DOCSLIB.ORG

Generation of Solutions of the Hamilton–Jacobi Equation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Generation of Solutions of the Hamilton–Jacobi Equation RESEARCH Revista Mexicana de F´ısica 60 (2014) 75–79 JANUARY-FEBRUARY 2014 Generation of solutions of the Hamilton–Jacobi equation G.F. Torres del Castillo Departamento de F´ısica Matematica,´ Instituto de Ciencias Universidad Autonoma´ de Puebla, 72570 Puebla, Pue., Mexico.´ Received 17 September 2013; accepted 7 October 2013 It is shown that any function G(qi; pi; t), defined on the extended phase space, defines a one-parameter group of canonical transformations which act on any function f(qi; t), in such a way that if G is a constant of motion then from a solution of the Hamilton–Jacobi (HJ) equation one obtains a one-parameter family of solutions of the same HJ equation. It is also shown that any complete solution of the HJ equation can be obtained in this manner by means of the transformations generated by n constants of motion in involution. Keywords: Hamilton–Jacobi equation; canonical transformations; constants of motion. Se muestra que cualquier funcion´ G(qi; pi; t), definida en el espacio fase extendido, define un grupo uniparametrico´ de transformaciones canonicas´ las cuales actuan´ sobre cualquier funcion´ f(qi; t), de tal manera que si G es una constante de movimiento entonces a partir de cualquier solucion´ de la ecuacion´ de Hamilton–Jacobi (HJ) uno obtiene una familia uniparametrica´ de soluciones de la misma ecuacion´ de HJ. Se muestra tambien´ que cualquier solucion´ completa de la ecuacion´ de HJ puede obtenerse de esta manera por medio de las transformaciones generadas por n constantes de movimiento en involucion.´ Descriptores: Ecuacion´ de Hamilton–Jacobi; transformaciones canonicas;´ constantes de movimiento. PACS: 45.20.Jj; 02.30.Jr; 02.20.Qs 1. Introduction way that if G is a constant of motion, then a solution of the corresponding HJ equation is mapped into a one-parameter In the Hamiltonian formulation of classical mechanics, the family of solutions of this equation. In this manner, each canonical transformations, given locally by expressions of constant of motion adds a continuous parameter to a given the form Qi = Qi(qj; pj; t), Pi = Pi(qj; pj; t), act on the ex- solution of the HJ equation. We also show that any complete tended phase space (the Cartesian product of the phase space solution of the HJ equation can be obtained from a solution and the time axis) and, therefore, there is a naturally defined without parameters by means of the action of the groups of action of a canonical transformation on any function defined canonical transformations generated by n constants of mo- on the extended phase space (that is, on any function of qi, tion in involution. pi, and t). On the other hand, Hamilton’s principal function, Throughout this paper, several examples are given in or- usually denoted by an S, is a function defined on the extended der to illustrate the definitions and results presented here. configuration space (that is, a function of qi and t), just like the wave function in the elementary Schrodinger¨ equation, 2. The action of a one-parameter group of but there is no obvious way to define the action of a canoni- canonical transformations on functions de- cal transformation on a function of qi and t. fined on the extended configuration space However, in Ref. [1] it was shown that, under certain conditions, one can define an action of a time-independent We start by reviewing the HJ equation, which will give us the canonical