Physics
Hamiltonian Mechanics
Hamiltonian mechanics is a formulation of classical mechanics that describes the evolution of a physical system over time. It is based on the concept of the Hamiltonian, which is a function that encapsulates the system's energy and dynamics. Unlike the more traditional Lagrangian mechanics, Hamiltonian mechanics emphasizes the use of generalized coordinates and momenta to describe the system's behavior.
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12 Key excerpts on "Hamiltonian Mechanics"
- Sow-Hsin Chen, Michael Kotlarchyk;;;(Authors)
- 2007(Publication Date)
- WSPC(Publisher)
Chapter 2 AN OVERVIEW OF CLASSICAL MECHANICS Theprimarymotivationforthischapteristolaythegroundworknecessaryformaking connections between the classical and quantum-mechanical description of a physical system. In particular, the dynamic description of mechanical systems in terms of La-grangian and Hamiltonian Mechanics is presented, with a special emphasis on oscilla-tors. It is only by introducing the so-called canonicalcoordinates and by representing the total energy in Hamilton’s form that one can ultimately make the transition from a classical to a quantum mechanical description of both atomic systems and the interacting electromagnetic radiation field. 2.1 The Lagrangian Formulation In many simple mechanical systems, the most natural way to obtain the equations of motion is to apply Newton’s Second Law to each particle, namely ˙ p i = F i , (2.1) which relates the force F i acting on the i th particle in the system to the rate of change * of the linear momentum p i = m i v i of that particle, where v i is the velocity of particle i . In most cases, the mass m i of each particle is fixed and Newton’s Second Law reduces to the more familiar form F i = m a i , (2.2) where a i = ˙ v i is the acceleration of a given particle in the system. Consider, for example, the simple case of a single particle under the influence of a linear restoring force along the x -axis. The equation of motion is that for a simple harmonic oscillator, i.e., m ¨ x = -Kx, (2.3) where K is a positive (spring) constant. Alternatively, this can be written as d dt parenleftbigg 1 2 m ˙ x 2 + 1 2 Kx 2 parenrightbigg =0 , (2.4) * Each dot above a letter indicates a time derivative. 6 An Overview of Classical Mechanics which means that the expression in the parentheses remains fixed (or conserved) as the system evolves in time. This constant of the motion can be defined as E , the total energy of the system: E = 1 2 m ˙ x 2 + 1 2 Kx 2 .
eBook - PDF- Antonio Giorgilli(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
A major role in the development of the theory of classical dynamical sys- tems is played by the Hamiltonian formulation of the equations of dynamics. This chapter is intended to provide a basic knowledge of the Hamiltonian formalism, assuming that the Lagrangian formalism is known. A reader al- ready familiar with the canonical formalism may want to skip the present chapter. The canonical equations were first written by Giuseppe Luigi Lagrangia (best known by the French version of his name, Joseph Louis Lagrange) as the last improvement of his theory of secular motions of the planets [140]. The complete form, later developed in what we now call Hamiltonian formalism, is due to William Rowan Hamilton [106][107][108]. A short sketch concerning the anticipations of Hamilton’s work can be found in the treatise of Edmund Taylor Whittaker [209], §109. In view of the didactical purpose of the present notes, the exposition in this chapter follows the traditional lines. The chapter includes some basic tools: the algebra of Poisson brackets and the elementary integration meth- ods. Many examples are also included in order to illustrate how to write the Hamiltonian function for some models, often investigated using Newton’s or Lagrange’s equations. 1.1 Phase Space and Hamilton’s Equations The dynamical state of a system with n degrees of freedom is identified with a point on a 2n-dimensional differentiable manifold, denoted by F , endowed with canonically conjugated coordinates (q, p) ≡ (q 1 , . . . , q n , p 1 , . . . , p n ). The object of investigation is the evolution of the state of the system. The man- ifold F was named a phase space by Josiah Willard Gibbs [75]. The evolution of the system is determined by a real-valued Hamiltonian function H : F → R through the vector field defined by Hamilton’s equations (also called canonical equations), 2 Hamiltonian Formalism (1.1) ˙ q j = ∂H ∂p j , ˙ p j = - ∂H ∂q j , j = 1, .
