An Exactly Solvable 3 Body Problem
|
|
- Daniela McCoy
- 5 years ago
- Views:
Transcription
1 An Exactly Solvable 3 Body Problem The most famous n-body problem is one where particles interact by an inverse square-law force. However, there is a class of exactly solvable n-body problems in which interactions are harmonic. In this section a Lie-algebraic method is illustrated by treating a solvable 3-body problem. Lagrange s equations of motion are so, defining canonical momentum d L L = 0 () dt ẋ i x i p i = L ẋ i, the Hamiltonian H(x, p, t) = p i ẋ i L i is a function of phase-space coordinates x and p. In general, one has 2n coordinates, x = {x,..., x n } and p = {p,..., p n }. This allows one to obtain Hamilton s Canonical Equations: ẋ i = H p i, ṗ i = H x i. (2) The time derivative of any function A( x, p ) of phase-space coordinates is da dt = n i= ( A ẋ i + A ) ṗ i x i p i So, following the sign convention of Perelomov [], one can define the Poisson bracket between any two functions of phase-space variables A and B [ A, B ] = ( A B B ) A. (3) p i i x i p i x i. So using Eqs. (2) and (3), da dt is the bracket of the Hamiltonian H with A da dt = [ H, A ].
2 If [ H, A ] = 0, then A is a constant of the motion, or a conserved quantity. The Poisson bracket is another example of a Lie bracket [ x, y ] which, as the reader will recall, has three main properties:. [x, y] = [y, x] -antisymmetry 2. [x + αy, z] = [x, z] + α[y, z] -linearity 3. [[x, y], z] + [[y, z], x] + [[z, x], y] = 0 -Jacobi identity As noted before, a Lie algebra is an algebra with a Lie bracket product. Mechanics concerns the Lie algebra A over C generated by functions of the phase space variables including the Hamiltonian function, with the Poisson bracket as the product. (So it is infinite dimensional.) The Poisson bracket has the following additional property: [x, yz] = y[x, z] + [x, y]z -Leibniz rule. The Three Body Problem: The three-body problem involves three particles, interacting with each other via harmonic forces. I choose these particles to have equal masses in order not to obscure the important points of the following discussion with inessential details. Thus, m = m 2 = m 3 = m, which have respective positions r, r 2, and r 3. (Here harmonic forces are forces associated with harmonic motion, where the Hamiltonian is of quadratic form.) There are three particles in this problem each with three degrees of freedom, corresponding to six phase space coordinates. Also, there is an external force similar to the force caused by a magnetic field acting on moving charges. (The actual magnetic force is not considered, since if the origin of the force were actually through electric charge, there would also be mutual electric repulsion of inverse square type, rendering the problem unsolvable.) So, the potential energy is V ( r) = k ( r α r β ) 2, 2 α β where k is like a spring constant. This potential leads to a homogeneous quadratic Hamiltonian, so that the motion is governed by a set of linear 2
3 homogenous differential equations with constant coefficients. This is what makes the problem exactly solvable. The Hamiltonian is H = 2m 3 p α q A c α = α β k ( r α r β 2 ), (4) where again the symmetric vector potential A = Bo (xŷ yˆx) is chosen. The 2 total angular momentum L is ( ) L = ( r 3 + r 2 + r 3 ) ( p + p 2 + p 3 ). (5) In order to solve the problem, it is helpful, as usual, to identify the conserved quantities. Due to the magnetic field, the vector L is not conserved. However, since B is parallel to the z-axis, the problem is invariant with respect to rotation about this axis. Recall Nöther s theorem states that to each symmetry of the the Lagrangian