transformation [that is, a canonical transforma- pattern to follow in the definition of the action of any one- tion that does not involve the time, Qi = Qi(qj; pj), parameter group of canonical transformations on functions Pi = Pi(qj; pj)] on functions defined on the configuration defined on the extended configuration space. space, in such a way that if a (time-independent) Hamilto- nian is invariant under the canonical transformation, a solu- 2.1. The Hamilton–Jacobi as an evolution equation tion of the corresponding time-independent Hamilton–Jacobi (HJ) equation is mapped into another solution. Furthermore, Given a Hamiltonian H(qi; pi; t) of a system with n degrees it was shown that the action of the one-parameter group of of freedom, the corresponding HJ equation is the first-order canonical transformations generated by an arbitrary function partial differential equation G(q ; p ) @S i i is determined by an equation similar to the HJ equa- H(q ; @S=@q ; t) + = 0: (1) tion. i i @t In this paper, we consider systems with a Hamiltonian Usually one is interested in complete solutions of the HJ that can depend on the time, we show that one can define the equation (1), that is, solutions S(qi; t; ®i) of Eq. (1), con- action of the one-parameter group of canonical transforma- taining n arbitrary parameters ®i, such that µ 2 ¶ tions generated by an arbitrary function G(qi; pi; t) on func- @ S det 6= 0; (2) tions defined on the extended configuration space, in such a @qi@®j 76 G.F. TORRES DEL CASTILLO because they yield the solution of the Hamilton equations The expression (7) is a (complete, R-separable) solution of (see, e.g., Refs. [2] and [3]). However, one can consider the the HJ equation that reduces to the specified function (5) for HJ equation as an evolution equation, which determines the t = 0. (A solution is R-separable if it is the sum of a function function S(qi; t) that reduces to a given function f(qi), for of two or more variables and functions of one variable.) t = 0 (or any other initial value, t0, of t). (This fact does not We close this subsection with a second example. The sound very strange if we consider the relationship between solution of the Hamilton equations for the time-dependent the HJ equation and the Schrodinger¨ equation.) In fact, if Hamiltonian p2 we have the solution of the Hamilton equations, we can use H = ¡ ktq; (8) it to find the solution of the HJ equation that satisfies any 2m initial condition S(qi; t0) = f(qi). (Similar ideas are often where m and k are constants, is given by mentioned in the standard textbooks on analytical mechanics, 3 tp0 kt 1 2 without presenting, however, a clear statement or an explicit q = q0 + + ; p = p0 + kt ; (9) procedure.) m 6m 2 For example, in the case of the Hamiltonian where q0 and p0 denote the values of q and p at t = 0, re- 2 2 spectively. Letting p1 + p2 H = + mgq2; (3) 2m S(q; 0; ®) = ®q; (10) where m and g are constants, the solution of the correspond- ® ing Hamilton equations can be readily obtained and is given where is an arbitrary constant, with the aid of Eqs. (9), we have ¯ by ¯ @S @S ¯ 1 2 1 2 = ¯ + kt = ® + kt : (11) (p ) t (p ) t gt2 @q @q t=0 2 2 q = (q ) + 1 0 ; q = (q ) + 2 0 ¡ ; 1 1 0 m 2 2 0 m 2 Thus, from Eqs. (1), (8), and (11), we have p = (p ) ; p = (p ) ¡ mgt; (4) µ ¶2 1 1 0 2 2 0 @S 1 @S = ¡ + ktq @t 2m @q where (q1)0 denotes the value of q1 at t = 0, and so on. As µ ¶ the initial condition we choose 1 1 2 = ¡ ® + kt2 + ktq: (12) 2m 2 S(q1; q2; 0) = ®1q1 + ®2q2; (5) Then, from Eqs. (10)–(12), we readily obtain where ® , ® are arbitrary