Available until 25 Jan |Learn more- 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 .
eBook - PDF- John H. Lowenstein(Author)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
1 Fundamentals of classical dynamics In this first chapter, we review the fundamentals and rather quickly introduce the central analytical and geometrical concepts of the Hamiltonian approach. We then illustrate these ideas in the context of a number of familiar examples. 1.1 Newtonian mechanics The physical system we will be dealing with throughout this book consists of N pointlike particles with masses m k and position vectors x k = (x k , y k , z k ), con- strained at each time t by C independent equations f j (x 1 , x 2 , . . . , x N , t ) = 0, j = 1, 2, . . . , C, (1.1) and moving under the influence of a potential-energy function V (x 1 , x 2 , . . . , x N , t ). Such a system is said to have n degrees of freedom, where n = 3 N − C. According to the principles of Newtonian mechanics, the instantaneous state of the system is prescribed by a 3 N -dimensional vector X and its time derivative, X = (x 1 , x 2 , . . . , x N ), ˙ X = ( ˙ x 1 , ˙ x 2 , . . . , ˙ x N ), (1.2) which evolve in time according to the system of second-order differential equations (k = 1, 2, . . . , N ) m k .. x k = total force on k th particle = −∇ k V (x 1 , x 2 , . . . , x N , t ) + constraint forces. (1.3) Given the state of the system at any time t = t 0 , the equations (1.3) determine the state for all later times. We will hardly ever be interested in the explicit representa- tion of the forces of constraint, so in (1.3) we suppress the details. In fact, we will very quickly adopt a more efficient dynamical formalism in which the constraints 1 2 Fundamentals of classical dynamics are completely absent from the equations of motion. In those rare instances in which we are curious about the constraints, we will always be able to calculate them by first determining x k (t ), k = 1, . . . , N , and then solving (1.3) for the forces. 1.2 Configuration space Unless otherwise specified, our constraint functions f j (X, t ), X ∈ R 3 N , j = 1, .
eBook - PDF- Dietrich Marcuse(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
C H A P T E R R E V I E W O F Q U A N T U M M E C H A N I C S 1.1 H a m i l t o n i a n M e c h a n i c s Formulation of Hamiltonian Mechanics The transition from classical mechanics to quantum mechanics can be made most conveniently and easily by using the Hamiltonian formulation of classical mechanics. Before giving a brief outline of quantum mechanics we, therefore, review a few of the most important equations of the classical Hamiltonian Mechanics. The methods of classical mechanics and quantum mechanics are vastly different. Classical mechanics is based on the assumption that any physically interesting variable connected with a particle, such as its position, its velocity, or its energy, can be measured with arbitrary precision and without mutual interference from any other such measurement. Classical mechanics, there-fore, uses sets of variables and functions of these variables to enable us to predict the behavior of physical systems by providing us with differential equations that determine the changes of these functions in space and time. Quantum mechanics is based on the realization that the measuring process may affect the physical system. It is, therefore, impossible in principle to measure simultaneously certain pairs of variables with arbitrary precision. The measurement of one variable affects other variables in such a way that it prevents us from knowing what their values might have been. The mathe-matical formulation of the laws of physics that takes this basic idea into account is very different from the mathematical formulation of classical mechanics, as we shall see later in this chapter. The laws of classical mechanics can be expressed in various mathematical forms. The simplest formulation is based upon Newton's law stating that the 1 2 Review of Quantum Mechanics CHAPTER ONE force acting on a body is equal to the product of its mass times its acceleration.
