Canonical Transformations

11 Key excerpts on "Canonical Transformations"

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  Notes on Hamiltonian Dynamical Systems

  • Antonio Giorgilli(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  In this respect it represents one of the highest points in the development of Mechanics in the nineteenth century, before the work of Poincar´ e. A final remark is in order. Nowadays it is common to place the canonical formalism in the framework of symplectic geometry. It is also common to call symplectic transformations the class which is traditionally called canonical (with some caveats concerning the local and global aspects). Actually, a 32 Canonical Transformations moderate use of the language of symplectic geometry helps to simplify some particular discussions. For this reason a short introduction to symplectic geometry is reported in Appendix B. 2.1 Preserving the Hamiltonian Form of the Equations It may be useful to outline again a connection with the Lagrangian for- malism. It is well known that Lagrange’s equations have the nice property of being invariant with respect to point transformation (i.e., changes of the coordinates in configuration space, which by differentiation generate the cor- responding transformations on the generalized velocities). The Hamiltonian formalism removes the tie between generalized coordinates and velocities, so that arbitrary transformations involving all the canonical coordinates may be devised. However, an arbitrary transformation will likely produce equa- tions which are not in Hamiltonian form. The problem then is to characterize a restricted class of transformation which preserves the form of Hamilton’s equations. The first approach consists in looking for a class of transformations (q, p) = C (Q, P ) satisfying the following Condition 1: To every Hamiltonian function H(q, p) one can associate an- other function K(Q, P ) such that the canonical system of equations ˙ q j = ∂H ∂p j , ˙ p j = - ∂H ∂q j , j = 1, . . . , n is changed into the system ˙ Q j = ∂K ∂P j , ˙ P j = - ∂K ∂Q j , j = 1, . . . , n , which is still canonical. We shall say that such a transformation preserves the canonical form of the equations.

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  The Variational Principles of Mechanics

  • Cornelius Lanczos(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  variables, They are the coordinates of a point of the phase space and nothing else. The specific problem of motion is entirely eliminated. However, it is important that our transformation shall preserve the canonical equations. This involves the differential form which appears in the canonical integral. If this differential form is preserved, the whole set of canonical equations is preserved. And hence we have a trans- formation problem which is characterized by the invariance of a certain differential form.

  We call transformations which preserve the canonical equations “Canonical Transformations.” The general theory of these transformations is the achievement of Jacobi.

  Summary.   The most effective tool in the investigation and solution of the canonical equations is the transformation of coordinates. Instead of trying to integrate the equations directly, we try to introduce a new coordinate system which is better adapted to the solution of the problem than the original one. There is an extended class of transformations at our disposal in this procedure. They are called “Canonical Transformations.”

           2.   The Lagrangian point transformations.  The position coordinates of Lagrangian mechanics are the quantities qi . The Lagrangian equations of motion remain invariant with respect to arbitrary point transformations of these coordinates. In Hamiltonian mechanics we again have a Lagrangian problem, but now in the 2n variables qi and pi . The configuration space of Hamiltonian mechanics is the 2n-dimensional phase space. Hence at first sight we might think that arbitrary point transformations of the phase space are now at our disposal. This would mean that the 2n coordinates qi and pi can be transformed into some new and by any functional relations we please. This, however, is not the case. The canonical equations

  are the result of a very special Lagrangian problem, namely a Lagrangian problem whose Lagrangian function is normalized to the canonical form

  An arbitrary point transformation of the qi and pi into and would destroy the normal form of the canonical integral, and with it the canonical equations. We thus restrict ourselves to transformations which preserve the canonical form of these equations, which is guaranteed if the variational integrand has the form (72.2 ). Any transformation which leaves the canonical integrand (72.2 ) invariant, leaves also the canonical equations (72.1 ) invariant

