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Contents
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(Top)
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1Notation
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2Conditions for restricted canonical transformation
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3Liouville's theorem
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4Generating function approach
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5Canonical transformation conditions
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6Extended Canonical Transformation
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7Infinitesimal canonical transformation
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8Motion as canonical transformation
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9Examples
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10Modern mathematical description
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11History
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12See also
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13Notes
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14References
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
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.,
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.
Conditions for restricted canonical transformation
Restricted canonical transformations are coordinate transformations where transformed coordinates Q and P do not have explicit time dependance, ie. and . The functional form of Hamilton's equations is
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
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]
Invariance of Poisson Bracket
The Poisson bracket which is defined as:
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:
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:
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
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:
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
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
In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let
Type 2 generating function
The type 2 generating function depends only on the old generalized coordinates and the new generalized momenta
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
In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let
Type 3 generating function
The type 3 generating function depends only on the old generalized momenta and the new generalized coordinates
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
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
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
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]
Canonical transformation conditions
Canonical transformation relations
From: , calculate :
Similarly:
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:
Symplectic Condition
Applying transformation of co-ordinates formula for , in Hamiltonian's equations gives:
Similarly for :
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.
Extended Canonical Transformation
Canonical transformation relations
By solving for:
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:
Symplectic condition
Following the same steps to derive the symplectic conditions, as:
where using instead gives:
Poisson and Lagrange Brackets
The Poisson brackets are changed as follows:
Infinitesimal canonical transformation
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:
Active canonical transformations
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,
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 as canonical transformation
Motion itself (or, equivalently, a shift in the time origin) is a canonical transformation. If and , then Hamilton's principle is automatically satisfied
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
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
- Symplectomorphism
- Hamilton–Jacobi equation
- Liouville's theorem (Hamiltonian)
- Mathieu transformation
- Linear canonical transformation
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.