corresponds a conserved quantity of the system. This guarantees that L z is conserved. Also, one finds that the z-components L αβz of the relative angular momentum vector of two particles α and β with respect to one another is conserved L αβ = ( r β r α ) ( p β p α ). (6) Finally, the z-component of the total linear momentum vector of the system is also conserved p z = (p z + p 2z + p 3z ). (7) Thus, in Poisson bracket notation [ H, L z ] = 0, [ H, L αβz ] = 0, [ H, p z ] = 0. (8) Consider first A, the finite dimensional sub-algebra of A generated by H and the 2n phase space variables. All of the elements such that the bracket between any two gives zero, form a sub Lie algebra of A which is the Cartan sub-algebra. In the three-body problem, the Cartan sub-algebra contains five elements L z, L 2z, p z,, and H. As usual, a linear combination of all the basis elements of the algebra A can be constructed such that it is an eigenvector of the Poisson bracket operation with each generator of the Cartan sub-algebra {K}, including the Hamiltonian. One defines the set of 8-dimensional phase-space coordinates as ξ = {ξ, ξ 2,..., ξ 8 } = {x, y, z, x 2,..., p 3y, p 3z }. 3
4 Thus, an eigenvector of adk (where k K) constructed from combinations of ξ is of the form 8 u = a k ξ k, k = where {a k } are complex coefficients to be determined. Thus, as before, treat the set of generators ξ = {ξ, ξ 2,..., ξ 8 } as a basis of independent vectors and compare coefficients. In this way, one determines what the coefficients must be in order to form, for example, an eigenvector of the eigenvalue equation [ H, u α ] = λ α u α. (9) In other words, the motivation here is to solve the differential equation u = λu, with elementary solutions that have exponential time dependence u(t) = u(0)e λt (since, again, this is a set of linear homogeneous DE s with constant coefficients). This differential equation describes the motion of u throughout time. Since u is a linear combination of the phase-space coordinates of the system, the solutions will allow one to find the position and momentum of each particle at any given time. Using the oscillator frequency ω o and cyclotron frequency ω b ω o = k m and making the substitution ω b = qb o 2mc, Ω = ω b2 + 3ω o2, there are two roots that appear in the solution to the eigenvalue equation, Eq.(9), namely r = ω b + Ω and r + = ω b + Ω. Therefore, solving the set of DE s, for the Hamiltonian alone, one obtains the following table of eigenvalues and eigenvectors: 4
5 λ eigenvector. 0 u = p z + p z2 + p z u 2 = p y + p y2 + p y3 + mx ω b + mx 2 ω b + mx 3 ω b 3. 0 u 3 = p x + p x2 + p x3 my ω b my 2 ω b my 3 ω b 4. 0 u 4 = z + z 2 + z 3 (generalized) 5. 2iω b u 5 = p y + p y2 + p y3 i(p x + p x2 + p x3 ) mω b [(x + x 2 + x 3 ) + i(y + y 2 + y 3 )] 6. 2iω b u 6 = p y + p y2 + p y3 + i(p x + p x2 + p x3 ) mω b [(x + x 2 + x 3 ) i(y + y 2 + y 3 )] 7. i 3ω o u 7 = p z3 p z + i 3mω o (z z 3 ) 8. i 3ω o u 8 = p z2 p z + i 3mω o (z z 2 ) 9. i 3ω o u 9 = p z3 p z i 3mω o (z z 3 ) 0. i 3ω o u 0 = p z2 p z i 3mω o (z z 2 ). ir + u = p y3 p y i(p x3 p x ) + mω(x x 3 ) + imω(y y 3 ) 2. ir + u 2 = p y2 p y i(p x2 p x ) + mω(x x 2 ) + imω(y y 2 ) 3. ir + u 3 = p y3 p y + i(p x3 p x ) + mω(x x 3 ) imω(y y 3 ) 4. ir + u 4 = p y2 p y + i(p x2 p x ) + mω(x x 2 ) imω(y y 2 ) 5. ir u 5 = p y3 p y + i(p x3 p x ) mω(x x 3 ) + imω(y y 3 ) 6. ir u 6 = p y2 p y + i(p x2 p x ) mω(x x 2 ) + imω(y y 2 ) 7. ir u 7 = p y3 p y i(p x3 p x ) mω(x x 3 ) imω(y y 3 ) 8. ir u 8 = p y2 p y i(p x2 p x ) mω(x x 2 ) imω(y y 2 ) 5
