have a non-singular
matrix whereas type 2 and type 3 generating functions always have a non-singular
matrix. Hence, the canonical transformations resulting from these generating functions are not completely general.[14]
In other words, since (Q, P) and (q, p) are each 2N independent functions, it follows that to have generating function of the form
and
or
and
, the corresponding Jacobian matrices
and
are restricted to be non singular, ensuring that the generating function is a function of 2N + 1 independent variables. However, as a feature of canonical transformations, it is always possible to choose 2N such independent functions from sets (q, p) or (Q, P), to form a generating function representation of canonical transformations, including the time variable. Hence, it can be proved that every finite canonical transformation can be given as a closed but implicit form that is a variant of the given four simple forms.[15]
From:
, calculate
:
Since the left hand side is
which is independent of dynamics of the particles, equating coefficients of
and
to zero, canonical transformation rules are obtained. This step is equivalent to equating the left hand side as
.
Similarly:
Similarly the canonical transformation rules are obtained by equating the left hand side as
.
The above two relations can be combined in matrix form as:
(which will also retain same form for extended canonical transformation) where the result
, has been used. The canonical transformation relations are hence said to be equivalent to
in this context.
The canonical transformation relations can now be restated to include time dependance:![{\displaystyle {\begin{aligned}\left({\frac {\partial Q_{m}}{\partial p_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=-\left({\frac {\partial q_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\\\left({\frac {\partial Q_{m}}{\partial q_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=\left({\frac {\partial p_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7e8b647130295eb2bffb139f2da83b919e18cb)
Since
and
, if Q and P do not explicitly depend on time,
can be taken. The analysis of restricted canonical transformations is hence consistent with this generalization.
Symplectic Condition
Applying transformation of co-ordinates formula for
, in Hamiltonian's equations gives:
Similarly for
:
or:
Where the last terms of each equation cancel due to
condition from canonical transformations. Hence leaving the symplectic relation:
which is also equivalent with the condition
. It follows from the above two equations that the symplectic condition implies the equation
, from which the indirect conditions can be recovered. Thus, symplectic conditions and indirect conditions can be said to be equivalent in the context of using generating functions.
Invariance of Poisson and Lagrange Bracket
Since
and
where the symplectic condition is used in the last equalities. Using
, the equalities
and
are obtained which imply the invariance of Poisson and Lagrange brackets.
By solving for:
with various forms of generating function, the relation between K and H goes as
instead, which also applies for
case.
All results presented below can also be obtained by replacing
,
and
from known solutions, since it retains the form of Hamilton's equations. The extended canonical transformations are hence said to be result of a canonical transformation (
) and a trivial canonical transformation (
) which has
(for the given example,
which satisfies the condition).[16]
Using same steps previously used in previous generalization, with
in the general case, and retaining the equation
, extended canonical transformation partial differential relations are obtained as:![{\displaystyle {\begin{aligned}\left({\frac {\partial Q_{m}}{\partial p_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=-\lambda \left({\frac {\partial q_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\\\left({\frac {\partial Q_{m}}{\partial q_{n}}}\right)_{\mathbf {q} ,\mathbf {p} ,t}&=\lambda \left({\frac {\partial p_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fefe8bfda647d06c9256b23ecf1383dfe4b5cf39)
Symplectic condition
Following the same steps to derive the symplectic conditions, as:
and
where using
instead gives:
The second part of each equation cancel. Hence the condition for extended canonical transformation instead becomes:
.[17]
Poisson and Lagrange Brackets
The Poisson brackets are changed as follows:
whereas, the Lagrange brackets are changed as:
Hence, the Poisson bracket scales by the inverse of
whereas the Lagrange bracket scales by a factor of
.[18]
Consider the canonical transformation that depends on a continuous parameter
, as follows:
For infinitesimal values of
, the corresponding transformations are called as infinitesimal canonical transformations which are also known as differential canonical transformations.
Consider the following generating function:
Since for
,
has the resulting canonical transformation,
and
, this type of generating function can be used for infinitesimal canonical transformation by restricting
to an infinitesimal value. From the conditions of generators of second type:
Since
, changing the variables of the function
to
and neglecting terms of higher order of
, gives:[19]
Infinitesimal canonical transformations can also be derived using the matrix form of the symplectic condition.[20]
In the passive view of transformations, the coordinate system is changed without the physical system changing, whereas in the active view of transformation, the coordinate system is retained and the physical system is said to undergo transformations. Thus, using the relations from infinitesimal canonical transformations, the change in the system states under active view of the canonical transformation is said to be:
or as
in matrix form.
For any function
, it changes under active view of the transformation according to:
Considering the change of Hamiltonians in the active view, ie. for a fixed point,
where
are mapped to the point,
by the infinitesimal canonical transformation, and similar change of variables for
to
is considered up-to first order of
. Hence, if the Hamiltonian is invariant for infinitesimal canonical transformations, its generator is a constant of motion.