constants. We want to find the 1 2 µ ¶ solution of the HJ equation [see Eqs. (1) and (3)] 1 1 1 k2t5 S(q; t; ®) = ®q + kt2q ¡ ®2t + ®kt3 + ; "µ ¶ µ ¶ # 2 2m 3 20 @S 1 @S 2 @S 2 = ¡ + ¡ mgq2; (6) R @t 2m @q1 @q2 which is an -separable complete solution of the HJ equa- tion. (We might start with expressions more complicated than that satisfies the initial condition (5). Recalling that (5) and (10), but these simple expressions are enough to ob- @S=@qi = pi, making use of Eqs. (4) and (5) we have tain complete solutions of the HJ equation.) ¯ It should be clear that a similar construction can be de- @S @S ¯ ¯ vised using an arbitrary function of (qi; pi; t) instead of a = ¯ = ®1; @q1 @q1 t=0 Hamiltonian. ¯ @S @S ¯ = ¯ ¡ mgt = ® ¡ mgt: 2.2. Families of solutions of the HJ equation and con- @q @q ¯ 2 2 2 t=0 stants of motion Substituting these expressions into the right-hand side of Any differentiable function, G(q ; p ; t), defines a (possibly Eq. (6) we have i i local) one-parameter group of canonical transformations, de- @S 1 £ ¤ termined by the (autonomous) system of first-order ordinary = ¡ ® 2 + (® ¡ mgt)2 ¡ mgq : @t 2m 1 2 2 differential equations Combining the last three equations one readily finds that, dqi @G(qj; pj; t) dpi @G(qj; pj; t) = ; = ¡ ; (13) choosing the integration constant so that Eq. (5) is satisfied, d® @pi d® @qi which are of the form of the Hamilton equations. S(q1; q2; t) = ®1q1 + ®2q2 ¡ mgtq2 For example, the function ® 2t (® ¡ mgt)3 ¡ ® 3 ¡ 1 + 2 2 : (7) p p 2m 6m2g G(q ; q ; p ; p ; t) = 1 2 + mgq ; (14) 1 2 1 2 m 1 Rev. Mex. Fis. 60 (2014) 75–79 GENERATION OF SOLUTIONS OF THE HAMILTON–JACOBI EQUATION 77 where m and g are constants, leads to the system of equations the examples of Sec. 2.1. Making use of Eqs. (16) and (19) we have [with S(qi; t; 0) = f(qi; t)] dq1 p2 dq2 p1 = ; = ; ¯ d® m d® m @S @S ¯ = ¯ ¡ mg® = ¡mg®; dp1 dp2 ¯ = ¡mg; = 0: (15) @q1 @q1 ®=0 d® d® ¯ ¯ @S @S ¯ From the last two equations we find that = ¯ = ®2 ¡ mgt: (20) @q2 @q2 ®=0 p = (p ) ¡ mg®; p = (p ) ; (16) 1 1 0 2 2 0 Substituting these expressions into Eq.
Load more

Recommended publications
  • 1 the Basic Set-Up 2 Poisson Brackets
    MATHEMATICS 7302 (Analytical Dynamics) YEAR 2016–2017, TERM 2 HANDOUT #12: THE HAMILTONIAN APPROACH TO MECHANICS These notes are intended to be read as a supplement to the handout from Gregory, Classical Mechanics, Chapter 14. 1 The basic set-up I assume that you have already studied Gregory, Sections 14.1–14.4. The following is intended only as a succinct summary. We are considering a system whose equations of motion are written in Hamiltonian form. This means that: 1. The phase space of the system is parametrized by canonical coordinates q =(q1,...,qn) and p =(p1,...,pn). 2. We are given a Hamiltonian function H(q, p, t). 3. The dynamics of the system is given by Hamilton’s equations of motion ∂H q˙i = (1a) ∂pi ∂H p˙i = − (1b) ∂qi for i =1,...,n. In these notes we will consider some deeper aspects of Hamiltonian dynamics. 2 Poisson brackets Let us start by considering an arbitrary function f(q, p, t). Then its time evolution is given by n df ∂f ∂f ∂f = q˙ + p˙ + (2a) dt ∂q i ∂p i ∂t i=1 i i X n ∂f ∂H ∂f ∂H ∂f = − + (2b) ∂q ∂p ∂p ∂q ∂t i=1 i i i i X 1 where the first equality used the definition of total time derivative together with the chain rule, and the second equality used Hamilton’s equations of motion. The formula (2b) suggests that we make a more general definition. Let f(q, p, t) and g(q, p, t) be any two functions; we then define their Poisson bracket {f,g} to be n def ∂f ∂g ∂f ∂g {f,g} = − .