6 Physically, the eigenvectors {u α } correspond to the normal mode coordinates of the three-mass system, providing those solutions to the motion that have exponential time dependence. Each eigenvector defines a mode of the system, the frequency of which is given by the corresponding eigenvalue λ = iω o. From the table, it is clear that in the 3-body problem, the eigenvectors appear in complex conjugate pairs. These pairs each correspond to two possible motions, both of which have the same frequency. Complete solutions to the equations of motion for this system involve combinations of the normal modes weighted with appropriate amplitude and phase factors [2]. A solution to the physical motion of the system is the real or imaginary part of the complex conjugate combination. From Table I (of solutions to Eq.(9)), it is clear that the problem decouples into xy and z parts. Thus, the Hamiltonian can be rewritten as H = ( (p 2 xα + p 2 2m yα) 2q ) c L zα + q2 c 2 (A2 xα + A 2 yα) + 2 α β α k ( x αˆx y α ŷ x β ˆx y β ŷ ) 2 + 2m p 2 zα + k( z α ẑ z β ẑ ) 2. 2 The motion along the z-axis is completely determined by eigenvectors u, u 4, u 7, u 8, u 9, and u 0. The eigenvector u corresponds to a translational mode along the z-axis, unaffected by the field. The eigenvectors u 7, u 8, u 9, and u 0 describe an oscillatory motion of two particles with respect to a third, confined to the z-axis moving with oscillator frequency ±ω o. The motion corresponding to the vector u 4 does not have an exponential solution. This vector is a generalized eigenvector of order greater than one. The occurrence of this generalized eigenvector means the entire Lie algebra fails to be semisimple. This adds unnecessary complication to the analysis of the problem. Thus, for simplicity, I consider only the xy portion in the following discussion. Motivated by the desire to construct raising and lowering operators for this system in the corresponding quantum mechanical problem (see section I), one looks for a set of eigenvectors that satisfy simultaneously the eigenvalue α α β 6
7 equations corresponding to all of the elements in the Cartan subalgebra [ H, u α ] = λ α u α, [ L z, u α ] = γ α u α, [ L 2z, u α ] = µ α u α. (0) In order to construct such a set of eigenvectors, start by obtaining solutions to the eigenvalue equation corresponding to L z by the same method as was employed for H. Then, collect the eigenvectors {u α } degenerate in both λ and γ. Finally, combine these degenerate eigenvectors into linear combinations v β = α a β α u α to form the desired set {v β } of vectors that satisfy the entire system. (The coefficient a may be zero.) In the process, one finds twelve ladder operators for the spectrum listed in Table II. 7
8 Table II λ γ µ operators. 0 i 0 v 2 = p y + p y2 + p y3 + mω b (x + x 2 + x 3 ) i(p x + p x2 + p x3 mω b (y + y 2 + y 3 )) 2. 0 i 0 v 3 = p y + p y2 + p y3 + mω b (x + x 2 + x 3 ) +i(p x + p x2 + p x3 mω b (y + y 2 + y 3 )) 3. 2iω b i 0 v 5 = i(p x + p x2 + p x3 ) + p y + p y2 + p y3 mω b ((x + x 2 + x 3 ) + i(y + y 2 + y 3 )) 4. 2iω b i 0 v 6 = i(p x + p x2 + p x3 ) + p y + p y2 + p y3 mω b ((x + x 2 + x 3 ) i(y + y 2 + y 3 )) 5. i(ω b + Ω) 0 0 v = i(p x + p x2 2p x3 ) + p y + p y2 2p y3 mω((x + x 2 2x 3 ) + i(y + y 2 2y 3 )) 6. i(ω b + Ω) 0 2i v 2 = i(p x p x2 ) p y + p y2 +mω((x x 2 ) + i(y y 2 )) 7. i(ω b + Ω) 0 0 v 3 = i(p x + p x2 2p x3 ) + p y + p y2 2p y3 mω((x + x 2 2x 3 ) i(y + y 2 2y 3 )) 8. i(ω b + Ω) 0 2i v 4 = i(p x p x2 ) p y + p y2 +mω((x x 2 ) i(y y 2 )) 9. i(ω ω b ) 0 0 v 5 = i(p x + p x2 2p x3 ) + p y + p y2 2p y3 +mω((x + x 2 2x 3 ) i(y + y 2 2y 3 )) 0. i(ω ω b ) 0 2i v 6 = i(p x p x2 ) p y + p y2 +mω((x 2 x ) + i(y y 2 )). i(ω ω b ) 0 0 v 7 = i(p x + p x2 2p x3 ) + p y + p y2 2p y3 +mω((x + x 2 2x 3 ) + i(y + y 2 2y 3 )) 2. i(ω ω b ) 0 2i v 8 = i(p x p x2 ) p y + p y2 +mω((x 2 x 2 ) i(y y 2 )) 8