Examples of ICT
Time evolution
Taking
and
, then
. Thus the continuous application of such a transformation maps the coordinates
to
. Hence if the Hamiltonian is time translation invariant, i.e. does not have explicit time dependance, its value is conserved for the motion.
Translation
Taking
,
and
. Hence, the canonical momentum generates a shift in the corresponding generalized coordinate and if the Hamiltonian is invariant of translation, the momentum is a constant of motion.
Rotation
Consider an orthogonal system for an N-particle system:
Choosing the generator to be:
and the infinitesimal value of
, then the change in the coordinates is given for x by:
and similarly for y:
whereas the z component of all particles is unchanged:
.
These transformations correspond to rotation about the z axis by angle
in its first order approximation. Hence, repeated application of the infinitesimal canonical transformation generates a rotation of system of particles about the z axis. If the Hamiltonian is invariant under rotation about the z axis, the generator, the component of angular momentum along the axis of rotation, is an invariant of motion.[20]
Motion itself (or, equivalently, a shift in the time origin) is a canonical transformation. If
and
, then Hamilton's principle is automatically satisfied
since a valid trajectory
should always satisfy Hamilton's principle, regardless of the endpoints.
Examples
- The translation
where
are two constant vectors is a canonical transformation. Indeed, the Jacobian matrix is the identity, which is symplectic:
. - Set
and
, the transformation
where
is a rotation matrix of order 2 is canonical. Keeping in mind that special orthogonal matrices obey
it's easy to see that the Jacobian is symplectic. However, this example only works in dimension 2:
is the only special orthogonal group in which every matrix is symplectic. Note that the rotation here acts on
and not on
and
independently, so these are not the same as a physical rotation of an orthogonal spatial coordinate system. - The transformation
, where
is an arbitrary function of
, is canonical. Jacobian matrix is indeed given by
which is symplectic.
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
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 q is written here as a superscript (
), not as a subscript as done above (
). 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.
See also
Notes
- ^ Goldstein, Poole & Safko 2007, p. 370
- ^ Goldstein, Poole & Safko 2007, p. 381-384
- ^ a b c Giacaglia 1972, p. 8-9
- ^ Lemos 2018, p. 255
- ^ Hand & Finch 1999, p. 250-251
- ^ Lanczos 2012, p. 121
- ^ Gupta & Gupta 2008, p. 304
- ^ Lurie 2002, p. 337
- ^ Lurie 2002, p. 548-550
- ^ Goldstein, Poole & Safko 2007, p. 373
- ^ Johns 2005, p. 438
- ^ Lurie 2002, p. 547
- ^ Sudarshan & Mukunda 2010, p. 58
- ^ Johns 2005, p. 437-439
- ^ Sudarshan & Mukunda 2010, pp. 58–60
- ^ Giacaglia 1972, p. 18-19
- ^ Goldstein, Poole & Safko 2007, p. 383
- ^ Giacaglia 1972, p. 16-17
- ^ Johns 2005, p. 452-454
- ^ a b Hergert, Heiko (December 10, 2021). "PHY422/820: Classical Mechanics" (PDF). Archived (PDF) from the original on December 22, 2023. Retrieved December 22, 2023.
References
- Goldstein, Herbert; Poole, Charles P.; Safko, John L. (2007). Classical mechanics (3rd ed.). Upper Saddle River, N.J: Pearson [u.a.] ISBN 978-0-321-18897-7.
- Landau, L. D.; Lifshitz, E. M. (1975) [1939]. Mechanics. Translated by Bell, S. J.; Sykes, J. B. (3rd ed.). Amsterdam: Elsevier. ISBN 978-0-7506-28969.
- Giacaglia, Georgio Eugenio Oscare (1972). Perturbation Methods in Non-Linear Systems. New York: Springer-Verlag. ISBN 3-540-90054-3. LCCN 72-87714.
- Lanczos, Cornelius (2012-04-24). The Variational Principles of Mechanics. Courier Corporation. ISBN 978-0-486-13470-3.
- Lurie, Anatolii I. (2002). Analytical Mechanics (1st ed.). Springer-Verlag Berlin. ISBN 978-3-642-53650-2.
- Gupta, Praveen P.; Gupta, Sanjay (2008). Rigid Dynamics (10th ed.). Krishna Prakashan Media.
- Johns, Oliver Davis (2005). Analytical Mechanics for Relativity and Quantum Mechanics. Oxford University Press. ISBN 978-0-19-856726-4.
- Lemos, Nivaldo A (2018). Analytical mechanics. Cambridge University Press. ISBN 978-1-108-41658-0.
- Hand, Louis N.; Finch, Janet D. (1999). Analytical Mechanics (1st ed.). Cambridge University Press. ISBN 978-0521573276.
- Sudarshan, E C George; Mukunda, N (2010). Classical Dynamics: A Modern Perspective. Wiley. ISBN 9780471835400.