    [Show full text]
  • Time-Dependent Hamiltonian Mechanics on a Locally Conformal
    Time-dependent Hamiltonian mechanics on a locally conformal symplectic manifold Orlando Ragnisco†, Cristina Sardón∗, Marcin Zając∗∗ Department of Mathematics and Physics†, Universita degli studi Roma Tre, Largo S. Leonardo Murialdo, 1, 00146 , Rome, Italy. ragnisco@fis.uniroma3.it Department of Applied Mathematics∗, Universidad Polit´ecnica de Madrid. C/ Jos´eGuti´errez Abascal, 2, 28006, Madrid. Spain. [email protected] Department of Mathematical Methods in Physics∗∗, Faculty of Physics. University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland. [email protected] Abstract In this paper we aim at presenting a concise but also comprehensive study of time-dependent (t- dependent) Hamiltonian dynamics on a locally conformal symplectic (lcs) manifold. We present a generalized geometric theory of canonical transformations and formulate a time-dependent geometric Hamilton-Jacobi theory on lcs manifolds. In contrast to previous papers concerning locally conformal symplectic manifolds, here the introduction of the time dependency brings out interesting geometric properties, as it is the introduction of contact geometry in locally symplectic patches. To conclude, we show examples of the applications of our formalism, in particular, we present systems of differential equations with time-dependent parameters, which admit different physical interpretations as we shall point out. arXiv:2104.02636v1 [math-ph] 6 Apr 2021 Contents 1 Introduction 2 2 Fundamentals on time-dependent Hamiltonian systems 5 2.1 Time-dependentsystems. ....... 5 2.2 Canonicaltransformations . ......... 6 2.3 Generating functions of canonical transformations . ................ 8 1 3 Geometry of locally conformal symplectic manifolds 8 3.1 Basics on locally conformal symplectic manifolds . ............... 8 3.2 Locally conformal symplectic structures on cotangent bundles............
    [Show full text]
  • Canonical Transformations (Lecture 4)
    Canonical transformations (Lecture 4) January 26, 2016 61/441 Lecture outline We will introduce and discuss canonical transformations that conserve the Hamiltonian structure of equations of motion. Poisson brackets are used to verify that a given transformation is canonical. A practical way to devise canonical transformation is based on usage of generation functions. The motivation behind this study is to understand the freedom which we have in the choice of various sets of coordinates and momenta. Later we will use this freedom to select a convenient set of coordinates for description of partilcle's motion in an accelerator. 62/441 Introduction Within the Lagrangian approach we can choose the generalized coordinates as we please. We can start with a set of coordinates qi and then introduce generalized momenta pi according to Eqs. @L(qk ; q_k ; t) pi = ; i = 1;:::; n ; @q_i and form the Hamiltonian ! H = pi q_i - L(qk ; q_k ; t) : i X Or, we can chose another set of generalized coordinates Qi = Qi (qk ; t), express the Lagrangian as a function of Qi , and obtain a different set of momenta Pi and a different Hamiltonian 0 H (Qi ; Pi ; t). This type of transformation is called a point transformation. The two representations are physically equivalent and they describe the same dynamics of our physical system. 63/441 Introduction A more general approach to the problem of using various variables in Hamiltonian formulation of equations of motion is the following. Let us assume that we have canonical variables qi , pi and the corresponding Hamiltonian H(qi ; pi ; t) and then make a transformation to new variables Qi = Qi (qk ; pk ; t) ; Pi = Pi (qk ; pk ; t) : i = 1 ::: n: (4.1) 0 Can we find a new Hamiltonian H (Qi ; Pi ; t) such that the system motion in new variables satisfies Hamiltonian equations with H 0? What are requirements on the transformation (4.1) for such a Hamiltonian to exist? These questions lead us to canonical transformations.