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  Analytical Mechanics

  • Nivaldo A. Lemos(Author)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  8 Canonical Transformations You boil it in sawdust: you salt it in glue: You condense it with locusts and tape: Still keeping one principal object in view − To preserve its symmetrical shape. Lewis Carroll, The Hunting of the Snark Lagrange’s equations are invariant under a general coordinate transformation in config- uration space: the form of the equations of motion stays the same whatever the choice of generalised coordinates. In the Hamiltonian formulation the coordinates and momenta are independent variables, and we are naturally led to study changes of variables in phase space that preserve the form of Hamilton’s equations, thus enormously enlarging the range of admissible transformations. This enlargement, in its turn, often makes possible the judicious choice of canonical variables that simplify the Hamiltonian, thereby making the equations of motion easy to solve. 8.1 Canonical Transformations and Generating Functions A change of variables in phase space is of our interest if it preserves the canonical form of the equations of motion. More precisely, given the canonical variables (q, p), the Hamiltonian H(q, p, t) and Hamilton’s equations ˙ q i = ∂ H ∂ p i , ˙ p i = − ∂ H ∂ q i , i = 1, . . . , n , (8.1) we are interested in the invertible transformation Q i = Q i (q, p, t) , P i = P i (q, p, t) , i = 1, . . . , n , (8.2) as long as it is possible to find a function K(Q, P, t) such that the equations of motion for the new variables take the Hamiltonian form: ˙ Q i = ∂ K ∂ P i , ˙ P i = − ∂ K ∂ Q i , i = 1, . . . , n . (8.3) It is worth stressing that the equations of motion for the new variables must have the Hamiltonian form no matter what the original Hamiltonian function H(q, p, t) may be. 1 1 See the Remark at the end of Section 8.2. 242

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  Quantum Mechanics In Phase Space: An Overview With Selected Papers

  An Overview with Selected Papers

  • Thomas L Curtright, David B Fairlie, Cosmas K Zachos(Authors)
  • 2005(Publication Date)
  • World Scientific

    (Publisher)

  111. LINEAR Canonical Transformations IN CLASSICAL MECHANICS Classical mechanics can be formulated in terms of the position and momentum variables. If Uand Vare two func- tions of x andp, then their Poisson bracket is defined asx a u a v auav ax ap ap ax [ U V ] ' X P Under the canonical transformation X=X(x,p), P= P(X,P), (11) the Poisson bracket remains invariant. For one pair of ca- nonical variables, this leads to the condition that the Jaco- bian determinant be 1: ax aP I I- - I aP aP I This means that the area element in phase space is pre- served under Canonical Transformations. We are interested in linear Canonical Transformations of the form X = a , i x + a i g + b i , P = a 2 ~ x + a 2 9 + b,. (13) The parameters b, and b, are for translations, which are area-preserving Canonical Transformations. If we do not consider this transformation by setting 6 , = 6, = 0, the above equations represent homogeneous linear transfor- mations. The most familiar linear transformation is the ro- tation around the origin: (14) cos(B/2) - sin(B/2)) (x) We use the angle B/2 instead of B for later convenience. Another area-preserving linear transformation is the squeeze along the x axis: We are now ready to formulate the group linear canoni- cal transformations in phase space. This group consists of translations, rotations, and squeezes in phase space. The coordinate transformation representing translations X = x + b , , P=p+b,, (16) can be written as The matrix performing the rotation around the origin by 8/2 takes the form (cos(:/2) -sin(B/2) 0 0 1 R ( 8 ) = sin(B/2) cos(B/2) 0). (18) It is possible to define the olar variables r and 4 in phase space, with r = ( x 2 +p2)IR and 4 = tan-'(p/x). The x direction in this case is the 4 = 0 direction. We can thus write S, (7) = S( 4 = 0 , ~ ) . Then, The elongation along the x axis is necessarily the contrac- tion along thep axis.

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  The Classical Dynamics of Particles

