Hamilton's Equations of Motion

12 Key excerpts on "Hamilton's Equations of Motion"

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  eBook - ePub

  Emergence of the Quantum from the Classical

  Mathematical Aspects of Quantum Processes

  • Maurice de Gosson(Author)
  • 2017(Publication Date)
  • WSPC (EUROPE)

    (Publisher)

  Like the movement of a symphony, a Hamiltonian flow involves a total ordering which implies the whole movement: Past, present, and future are actively present in any one movement. When we are listening to music we are actually directly perceiving an implicate order. This order is active because it is continuously flowing in emotional responses which are inseparable from the flow itself. Similarly, the solutions of Hamilton’s equations are uniquely determined for all bounded times and all locations close to the original one, exactly as in the symphony metaphor: if we observe during a tiny time interval the motion of a particle moving under the influence of a Hamiltonian flow, we see an unfoldment of the totality of the flow, which is uniquely determined by the past — and the future!

  There are nowadays many texts, at various levels, presenting Hamiltonian mechanics from the symplectic point of view; for instance the books by Arnold [5 ], Abraham et al. [2 ], Guillemin and Sternberg [139 , 140 ] are classical references; for good introductions to symplectic geometry from the modern point of view see [23 ].

  1.1. Hamilton’s Equations

  1.1.1. The origins; examples

  Most physical systems can be studied by using two specific theories originating from Newtonian mechanics, and having overlapping — but not identical — domains of validity. The first of these theories is “Lagrangian mechanics”, which essentially uses variational principles (e.g., the “least action principle”); it will not be discussed at all in this book; we refer to Souriau [268 ] (especially p. 140) for an analysis of some of the drawbacks of the Lagrangian approach. The second theory, “Hamiltonian mechanics”, is based on Hamilton’s equations of motion

  where the Hamiltonian function

  is associated with the “vector” and “scalar” potentials A and V (both possibly depending on time t). We are writing r = (x, y, z) for the position vector, and p = (px , py , pz ) for the momentum vector. For A

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  Calculus of Variations

  • Robert Weinstock(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  lagrangian —as

    6-2. Hamilton’s Principle. Lagrange Equations of Motion

  (a ) Although Newton’s laws of motion are the most fundamental mathematical description of mechanical phenomena in general, it best suits the purposes of our study to assume the validity of Hamilton’s principle as the physical law which describes the motion of any system of the type considered in 6-1 above. The principle of Hamilton reads:

  The actual motion of a system whose lagrangian is

  is such as to render the (Hamilton’s) integral

   

  where t 1 and t 2 are two arbitrary instants of time , an extremum withrespect to continuously twice-differentiable functions q 1 (t ), q 2 (t ), …,

  qN

  (t )

  for which qi

  (t 1 )

  and qi

  (t 2 ) are prescribed for all i = 1, 2, …, N . We accept Hamilton’s principle as applicable to the motion of any conservative system.

  Although it is in some places stated that Hamilton’s principle may be used to replace Newton’s laws of motion as the fundamental starting point for mechanical systems possessing a lagrangian, it should be realized that Newton’s laws are implicitly employed in the preceding paragraphs in at least two ways: (i) The definition of mass resides in Newton’s third law. (ii) In the tacit assumption that our system of coordinates is fixed relative to an inertial frame of reference, we make use of Newton’s first law, by means of which an inertial frame is defined.

  (b ) With the result (57 ) of 3-8(a ) we conclude from Hamilton’s principle that the generalized coordinates describing the motion of a system of particles must satisfy the set of Euler-Lagrange equations

   

  The equations (8 ), Lagrange’s equations of motion, constitute a set of N simultaneous second-order differential equations, whose solution yields the functions q 1 (t ), q 2 (t ), …,

  qN

  (t ). The 2N constants involved in the general solution of (8 ) are evaluated when the initial (t = 0, for example) values of all the

  qi

  and (i = 1, 2, …, N ) are given. Once the initial state of the system is thus prescribed, its future motion is described in detail by the functions obtained through the solution of (8