    [Show full text]
  • Classical Mechanics Examples (Canonical Transformation)
    Classical Mechanics Examples (Canonical Transformation) Dipan Kumar Ghosh Centre for Excellence in Basic Sciences Kalina, Mumbai 400098 November 10, 2016 1 Introduction In classical mechanics, there is no unique prescription for one to choose the generalized coordinates for a problem. As long as the coordinates and the corresponding momenta span the entire phase space, it becomes an acceptable set. However, it turns out in prac- tice that some choices are better than some others as they make a given problem simpler while still preserving the form of Hamilton's equations. Going over from one set of chosen coordinates and momenta to another set which satisfy Hamilton's equations is done by canonical transformation. For instance, if we consider the central force problem in two dimensions and choose Carte- k sian coordinate, the potential is − . However, if we choose (r; θ) coordinates, px2 + y2 k the potential is − and θ is cyclic, which greatly simplifies the problem. The number of r cyclic coordinates in a problem may depend on the choice of generalized coordinates. A cyclic coordinate results in a constant of motion which is its conjugate momentum. The example above where we replace one set of coordinates by another set is known as point transformation. In the Hamiltonian formalism, the coordinates and momenta are given equal status and the dynamics occurs in what we know as the phase space. In dealing with phase space dynamics, we need point transformations in the phase space where the new coordinates (Qi) and the new momenta (Pi) are functions of old coordinates (pi) and old momenta (pi): Qi = Qi(fqjg; fpjg; t) Pi = Pi(fqjg; fpjg; t) 1 c D.
    [Show full text]
  • PHYS 705: Classical Mechanics Canonical Transformation 2
    1 PHYS 705: Classical Mechanics Canonical Transformation 2 Canonical Variables and Hamiltonian Formalism As we have seen, in the Hamiltonian Formulation of Mechanics, qj, p j are independent variables in phase space on equal footing The Hamilton’s Equation for q j , p j are “symmetric” (symplectic, later) H H qj and p j pj q j This elegant formal structure of mechanics affords us the freedom in selecting other appropriate canonical variables as our phase space “coordinates” and “momenta” - As long as the new variables formally satisfy this abstract structure (the form of the Hamilton’s Equations. 3 Canonical Transformation Recall (from hw) that the Euler-Lagrange Equation is invariant for a point transformation: Qj Qqt j ( ,) L d L i.e., if we have, 0, qj dt q j L d L then, 0, Qj dt Q j Now, the idea is to find a generalized (canonical) transformation in phase space (not config. space) such that the Hamilton’s Equations are invariant ! Qj Qqpt j (, ,) (In general, we look for transformations which Pj Pqpt j (, ,) are invertible.) 4 Invariance of EL equation for Point Transformation First look at the situation in config. space first: dL L Given: 0, and a point transformation: Q Qqt( ,) j j dt q j q j dL L 0 Need to show: dt Q j Q j L Lqi L q i Formally, calculate: (chain rule) Qji qQ ij i qQ ij L L q L q i i Q j i qiQ j i q i Q j From the inverse point transformation equation q i qQt i ( ,) , we have, qi q q q q 0 and i i qi Q i Q i k j Q j Qj k Qk t 5 Invariance of EL equation
    [Show full text]
  • Classical Mechanics
    Classical Mechanics Joseph Conlon April 9, 2014 Abstract These are the current notes for the S7 Classical Mechanics course as of 9th April 2014. They contain the complete text and diagrams of the notes. There will be no more substantive revisions of these notes until Hilary 2014-15 (small typographical revisions may occur inbetween). Contents 1 Course Summary 3 2 Calculus of Variations 5 3 Lagrangian Mechanics 10 3.1 Lagrange's Equations . 10 3.2 Rotating Frames . 16 3.3 Normal Modes . 22 3.4 Particle in an Electromagnetic Field . 25 3.5 Rigid Bodies . 27 3.6 Noether's Theorem . 30 4 Hamiltonian Mechanics 33 4.1 Hamilton's Equations . 33 4.2 Hamilton's Equations from the Action Principle . 35 4.3 Poisson Brackets . 36 4.4 Liouville's Theorem . 39 4.5 Canonical Transformations . 42 5 Advanced Topics 50 5.1 The Hamilton-Jacobi Equation . 50 5.2 The Path Integral Formulation of Quantum Mechanics . 55 5.3 Integrable systems and Action/Angle Variables . 60 2 Chapter 1 Course Summary This course is the S7 Classical Mechanics short option (for physicists) and also the B7 Classical Mechanics option for those doing Physics and Philosophy. It consists of 16 lectures in total, and aims to cover advanced classical me- chanics, and in particular the theoretical aspects of Lagrangian and Hamiltonian mechanics. Approximately, the first 12 lectures cover material that is examinable for both courses, whereas the last four lectures (approximately) cove material that is examinable only for B7. General Comments Why be interested in classical mechanics and why be interested in this course? Classical mechanics has a beautiful theoretical structure which is obscured simply by a presentation of Newton's laws along the lines of F = ma.