  Galilean and Lorentz Relativity

  • Ronald A. Mann(Author)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  122 4. THE CANONICAL FORMALISM variant. Once again we have established that intimate connection between conservation laws and the properties of the system under certain Canonical Transformations. We illustrate these results by considering the canonical transformation which performs a spatial displacement of the entire system by an infinitesimal amount, i.e., àq a = s (4.56) From (4.48b) it follows that the generator of such a transformation is G = ZPa (4-57) a The Poisson bracket of G with H is [H, G] = £ dH/dq a (4.58) a=l Therefore, if the Hamiltonian is cyclic in the coordinates q a , the total momentum (i.e., G) is conserved. This result agrees with that obtained from Noether's theorem. E. Hamilton-Jacobi Theory In our discussion of the harmonic oscillator we demonstrated that a canonical transformation could be used to simplify the solution of the dynamical equations. Hamilton-Jacobi theory provides a systematic method for determining generating functions which lead to equations of motion that can be solved easily. But now we will be more ambitious than we were in the harmonic oscillator problem, where we sought a canonical transfor-mation which made our Hamiltonian cyclic in Q. We now seek a canonical transformation to a new set of canonical coordinates for which our Hamil-tonian will be cyclic in both the coordinates and momenta. If we are suc-cessful, it follows immediately that the momenta and coordinates are constants, and if we identify the constants as the values of our original coordinates and momenta at the initial time t 0 , i.e., Q a = q 0a andP a =A) a , then our Canonical Transformations back to the original coordinates have the form 4a = tfa(?o & ,/V') (4.59a) Pa=Pa(Po b ,t) (4.59b) But these are just the solutions to our dynamical equations of motion.

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  The Semiclassical Way to Dynamics and Spectroscopy

  • Eric J. Heller(Author)
  • 2018(Publication Date)
  • Princeton University Press

    (Publisher)

  See section 2.4 for a more general discussion of composition of Canonical Transformations. EXAMPLE: THE HARMONIC OSCILLATOR We illustrate these points using the generator of harmonic oscillator motion, which for frequency ω, mass m, and time t is S (q t , q 0 , t ) = f 1 (q t , q 0 , t ) = mω 2 sin ωt (q 2 t + q 2 0 )cos ωt − 2 q 0 q t . (2.30) This is a quadratic form in q 0 and q t corresponding to a linear transformation, here a pure rigid clockwise rotation of the whole phase space about the origin, progressing steadily with angle ωt . The reader should verify that the usual motion p t = p 0 cos ωt − mωq 0 sin ωt and q t = q 0 cos ωt + p 0 / mω sin ωt follows from this gener- ator, and that S (q 2t , q 0 , 2t ) = S (q 2t , q t , t ) + S (q t , q 0 , t ) with ∂ S (q 2t , q 0 , 2t )/∂ q t = 0 (this follows from the composition rule for two successive Canonical Transformations). The harmonic oscillator action also contains the free particle, by taking the limit ω → 0. 2.3 Phase Space and Canonical Transformations Phase space, the 2 N dimensional collection of canonically conjugate coordinates and momenta (q 1 , q 2 , . . . , q N , p 1 , p 2 , . . . p N ) is an extremely important construct in both classical and quantum mechanics. QUESTION OF DISTANCE IN PHASE SPACE Position q and momentum p have different units, making the notion of distance in phase space ill defined. Areas in phase space have no problem: they have units of action—that is, momentum times position. Canonical Transformations distort phase space in such a way as to preserve areas. Defining a generating function f 2 (q , P ) = γ q P , where γ has units of kg/s, gives P = γ −1 p, Q = γ q , and gives both new variables dimensions of √ action, so lines would have that dimension too. This does not resolve the problem, because the magnitude of γ is arbitrary, scaling Q while also scaling P by the inverse amount.

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  A Student's Guide to Lagrangians and Hamiltonians

  • Patrick Hamill(Author)
  • 2013(Publication Date)
  • Cambridge University Press

    (Publisher)

  (This canonical transformation in time is some- times called a “contact” transformation.) Thus, the Hamiltonian is the genera- tor of the motion of the system in time; it generates the canonical variables at a later time. We now consider the change in a function u = u(q, p) under a canonical transformation. Note that there are two ways to interpret a canonical trans- formation. On the one hand, it changes the description of a system from an old set of canonical variables q, p to a new set of canonical variables Q, P. Under such a transformation a function u may have a different form but it will still have the same value. Thus, if q 1 (t), . . . , p n (t) are denoted by A and Q 1 (t), . . . , P n (t) are denoted by A  , the transformed function u(A  ) has the same value as the original function u(A): u(A  ) = u(A). (The value of the total angular momentum of a rotating body is the same regardless of the set of coordinates used to evaluate it.) Of course, u(A  ) will in general have a different mathematical form than u(A). (In Cartesian coordi- nates the angular momentum of a mass point moving in a circle is m(x ˙ y − y ˙ x) and in polar coordinates it is mr 2 ˙ θ.) On the other hand, if we consider an infinitesimal contact transformation that generates a translation in time, the interpretation is quite different. Now the transformation takes us from A, representing the values q 1 (t), . . . , p n (t), to B, representing q 1 (t + dt), . . . , p n (t + dt), in the same phase space (with the same set of phase space axes) but at a later time. Since we are in the same phase space and using the same set of coordinates, the form of u is conserved but its value changes. Let us represent such a change in the function u by ∂u = u(B) − u(A). 5.4 The equations of motion in terms of Poisson brackets 123 But since u = u(q, p) we write ∂u = ∂u ∂q i δq i + ∂u ∂p i δp i = ∂u ∂q i  ∂G ∂pi + ∂u ∂p i  − ∂G ∂q i  =   ∂u ∂q i ∂G ∂p i − ∂u ∂p i ∂G ∂q i  , or ∂u =  [u, G].