eBook - PDF- Mohsen Razavy(Author)
- 2011(Publication Date)
- World Scientific(Publisher)
Chapter 1 A Brief Survey of Analytical Dynamics 1.1 The Lagrangian and the Hamilton Principle We can formulate the laws of motion of a mechanical system with N degrees of freedom in terms of Hamilton’s principle. This principle states that for every motion there is a well-defined function of the N coordinates q i and N velocities ˙ q i which is called the Lagrangian, L , such that the integral S = Z t 2 t 1 L ( q i , ˙ q i , t ) dt, (1.1) takes the least possible value (or extremum) when the system occupies positions q i ( t 1 ) and q i ( t 2 ) at the times t 1 and t 2 [1],[2]. The set of N independent quantities { q i } which completely defines the position of the system of N degrees of freedom are called generalized coordinates and their time derivatives are called generalized velocities. The requirement that S be a minimum (or extremum) implies that L must satisfy the Euler–Lagrange equation ∂L ∂q i -d dt ∂L ∂ ˙ q i = 0 , i = 1 , · · · N. (1.2) The mathematical form of these equations remain invariant under a point trans-formation. Let us consider a non-singular transformation of the coordinates from the set of N { q i } s to another set of N { Q i } s given by the equations Q i = Q i ( q 1 , · · · , q N ) , i = 1 , · · · N, (1.3) 1 2 Heisenberg’s Quantum Mechanics and its inverse transform given by the N equations q j = q j ( Q 1 , · · · , Q N ) , j = 1 , · · · N. (1.4) Now let F ( q 1 , · · · , q N , ˙ q 1 , · · · , ˙ q N ) be a twice differentiable function of 2 N vari-ables q 1 , · · · , q N , ˙ q 1 , · · · , ˙ q N . We note that this function can be written as a function of Q j s and ˙ Q j s if we replace q i s and ˙ q i s by Q j s and ˙ Q j s using Eq. (1.4). Now by direct differentiation we find that ∂ ∂q i -d dt ∂ ∂ ˙ q i F q i ( Q j ) , ˙ q i ( Q j , ˙ Q j ) = N X j =1 ∂Q j ∂q i ∂ ∂Q j -d dt ∂ ∂ ˙ Q j ! F q i ( Q j ) , ˙ q i ( Q j , ˙ Q j ) , i = 1 , · · · N. (1.5) Thus if L ( Q 1 , · · · ˙ Q N ) has a vanishing Euler–Lagrange derivative i.e.
eBook - PDF- Yves Talpaert(Author)
- 2012(Publication Date)
- De Gruyter(Publisher)
306 Chapter 3 where A = l — du and so this case is related to the previous study. Remark. We mention that if L does not explicitly depend on t (that is, the one-parameter group of time translations
- Patrick Hamill(Author)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
A particle of mass m is attached to the lower end of the spring. Determine the Hamiltonian of the system. 4.10 A particle of mass m can slide freely on a horizontal rod of mass M. One end of the rod is attached to a turntable and moves in a circular path of radius a at angular speed ω. Write the Hamiltonian for the system. 5 Canonical transformations; Poisson brackets In this chapter we begin by considering canonical transformations. These are transformations that preserve the form of Hamilton’s equations. This is fol- lowed by a study of Poisson brackets, an important tool for studying canonical transformations. Finally we consider infinitesimal canonical trans- formations and, as an example, we look at angular momentum in terms of Poisson brackets. 5.1 Integrating the equations of motion In our study of analytical mechanics we have seen that the variational prin- ciple leads to two different sets of equations of motion. The first set consists of the Lagrange equations and the second set consists of Hamilton’s canonical equations. Lagrange’s equations are a set of n coupled second-order differ- ential equations and Hamilton’s equations are a set of 2n coupled first-order differential equations. The ultimate goal of any dynamical theory is to obtain a general solution for the equations of motion. In Lagrangian dynamics this requires integrating the equations of motion twice. This is often quite difficult because the Lagrangian (and hence the equations of motion) depends not only on the coordinates but also on their derivatives (the velocities). There is no known general method for integrating these equations. 1 You might wonder if it is possible to trans- form to a new set of coordinates in which the equations of motion are sim- pler and easier to integrate.