    [Show full text]
  • PHY411 Lecture Notes Part 2
    PHY411 Lecture notes Part 2 Alice Quillen October 2, 2018 Contents 1 Canonical Transformations 2 1.1 Poisson Brackets . 2 1.2 Canonical transformations . 3 1.3 Canonical Transformations are Symplectic . 5 1.4 Generating Functions for Canonical Transformations . 6 1.4.1 Example canonical transformation - action angle coordinates for the harmonic oscillator . 7 2 Some Geometry 9 2.1 The Tangent Bundle . 9 2.1.1 Differential forms and Wedge product . 9 2.2 The Symplectic form . 12 2.3 Generating Functions Geometrically . 13 2.4 Vectors generate Flows and trajectories . 15 2.5 The Hamiltonian flow and the symplectic two-form . 15 2.6 Extended Phase space . 16 2.7 Hamiltonian flows preserve the symplectic two-form . 17 2.8 The symplectic two-form and the Poisson bracket . 19 2.8.1 On connection to the Lagrangian . 21 2.8.2 Discretized Systems and Symplectic integrators . 21 2.8.3 Surfaces of section . 21 2.9 The Hamiltonian following a canonical transformation . 22 2.10 The Lie derivative . 23 3 Examples of Canonical transformations 25 3.1 Orbits in the plane of a galaxy or around a massive body . 25 3.2 Epicyclic motion . 27 3.3 The Jacobi integral . 31 3.4 The Shearing Sheet . 32 1 4 Symmetries and Conserved Quantities 37 4.1 Functions that commute with the Hamiltonian . 37 4.2 Noether's theorem . 38 4.3 Integrability . 39 1 Canonical Transformations It is straightforward to transfer coordinate systems using the Lagrangian formulation as minimization of the action can be done in any coordinate system.
    [Show full text]
  • Infinitesimal Canonical Transformations. Symmetries And
    Infinitesimal Canonical Transformations. Symmetries and Conservation Laws. Physics 6010, Fall 2010 Infinitesimal Canonical Transformations. Symmetries and Conservation Laws. Relevant Sections in Text: x8.2, 8.3, 9.1, 9.2, 9.5 Infinitesimal Canonical Transformations With the notion of canonical transformations in hand, we are almost ready to explore the true power of the Hamiltonian formalism. One final ingredient needs to be developed: continuous canonical transformations and their infinitesimal form. Consider a family of canonical transformations that depend continuously upon a pa- rameter λ such that for some value of that parameter the transformation is the identity. We write i i q (λ) = q (q; p; λ); pi(λ) = pi(q; p; λ): Let us adjust our parameter such that λ = 0 is the identity: i i q (q; p; 0) = q ; pi(q; p; 0) = pi: As examples, think about the point transformations corresponding to rotations about an axis, or translations along a direction (exercise). Let us consider the transformations of the variables arising when the parameter is nearly zero. We have, with λ << 1, i i i q (λ) ≈ q + λδq ; pi(λ) ≈ pi + λδpi; where we have denoted the first-order changes in the canonical variables by i i @q (q; p; λ) @pi(q; p; λ) δq = ; δpi = : @λ λ=0 @λ λ=0 We call δq and δp the infinitesimal change in q and p induced by the infinitesimal canonical transformation. For example, let q be the position of a particle in one dimension. Consider a transla- tional point canonical transformation (exercise): q(λ) = q + λ, p(λ) = p: Here we have the rather trivial infinitesimal transformations: δq = 1; δp = 0: 1 Infinitesimal Canonical Transformations.