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  Mechanics in Differential Geometry

  • Yves Talpaert(Author)
  • 2012(Publication Date)
  • De Gruyter

    (Publisher)

  370 Chapter 3 4. HAMILTON-JACOBI MECHANICS Symplectic mappings also called 'Canonical Transformations', and which enrich Hamiltonian mechanics, will be again considered. Such transformations of generalized coordinates and momenta preserve the form of Hamilton's equations; of course we recall that there are transformations which preserve the canonical shape of Hamilton's equations but are not Canonical Transformations (see Sect. 2.4.3 of this chapter). Generating functions for Canonical Transformations will be next considered. The so called Hamilton-Jacobi theory, famous for solving insoluble problems by Lagrangian and Hamiltonian formalisms, is then introduced in a natural way. Finally, a fundamental variational principle is recalled. 4.1 Canonical Transformations AND GENERATING FUNCTIONS If necessary the reader can refer to symplectic geometry set out in Ch.2. 4.1.1 Symplectic Matrix, Poisson and Lagrange Brackets, First Integrals Let us recall that the notion of Poisson brackets notably lets: - verify if a function is a first integral, - find new first integrals from known first integrals, - express the motion equations under a very simple form, and so on. Let us denote the coordinates of any point χ of the momentum phase space P = T'Q by x A =(q',P,) where Ae{,...,2n} and /e{!,...,«}. We recall the following (2« χ 2n) symplectic matrix [see Sect. 1.2 of Ch.2]: (3-74) The inverse matrix (3-75) is such that ω, η ω Β€ =δ •c (3-76) We know that J 2 =-/ and A Modern Exposition of Mechanics J-1 ='J = -J = The symplectic matrix is antisymmetric and of determinant +1. 371 0 -I 1 0 PR52 The 2η Hamilton's equations [e.g. Eqs. (3-27), (3-28)] are denoted by dH χ = -ω dx B (3-77) with^,5 e {l,...,2n} Proof.

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  Classical Mechanics

  • Tai L. Chow(Author)
  • 2013(Publication Date)
  • CRC Press

    (Publisher)

  Correspondingly, the motion of the system in a finite time interval from t 0 to t is represented by a succession of infinitesimal Canonical Transformations. As the result of two canonical transforma-tions applied one after the other is equivalent to a single canonical transformation, we see that any two points on a given trajectory in phase space are connected by a canonical transformation. Hence, the motion of a mechanical system can thus be regarded as the continuous unfolding of a canonical transformation generated by the Hamiltonian of the system. From this result follows an important theorem, which can be stated as follows: any invariants during the motion of a system are also invariants under canonical transformation, and conversely, invariants of Canonical Transformations are invariants of the motion of the system. In this light, the result of Equation 5.45 in Chapter 5 d d F t F H F t = + ∂ ∂ [ , ] is not surprising. The evolution of the system motion is determined by H , and this relationship shows how a particular F develops in time through the influence of H ; as a consequence, Equation 5.45 is 529 Hamilton–Jacobi Theory of Dynamics © 2010 Taylor & Francis Group, LLC sometimes called the unfolding theorem. It turns out that there is a neat, practical way of solving dynamical problems by means of this theorem. Expanding q ( t ) and p ( t ) as functions of time about their initial values q 0 and p 0 , we have q t q q t t q t t t t ( ) ! ... = +       +       + = = 0 0 2 2 0 2 2 d d d d p t p p t t p t t t t ( ) ! ... = +       +       + = = 0 0 2 2 0 2 2 d d d d . Now according to Equation 5.45, d d d d d d d d q t q H q t t q t q H H = = =     [ , ] [ , ], and 2 2 and so on. We have similar results for the p-derivative. It follows that q t q q t t q t t t t ( ) ! ... = +       +       + = = = 0 0 2 2 0 2 2 d d d d q q H q H H t t t 0 0 0 2 1 2 + +     + = = [ , ] [ , ], ...