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  Classical Mechanics

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

    (Publisher)

  125 © 2010 Taylor & Francis Group, LLC Hamiltonian Formulation of Mechanics Descriptions of Motion in Phase Spaces The Lagrangian dynamics have been shown to be elegant and straightforward. Half a century after Lagrange, William R. Hamilton introduced another way of writing the equations of motion of a sys-tem. Instead of a single differential equation of second order for each coordinate, Hamilton found a set of twice as many equations but only of the first order, that is, containing only first derivatives with respect to the time. How could Hamilton achieve this? In the Lagrangian formulation, we can transform to a new set of coordinates as well; once the coor-dinates have been chosen, the corresponding velocities are also determined. Hamilton removed this subordinate feature of velocities by eliminating them in favor of the generalized momenta. One reason for this change is that the momenta are often conservative quantities, and symmetries is even more explicit in Hamilton’s new formulation. Another reason is that the Lagrangian function has no useful physical meaning, but the Hamiltonian function, when conserved, is the energy of the system, a very important quantity. The Hamilton method is intimately connected with symmetry and conservation. 5.1 THE HAMILTONIAN OF A DYNAMIC SYSTEM As we learned in Chapter 4, the Lagrangian L for a holonomic system of n degrees of freedom is defined in terms of q i and dotnosp q i as L L q q q q q q t n n = ( , , , , , , ) 1 2 1 2 ..., ..., dotnosp dotnosp dotnosp and the equations of motion are Lagrange’s equations, a set of second-order differential equations, d d , t L q L q j n j j ∂ ∂ -∂ ∂ = = dotnosp … 0 1 2 , , . The generalized momentum conjugate to q j is defined as p L q j j = ∂ ∂ dotnosp . We are tempted, at this point, to search for a new way of describing the complete mechanical state of a system by giving q j and p j as functions of time, rather than q i and dotnosp q i .

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  Advanced Dynamics

  • Donald T. Greenwood(Author)
  • 2006(Publication Date)
  • Cambridge University Press

    (Publisher)

  5 Equations of motion: integral approach Integral principles and, in particular, Hamilton’s principle, have long occupied a prominent position in analytical mechanics. Hamilton’s principle, first announced in 1834, presents a variational principle as the basis for the dynamical description of a holonomic system. This approach tends to view the motion as a whole and involves a search for the path in configuration space which yields a stationary value for a certain integral. As a result, one obtains the differential equations of motion. The requirement of stationarity does not apply to nonholonomic systems. Nevertheless, one can use integral methods to obtain the equations of motion for nonholonomic systems. Here we use the integral of the variation rather than the variation of the integral. In this chapter, we shall discuss the derivation and application of these methods, particularly with respect to nonholonomic systems. 5.1 Hamilton’s principle Holonomic system Consider a dynamical system whose motion satisfies Lagrange’s principle , namely, n i = 1 d dt ∂ T ∂ ˙ q i − ∂ T ∂ q i − Q i δ q i = 0 (5.1) There are n generalized coordinates and the δ q s satisfy the instantaneous constraints. The kinetic energy T ( q , ˙ q , t ) is written for the unconstrained system, and is assumed to have at least two continuous derivatives in each of its arguments. Q i is the generalized applied force associated with q i . Now integrate (5.1) with respect to time over the fixed interval t 1 to t 2 . Using integration by parts, we find that t 2 t 1 n i = 1 d dt ∂ T ∂ ˙ q i δ q i dt = − t 2 t 1 n i = 1 ∂ T ∂ ˙ q i d dt ( δ q i ) dt + n i = 1 ∂ T ∂ ˙ q i δ q i t 2 t 1 (5.2) 290 Equations of motion: integral approach q O t 1 t 2 t t δ q varied path actual path Figure 5.1.

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  Introduction to Quantum Mechanics with Applications to Chemistry

  • Linus Pauling, E. Bright Wilson, E. Bright Wilson(Authors)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  Hamiltonian function .