    [Show full text]
  • Reduced Phase Space Optics for General Relativity: Symplectic Ray Bundle Transfer 2
    Reduced phase space optics for general relativity: Symplectic ray bundle transfer Nezihe Uzun Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University in Prague, Prague, Czech Republic Univ Lyon, Ens de Lyon, Univ Lyon1, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F69007, Lyon, France 10 June 2019 Abstract. In the paraxial regime of Newtonian optics, propagation of an ensemble of rays is represented by a symplectic ABCD transfer matrix defined on a reduced phase space. Here, we present its analogue for general relativity. Starting from simultaneously applied null geodesic actions for two curves, we obtain a geodesic deviation action up to quadratic order. We achieve this by following a preexisting method constructed via Synge’s world function. We find the corresponding Hamiltonian function and the reduced phase space coordinates that are composed of the components of the Jacobi fields projected on an observational screen. Our thin ray bundle transfer matrix is then obtained through the matrix representation of the Lie operator associated with this quadratic Hamiltonian. Moreover, Etherington’s distance reciprocity between any two points is shown to be equivalent to the symplecticity conditions of our ray bundle transfer matrix. We further interpret the bundle propagation as a free canonical transformation with a generating function that is equal to the geodesic deviation action. We present it in the form of matrix inner products. A phase space distribution function and the associated Liouville equation is also provided. Finally, we briefly sketch the potential applications of our construction. Those include reduced phase space and null bundle averaging; factorization of light arXiv:1811.10917v2 [gr-qc] 26 Jan 2020 propagation in any spacetime uniquely into its thin lens, pure magnifier and fractional Fourier transformer components; wavization of the ray bundle; reduced polarization optics and autonomization of the bundle propagation on the phase space to find its invariants and obtain the stability analysis.
    [Show full text]
  • Applications of Canonical Transformations in Hamiltonian Mechanics
    Applications of Canonical Transformations in Hamiltonian Mechanics Brian Tu Jan. 14, 2014 Contents 1 Introduction 2 2 Preliminaries 2 3 Poisson Bracket 3 3.1 Characterization of Canonical Transforms . 3 3.2 Application to Integrals of Motion . 5 4 Infinitesimal Canonical Transformations 5 5 Generating Functions 6 1 1 Introduction Canonical transformations were introduced in the theory of Hamiltonians as a class of transformations that preserve the form of the Hamiltonian equations. We explore these transformations in greater detail, looking at topics such as its invariants, generating such transformations, and time evolution of a Hamiltonian system as a canonical transforma- tion. We consider only time-independent Hamiltonians. 2 Preliminaries We begin with a few definitions for review. Definition 2.1. Recall that the Hamiltonian H is a function of 2n coordinates q = (q1; q2; : : : ; qn) and p = (p1; p2; : : : ; pn) that satisfies the 2n relations @H p_i = − @qi @H q_i = ; @pi where 1 ≤ i ≤ n, known as Hamilton's equations. Definition 2.2. A transformation of coordinates is canonical if it preserves the form of Hamiltonians equations [3, p. 14]. Formally, let x = (q1; : : : ; qn; p1; : : : ; pn). Let 0 1 J = −1 0 be a 2n × 2n block matrix, e.g., 0 is an n × n 0 matrix, 1 is In, and -1 is −In. Hamilton's equations then can be written as x_ = JrH; Let y : xi 7! yi(x) be a transformation of coordinates. We have 2n 2n X @yi X @yi @H y_ = x_ = J ; i @x j @x @x j=1 j j=1 j j and hence, y_ = (J JJ T )rH; where (J)ij = @yi=@xj is the Jacobian of y.