eBook - PDF- Nivaldo A. Lemos(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
As to (7.10), it is not an equation of motion but an important relationship between the explicit time dependence of the Lagrangian and the Hamiltonian. The first half of Hamilton’s equations expresses the generalised velocities in terms of the canonical variables, which means that they are the inverses of the equations (7.2) that define the canonical momenta p i . It must be stressed, however, that contrarily to the Lagrangian formulation in which there holds the a priori connexion ˙ q i = dq i /dt, in Hamilton’s dynamics there is no a priori connexion among the canonical variables: the qs and ps are entirely independent among themselves. This is why the two halves of Hamilton’s equations (7.9) must be treated on equal footing, comprising the complete set of equations of motion for the system. Except for certain cases in which the Hamiltonian can be written down directly, the construction of Hamilton’s equations involves the following steps: 2 The word “canonical” is used in the sense of “standard”. 218 Hamiltonian Dynamics (a) Choose generalised coordinates and set up the Lagrangian L(q, ˙ q, t). (b) Solve Eqs. (7.2) for the velocities ˙ q i as functions of q, p, t. (c) Construct H(q, p, t) by substituting into (7.5) the ˙ qs obtained in (b). (d) Once in possession of H(q, p, t), write down the equations of motion (7.9). Hamiltonian and the Total Energy The usual Lagrangian is L = T − V . If (1) T is a purely quadratic function of the velocities and (2) V does not depend on the velocities then Euler’s theorem on homogeneous functions (Appendix C) yields ∑ i ˙ q i ∂ L/∂ ˙ q i = ∑ i ˙ q i ∂ T /∂ ˙ q i = 2T . As a consequence H = T + V = E , (7.11) that is, the Hamiltonian is the total energy expressed as a function of the coordinates and momenta. Conditions (1) and (2) prevail in the vast majority of physically interesting cases, so the Hamiltonian has the extremely important physical meaning of being the total energy in most situations of physical relevance.
- W. D. Curtis, F.R Miller(Authors)
- 1985(Publication Date)
- Academic Press(Publisher)
2 C I assica I M ec ha n ics As discussed in Chapter 1, we want to develop Lagrangian mechanics and generalized coordinates (i.e., manifold theory). The purpose of this chapter is to introduce the reader to some of the basic ingredients of Lagrangian and Hamiltonian Mechanics for the special case when configuration space is an open set M in Euclidean space R and state space is M x R. We will gen- eralize these constructions to the manifold setting later. For now, we also restrict attention to time-independent systems. We begin with Newton's law of motion and show how to rewrite it so as to get Lagrange's equations. We then introduce phase space, defined to be M x (R)*, and obtain Hamilton's equations. We obtain vector fields on state and phase space whose integral curves are the trajectories of the system. We see that, in the Hamiltonian form, the vector field is obtained in a very simple way from the energy function. The trajectories of the Lagrangian and Hamil- tonian systems are related by the Legendre transformation. These construc- tions easily generalize to the manifold setting of Chapter 7. MECHANICS OF MANY-PARTICLE SYSTEMS Consider a system of n particles moving in space. Conjiguration space is a subset M c R3. A state is a pair (4, u) E M x R3, q = (q', . . . , q3) rep- resenting position of the n particles and u = (d, . . . , u3) representing ve- locities. If the masses of the particles are m,, . . . , m,, define M i , i = 1, . . . , 3n, by M 3 i -2 = M3i-1 = M 3 i = mi. This is just convenient notation. 5 6 2. CLASSICAL MECHANICS EXAMPLE 2.1 (Two-body problem) Consider a system of two parti- cles of masses m, and m, moving freely in space. If q, = (ql, q2, q3), q, = (q4, q5, q6) are positions, then we define configuration space to be M = {(ql, q2) E R~ x R ~ I ~ , z q,). The requirement q1 # q2 means that our model does not describe collisions. Given a position q E M the velocities are arbitrary so our state space is S = M x R6.