  2b. The Hamiltonian Function and Equations. —For conservative systems1 we shall show that the function H is the total energy (kinetic plus potential) of the system, expressed in terms of the

  pk

  's and

  qk

  's. In order to have a definition which holds for more general systems, we introduce H by the relation

  Although this definition involves the velocities k , H may be made a function of the coordinates and momenta only, by eliminating the velocities through the use of Equation 2–5 . From the definition we obtain for the total differential of H the equation

  or, using the expressions for

  pk

  and k given in Equations 2–5 and 2–6 (equivalent to Lagrange's equations),

  whence, if H is regarded as a function of the

  qk

  's and

  pk

  's, we obtain the equations

  These are the equations of motion in the Hamiltonian or canonical form .

  2c. The Hamiltonian Function and the Energy. —Let us consider the time dependence of H for a conservative system. We have

  using the same substitutions for

  pk

  and k (Eqs. 2–5 and 2–6) as before. H is hence a constant of the motion, which is called the energy of the system. For Newtonian systems, in which we shall be chiefly interested, the Hamiltonian function is the sum of the kinetic energy and the potential energy,

  expressed as a function of the coordinates and momenta. This is proved by considering the expression for T for such systems. For any set of coordinates, T will be a homogeneous quadratic function of the velocities

  where the

  aij

  's may be functions of the coordinates.

  Hence so that 2d. A General Example.

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  A Student's Guide to Analytical Mechanics

  • John L. Bohn(Author)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  This should look pretty familiar. We have already seen, a couple times, the change in sign of the kinetic energy if you take gradients at fixed velocity versus fixed momentum. Here it is again, in the more general context. And now – finally – we use a dynamical fact, not just a property of functions. From the Lagrange equations we started with, ∂ L /∂ q a is the time derivative of the momentum. We conclude that the equation of motion for the momentum, one of our goals, is now achieved: dp a dt = − ∂ H ∂ q a . And the other equation works the same way. Look at the variation of H when one of the momenta p a changes. Equating the terms in ( 6.20 ) and ( 6.21 ), we get dq a dt = ∂ H ∂ p a . 6.2 Moving Coordinate Systems 123 I know I said we needed to regard ˙ q a as a function of momenta for the expression of H to make sense, but look, if somebody hands us a differential equation for the very thing we’re looking for, as a first derivative with respect to time, we’ll take it. The function H that effects the transformation from velocities to momenta is called, as you might have anticipated, the Hamiltonian. For completeness, we rewrite here Hamilton’s equations of motion, derived now in full generality: dp a dt = − ∂ H ∂ q a dq a dt = ∂ H ∂ p a ∂ H ∂ t = − ∂ L ∂ t . These equations are succinct enough that you could have them tattooed on your arm or printed on a T-shirt (although Maxwell’s equations seem to be more popular for this purpose). 6.2.3 Examples The Legendre transformation is a very different way of achieving Hamilton’s equations than the explicit, matrix-based derivation we used in the simpler case of the nonmoving coordinate system. But the latter is there, as a special case of the former.

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  Dynamics for Engineers

  • Soumitro Banerjee(Author)
  • 2005(Publication Date)
  • Wiley

    (Publisher)