    [Show full text]
  • Contact Hamiltonian Mechanics
    Contact Hamiltonian Mechanics Alessandro Bravettia, Hans Cruzb, Diego Tapiasc aInstituto de Investigaciones en Matem´aticas Aplicadas y en Sistemas, Universidad Nacional Aut´onoma de M´exico, A. P. 70543, M´exico, DF 04510, M´exico. bInstituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, A. P. 70543, M´exico, DF 04510, M´exico. cFacultad de Ciencias, Universidad Nacional Aut´onoma de M´exico, A.P. 70543, M´exico, DF 04510, Mexico. Abstract In this work we introduce contact Hamiltonian mechanics, an extension of symplectic Hamiltonian mechanics, and show that it is a natural candidate for a geometric description of non-dissipative and dissipative systems. For this purpose we review in detail the major features of standard symplectic Hamiltonian dynamics and show that all of them can be generalized to the contact case. Keywords: Hamiltonian mechanics, Dissipative systems, Contact geometry arXiv:1604.08266v2 [math-ph] 15 Nov 2016 Email addresses: [email protected] (Alessandro Bravetti), [email protected] (Hans Cruz), [email protected] (Diego Tapias) Contents 1 Introduction 2 2 Symplectic mechanics of non-dissipative systems 5 2.1 Time-independent Hamiltonian mechanics . 5 2.2 Canonical transformations and Liouville’s theorem . 6 2.3 Time-dependent Hamiltonian systems . 7 2.4 Hamilton-Jacobi formulation . 10 3 Contact mechanics of dissipative systems 11 3.1 Time-independent contact Hamiltonian mechanics . 11 3.2 Time evolution of the contact Hamiltonian and mechanical energy ............................... 15 3.3 Contact transformations and Liouville’s theorem . 17 3.4 Time-dependent contact Hamiltonian systems . 20 3.5 Hamilton-Jacobi formulation .
    [Show full text]
  • Tutorial Problems
    Analytical Mechanics HT-18 Tutorial 1 Problem 1. The Hamiltonian of a relativistic particle of mass m has the form p H = m2 c4 + p 2c 2 ; where p is the 3-dimensional momentum and c is the speed of light. Find a Lagrangian corresponding to this Hamiltonian. Problem 2. A dynamical systems with the following Lagrangians 52 1 2 (a) L = q_1 + q_2 + q_1q_2 cos(q1 − q2) + 3 cos q1 + cos q2; 2 2 1 h 2 2 2 i (b) L = (_q1 − q_2) + aq_1 t − a cos q2: 2 Find a Hamiltonian corresponding to the above Lagrangians. Problem 3. Find a Lagrangian corresponding to the following Hamiltonian H = p1p2 + q1q2: Problem 4. The Hamiltonian for a system with one degree of freedom is given by p 2 ab kq2 H = − bqpe−αt + q2 e −αt �α + be−αt + ; 2a 2 2 where a; b; α and k are constants. (a) Find a Lagrangian corresponding to this Hamiltonian. (b) Find an equivalent Lagrangian that does not depend explicitly on time. (c) Find the Hamiltonian corresponding to the second Lagrangian. What is the relation between this and the original Hamiltonian? Problem 5. Find the Hamiltonian corresponding to a Lagrangian of the form 1 L (q; q_ ; t) = L (q; t) + q_ Ta + q_ TT q_ ; 0 2 0 1 0 1 q1 a1 B . CB . C where q = @ . A ; a = @ . A and T is a symmetric n × n matrix. qn an Analytical Mechanics HT-18 Tutorial 2 Problem 1. Find a Lagrangian corresponding to the following Hamiltonian �2 2 H = q1p2 − q2p1 + a p1 + p2 : Problem 2.
    [Show full text]