eBook - PDF- Michael T. Vaughn(Author)
- 2008(Publication Date)
- Wiley-VCH(Publisher)
In terms of ( X, P ) , the Hamiltonian is H = 1 2 ω ( P 2 + X 2 ) (3.272) 148 3 Geometry in Physics We can further introduce variables J, α by X ≡ √ 2 J sin α P ≡ √ 2 J cos α (3.273) corresponding to J = 1 2 ( P 2 + X 2 ) tanα = P X (3.274) In terms of these variables, the Hamiltonian is given simply by H = ωJ (3.275) The variables J, α are action-angle variables. Since the Hamiltonian (3.275) is indepen-dent of the angle variable α , the conjugate momentum J (the action variable ) is a constant of the motion, and ˙ α = ∂H ∂J = ω (3.276) Thus the motion in the phase space defined by the variables ( X, P ) is a circle of radius √ 2 J , with angular velocity given by ˙ α = ω . Note that for the special case of simple harmonic oscillator, the angular velocity ˙ α is independent of the action variable. This is not true in general (see Problem 20). Exercise 3.22. Show that the transformation to action-angle variables is canonical, i.e., show that dJ ∧ dα = dP ∧ dX Then explain the choice of √ 2 J , rather than some arbitrary function of J , as the “radius” variable in Eq. (3.273). 3.6 Fluid Mechanics A real fluid consists of a large number of atoms or molecules whose interactions are suffi-ciently strong that the motion of the fluid on a macroscopic scale appears to be smooth flow superimposed on the thermal motion of the individual atoms or molecules, the thermal motion being generally unobservable except through the Brownian motion of particles introduced into the fluid. An ideal fluid is characterized by a mass density ρ = ρ ( x, t ) and a velocity field u = u ( x, t ) , as well as thermodynamic variables such as pressure p = p ( x, t ) and temperature T = T ( x, t ) . If the fluid is a gas, then it is often important to consider the equation of state relating ρ , p , and T . For a liquid, on the other hand, it is usually a good approximation to treat the density as constant ( incompressible flow ).
- Kevin W. Cassel(Author)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
This distinction is largely philosophical as it is the differential form that is typically solved to obtain the stationary function of Hamilton’s principle; however, quantum theory, in the form of Feynman’s path-integral, offers a possible resolution (see Section 8.2.3). Implied by the above description of Hamilton’s principle and Newton’s second law is an important fundamental distinction between variational and differential forms of the governing equations that deserves highlighting. The differential form, in the form of the Euler equation, is posed as a local law that enforces the physical principle for each rigid particle in the system at each point within the domain. On the other hand, the variational form of Hamilton’s principle is posed as a global law that enforces the physical principle globally over the entire system or domain. Although the two forms are certainly equivalent mathematically, the global variational form leads to a greater intuitive understanding of the physical principle involved and provides a platform for determining certain properties of the physical 122 4 Hamilton’s Principle laws (for example, see Noether’s theorem in Section 4.7), while the local differential form is typically the one used to actually obtain solutions. Hamilton’s principle in the form (4.1), (4.5), or (4.6) may be extended to systems having multiple discrete particles through summation. For a system of N particles (point masses), the kinetic energy of the moving system is T = N k=1 T k = N k=1 1 2 m k ˙ r 2 k = N k=1 1 2 m k v 2 k . For conservative forces, the potential energy, which is the negative of the virtual work required to move the particles, is V = N k=1 V k = N k=1 δV k = − N k=1 r k r 1k f ck · δr k , where r 1k are the position vectors for the system at the initial time t 1 , and r k are the position vectors at the current time t .
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