  (5.15) Such symmetry in the form of dynamical equations has prompted generations of scientists to probe these systems in detail, and through this a body of knowledge has emerged. Dynamical systems where these equations hold are called Hamiltonian systems. We are, however, not much interested in such systems that are almost non-existent in engineering, and will use differential equations of the form ˙ q i = ∂H ∂p i , (5.16) ˙ p i = − ∂H ∂q i − ∂ ∂ ˙ q i . (5.17) Example 5.3 Let us take the system in the Example 5.2 and derive the system equations by the Hamiltonian method. H = T + V, = 1 2 L 1 ( ˙ q 1 − ˙ q 2 ) 2 + 1 2 L 2 ˙ q 2 2 + 1 2C q 2 1 − q 1 E, = 1 2 L 1 p 1 L 1 2 + 1 2 L 2 p 1 + p 2 L 2 2 + 1 2C q 2 1 − q 1 E, = 1 2L 1 p 2 1 + 1 2L 2 (p 1 + p 2 ) 2 + 1 2C q 2 1 − q 1 E. Hence, the Hamiltonian equations are ˙ q 1 = ∂H ∂p 1 = p 1 L 1 + p 1 + p 2 L 2 , ˙ q 2 = ∂H ∂p 2 = p 1 + p 2 L 2 , ˙ p 1 = − ∂H ∂q 1 − ∂ ∂ ˙ q 1 = − q 1 C + E, ˙ p 2 = − ∂H ∂q 1 − ∂ ∂ ˙ q 2 = −R ˙ q 2 = − R L 2 (p 1 + p 2 ). Note that these equations are the same as those derived in Example 5.2. 92 OBTAINING FIRST-ORDER EQUATIONS E C R L 0 1 2 q 1 q 2 Figure 5.2 The circuit pertaining to Example 5.4. Example 5.4 The simple circuit of Fig. 5.2 was taken up in Chapter 3. If we want to obtain the first-order equations by the Hamiltonian method, we note L = 1 2 L ˙ q 2 1 − 1 2C (q 1 − q 2 ) 2 + Eq 1 and = 1 2 R ˙ q 2 2 . Therefore, p 1 = ∂ L ∂ ˙ q 1 = L ˙ q 1 or ˙ q 1 = p 1 /L, and p 2 = ∂ L ∂ ˙ q 2 = 0. The Hamiltonian function can then be expressed as H = 1 2 L ˙ q 2 1 + 1 2C (q 1 − q 2 ) 2 − Eq 1 , = 1 2L p 2 1 + 1 2C (q 1 − q 2 ) 2 − Eq 1 . In terms of this Hamiltonian function, ˙ q 1 = ∂H ∂p 1 = p 1 /L, (5.18) and since p 2 = 0, ∂H/∂p 2 cannot be evaluated. The momentum equations are ˙ p 1 = − ∂H ∂q 1 − ∂ ∂ ˙ q 1 = − 1 C (q 1 − q 2 ) + E, (5.19) ˙ p 2 = − ∂H ∂q 2 − ∂ ∂ ˙ q 2 = 1 C (q 1 − q 2 ) − R ˙ q 2 . OBTAINING FIRST-ORDER EQUATIONS 93 Since p 2 = 0, ˙ p 2 = 0.

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  Classical Dynamics of Particles and Systems

  • Jerry B. Marion(Author)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  It is just this fact that makes Hamilton's Principle useful for quantum-mechanical systems in which the forces are sometimes not known but the energies are known. The differential statement of mechanics contained in Newton's equations or the integral statement embodied in Hamilton's Principle (and the result-ing Lagrangian equations) have been shown to be entirely equivalent. Hence, there can be no distinction between these viewpoints which are based on the description of physical effects. From a philosophical standpoint, however, it is possible to make a distinction. In the Newtonian formulation, * Lagrange's equations, 1788; Hamilton's Principle, 1834. 234 9 · HAMILTON'S PRINCIPLE a certain force on a body is considered to produce a definite motion ; that is, there are associated a certain cause and a definite effect. According to Hamilton's Principle, however, the motion of a body may be considered to result from the attempt of Nature to achieve a certain purpose, namely, to minimize the time integral of the difference between the kinetic and potential energies. Clearly, the operational solving of problems in mechanics is not dependent on adopting one or the other of these views, but historically such considerations had a profound influence on the development of dynamics (as, for example, in Maupertuis' principle, mentioned in Section 9.2). The interested reader is referred to Margenau's excellent book (Ma50, Chapter 19) for a discussion of these matters. 9.8 A Theorem Concerning the Kinetic Energy If the kinetic energy is expressed in fixed, rectangular coordinates, the result is a homogeneous, quadratic function of the x ai : Τ = ΪΣ Σ m A (9-26) a = 1 i= 1 We now wish to determine the dependence of T on the generalized co-ordinates and velocities.