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  Quantum Field Theory

  An Integrated Approach

  • Eduardo Fradkin(Author)
  • 2021(Publication Date)
  • Princeton University Press

    (Publisher)

  4 Canonical Quantization We now begin the discussion of our main subject of interest: the role of quantum mechanical fluctuations in systems with infinitely many degrees of freedom. We begin with a brief overview of quantum mechanics of a single particle. 4.1 Elementary quantum mechanics Elementary quantum mechanics describes the quantum dynamics of systems with a finite number of degrees of freedom. Two axioms are involved in the standard procedure for quantizing a classical system. Let L ( q , ˙ q ) be the Lagrangian of an abstract dynamical system described by the generalized coordinate q . In chapter 2, we recalled that the canonical formalism of classical mechanics is based on the concept of canonical pairs of dynamical variables. So the canonical coordinate q has for its partner the canonical momentum p : p = ∂ L ∂ ˙ q (4.1) In the canonical formalism, the dynamics of the system is governed by the classical Hamiltonian H ( q , p ) = p ˙ q − L ( q , ˙ q ) (4.2) which is the Legendre transform of the Lagrangian. In the canonical (Hamiltonian) formal-ism, the equations of motion are just Hamilton’s equations: ˙ p = − ∂ H ∂ q , ˙ q = ∂ H ∂ p (4.3) The dynamical state of the system is defined by the values of the canonical coordinates and momenta at any given time t . As a result of these definitions, the coordinates and momenta satisfy a set of Poisson bracket relations, { q , p } PB = 1, { q , q } PB = { p , p } PB = 0 (4.4) Canonical Quantization . 77 where { A , B } PB ≡ ∂ A ∂ q ∂ B ∂ p − ∂ A ∂ p ∂ B ∂ q (4.5) In quantum mechanics, the primitive (or fundamental) notion is the concept of a physical state . A physical state of a system is represented by a state vector in an abstract vector space, which is called the Hilbert space H of quantum states.

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  Lectures on Quantum Mechanics

  • Steven Weinberg(Author)
  • 2015(Publication Date)
  • Cambridge University Press

    (Publisher)

  The great advantage of the Lagrangian formalism, described in Section 9.2, is that it allows us to derive the existence of conserved quantities from symmetry principles. One of these con- served quantities is the Hamiltonian, discussed in Section 9.3. The Hamiltonian is expressed in terms of generalized coordinates and generalized momenta. As shown in Section 9.4, these variables must satisfy certain commutation relations in order for the conserved quantities provided by the Lagrangian formalism to act as the generators of symmetry transformations with which they are asso- ciated, and in particular for the Hamiltonian to act as the generator of time translations. 325 326 9 The Canonical Formalism I will illustrate all these points by reference to the theory of non-relativistic particles in a local potential. In this case, the application of the canonical formalism is pretty simple. It becomes more complicated for systems satisfying a constraint, such as a particle constrained to move on a surface. Constrained systems are discussed in Section 9.5. An alternative version of the canonical formalism, the path-integral formalism, is derived in Section 9.6. 9.1 The Lagrangian Formalism It is common to find that the dynamical equations that govern the general coordi- nate variables q N (t ) describing a classical physical system can be derived from a variational principle, which states that an integral I [q ] ≡  ∞ −∞ L  q (t ), ˙ q (t ), t  dt (9.1.1) is stationary with respect to all infinitesimal variations q N (t )  → q N (t ) + δq N (t ), for which all δq N (t ) vanish at the end-points of the integral, t → ±∞. The function or functional L is known as the Lagrangian of the theory, while the functional I [q ] is called the action. In a theory of particles, N is a compound index ni , with q N (t ) the i th component x ni (t ) of the position of the nth particle at time t .

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