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  Advanced Theoretical Mechanics

  A Course of Mathematics for Engineers and Scientists

  • Brian H. Chirgwin, Charles Plumpton(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  11:2 Hamilton's principle We consider a holonomic system specified by n generalised coordinates α ι> α 2> --> α η· For our present purposes it is useful to represent the con-figuration of the system by a point in an ^-dimensional configuration space in which q 1 ,q 2 , ···,#/, a r e ^ n e coordinates of the point. All pos-456 Ml:2| ANALYTICAL DYNAMICS 457 sible states of the system are represented by a set of points and as the system evolves with time the representative point traces out a curve, called a trajectory, in the space. Since, in general, the coordinates q :i are continuous functions of the time this curve is continuous and, if we rule out impulsive changes of velocity, the curve is also smooth since the velocities q t are continuous. A given system, then, in its motion between the instants l ± and i 2 follows some curve MN in the configuration space, M representing the initial configuration and N the final configuration. We can join the points M and N by curves other than the natural trajectory; this cor-responds to varying the motion of the system e.g., by the imposition of suitable constraints. (The reader will recall the similar situation with comparison motions in Chapter VIII.) Hamilton's principle states that the integral S = f'Ldt, (11.1) where L is the Lagrangian function L(q li q 2 , ...,#„, q l9 q i9 ···, q n , t) of the given system, has a stationary value if the trajectory is varied slightly from the natural trajectory. The variations considered must all correspond to motions leaving M at time t x , and reaching N at time t 2 . This principle is written concisely t 2 OS = δ flat = 0, (11.2) the symbol ô standing for the variations of the trajectory. When we use Hamilton's principle to find the motion of the system we must find which of all the trajectories joining M and N gives S a sta-tionary value. The mathematical problem is that of determining n functions q t (t) which, when substituted into L, give S a stationary value.

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  Modelling of Mechanical Systems: Discrete Systems

  • Francois Axisa(Author)
  • 2003(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  Chapter 3

  Hamilton’s principle and Lagrange’s equations of unconstrained systems

  One of the most famous fundamental principles of theoretical physics is certainly the law of least action enunciated as a universal principle for the first time by Maupertuis (1746): “when a change occurs in nature, the quantity of action necessary for the change is the least possible”. Here, we shall introduce the exact formulation of this principle as made by Hamilton (1834), which states that the actual motion of a mechanical system between two arbitrarily fixed times t 1 , t 2 makes stationary the time integral over t 1 , t 2 of the extended Lagrangian of the system. Since this integral is suitably identified with the action of the Lagrangian, Hamilton’s principle is thus a principle of stationary action, historically understood as a principle of least action. It can be either postulated as a first principle, or derived from the principle of virtual work. One interesting point for applications in mechanics is that it allows one to introduce kinetic energy in a quite natural way through the Lagrangian. Furthermore, starting from it, Lagrange’s equations can be established by using a few mathematical procedures which are also of major interest in other problems in mechanics, which deal with continuous material systems provided with boundaries.

  3.1 Introduction

  In the preceding chapter, Lagrange’s equations of discrete and unconstrained mechanical systems were formulated starting from the variational principle of virtual work. The latter may be considered as being differential in nature, because it deals with variations that are taken at a fixed time t .

  Here, we shall introduce another variational principle, where we consider the actual motion of the system between two arbitrarily fixed times t 1 , t 2 and small virtual variations about it, see Figure 3.1 . This is the principle of least action in its exact formulation given by Hamilton in 1834 and widely known as Hamilton’s principle,

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  Quantum Mechanics: A Complete Introduction: Teach Yourself

  • Alexandre Zagoskin(Author)
  • 2015(Publication Date)
  • Teach Yourself

    (Publisher)

  . Remember that when we take a partial derivative with respect to one independent variable, we keep all others constant. Then indeed

  Moreover, for any function f of x, px (e.g. a probability distribution function) its time derivative will be, according to the chain rule and the Hamilton’s equations, a Poisson bracket of this function with the Hamilton function:

  If there are many pairs of conjugate variables, the formula remains the same. Only the Poisson bracket will now be the sum of contributions from all these pairs.

  The important physical insight from this equation is that the Hamilton function – that is, energy – determines the time evolution of any physical system. Time and energy turn out to be related in classical mechanics, almost like (but not quite) the position and momentum.

  What has this to do with quantum mechanics? A while ago Werner Heisenberg found out that quantum equations of motion for operators (Heisenberg equations) have the following form:1

  In particular, . Here the brackets denote the commutator of two operators.

  Key idea: Rotations

  A vector r can be rotated to become vector r’. This operation can be written as the action of the rotation operator Ô on vector r:

  It transforms the old coordinates (e.g. (x,y,z)) into the new ones (x’,y’,z’).

  This is a particularly simple operation in a two-dimensional space; a planar rotation by an angle α. Then the vector r can be represented as a column of two numbers, x and y, and the operator Ô as a two-by-two matrix.

  The Hamiltonian operator Ĥ is the quantum operator, which corresponds to the classical Hamilton function, i.e. the energy of the system. For example, for a harmonic oscillator it is simply

  One of the great insights of Paul Dirac was that these Heisenberg equations of motion are directly analogous to classical equations of motion written using Poisson brackets. One only needs to replace the classical functions and variables with quantum operators, and instead of a classical Poisson bracket put in the commutator divided by iħ:2

  And voilà! You have a quantum equation of motion. This is a much more consistent scheme of quantization than the one of Bohr–Sommerfeld.

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  Mechanical Vibration

  Analysis, Uncertainties, and Control, Fourth Edition

  • Haym Benaroya, Mark Nagurka, Seon Han(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  Path and varied path for Hamilton’s principle. Note that here, unlike in Figure 5.2, the paths are functions of time. since δ r i = 0 at t1 and t2 because there are no variations at t 1 and t 2 Therefore, (5.30) ∫ t 1 t 2 (δ T + δ W) d t = 0 This is called the extended Hamilton’s principle with the term extended implying that includes both conservative and nonconservative work. If the forces are only conservative, then = and we find Hamilton’s principle, (5.31) δ ∫ t 1 t 2 (T - V) d t = 0. This equation may be interpreted as nature trying to equalize the kinetic and potential energies of a system, in absence of dissipation. Lagrange’s equation can also be derived beginning with Hamilton’s principle. To see this, for each generalized coordinate in Equation 5.30, we vary T (q i, q i)) (5.31) δ T = ∑ i = 1 N δ T δ q i δ q i + ∑ i = 1 N δ T δ q ˙ i δ q ˙ i. Letting δ q ˙ i = d (δ q i) d t and integrating by parts, the i th component of the second term in Equation. 5.31, ∫ t 1 t 2 ∂ T ∂ q ˙ i δ q i d t d t = ∂ T ∂ q ˙ i δ q i | t 1 t 2 - ∫ t 1 t 2 ∂ T ∂ q ˙ i d t, where ∂ T ∂ q ˙ i δ q i | t 1 t 2 = 0 at the end times. Equation 5.30 then. becomes (5.32) ∫ t 1 t 2 ∑ i = 1 N ∂ T ∂ q i - d d t ∂ T ∂ q ˙ i + Q i δ q i d t = 0, where δ W = - ∑ i (∂ V / ∂ q i â € “Q i n c) δ q i ≡ ∑ i Q i δ q i. Since all the δq i are arbitrary except at the end times, for each i in Equation. 5.32 the expression within the braces equals zero. This gives us Lagrange’s equation, one for each generalized coordinate, as in Equation 5.26. Example 5.14 Hamilton’s Principle for the Derivation of the Equations of Motion of an Elastic Pendulum Derive the equation of motion of the spring‐pendulum system depicted in Figure 5.19. Demonstrate the utility of Hamilton’s principle for the derivation of the equations of motion of complex dynamic systems. The system is assumed to have no dissipation. Figure 5.19

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