9 Once one has the list of eigenvectors which satisfy by all of the bracket relations simultaneously, Eq.(0), the Hamiltonian can be rewritten in terms of these vectors. Classically, these eigenvectors are the elementary solutions of the system. The new eigenvectors of the total system appear in complex conjugate pairs v 2 = v v 4 = v3 v 7 = v5 v 8 = v6 v = v9 v 2 = v0 With this in mind, the Hamiltonian of the system becomes: H xy = 24mΩ 2 [ ω 2 b v 9 v 9 + 3ω 2 b v 0 v 0 + 4ω 2 b v 3 v 3 ω b Ωv 9 v 9 3ω b Ωv 0 v 0+ ω b (Ω + ω b )v 5 v 5 + 3ω b (Ω + ω b )v 6 v 6 + 3ω 2 o(v 5 v 5 + 3v 6 v 6 + v 9 v 9 + 3v 0 v 0 + 4v 3 v 3) ]. Converting each of the phase-space coordinates into combinations of the eigenvectors {v β }, one obtains Table III. As was the case with the raising and lowering operators in section I, these elementary solutions with exponential time dependence can be used to determine the motion of the system corresponding to each mode. (To make this a quantum mechanical problem, the Hamiltonian must be rewritten in terms of symmetrized combinations of the eigenvectors 2 α v αv α + v αv α.) In order to trace out this motion, begin by collecting the appropriate elementary solutions. For example when analyzing the mode corresponding to frequency λ = (ω b + Ω), one must form linear combinations of the elementary solutions v, v 2, v 3, and v 4. When combined properly, these elementary solutions lead to real functions, which constitute actual motion. So, including the proper time dependence, v α = c α e λαt (α =, 2,..., 8), one can graph the motion of each particle as a function of time. Therefore, the coefficients c α are the initial conditions of the system v(t) = v(0) e λ αt. In this way, the entire path of each particle is determined over all time. A 2-D parametric plot of the motion corresponding to each mode of the system is shown in Figure. From fig., it is clear that the solutions v 5 and v 6 represent a mode of the system in which all three particles move in a circle 9
10 about the center of mass. Analysis of initial conditions for this mode reveals that all three particles begin at the same position on the x or y axis, and move in unison around the circle. The mode corresponding to solutions v, v 2, v 3, and v 4 is slightly more complicated. There are a number of possible situations for motion in this mode, depending on the initial conditions given. The final oscillatory mode in the xy plane is determined by elementary solutions v 5, v 6, v 7, and v 8. From the figure, it is clear that this mode is characterized by the same motion as was displayed for the previous mode. The only difference is in the angular frequency of rotation and the direction of motion. The final mode corresponding to xy motion of the particles (not included in the figure) is the zero frequency mode given by solutions v 2 and v 3. Here, the particles are stationary relative to each other throughout time, only translating in the xy plane effected by the magnetic field. All possible motion of this system can be described by appropriate linear combinations of these normal mode solutions. Thus, by analyzing the elementary solutions that satisfy the entire Cartan subalgebra (Eq.(0)) simultaneously, one can solve this harmonic 3-body problem completely. 0
11 Table III x 24mω b Ω (ω b(v + 3v 2 + v 3 + 3v 4 v 5 3v 6 v 7 3v 8 ) +2Ω(v 2 + v 3 v 5 v 6 )) x 2 24mω b Ω (ω b(v 3v 2 + v 3 3v 4 v 5 + 3v 6 v 7 + 3v 8 ) +2Ω(v 2 + v 3 v 5 v 6 )) x 3 2mω b Ω (ω b( v v 3 + v 5 + v 7 ) +Ω(v 2 + v 3 v 5 v 6 )) y y 2 y 3 p x p x2 p x3 i (ω 24mω b Ω b(v + 3v 2 v 3 3v 4 + v 5 + 3v 6 v 7 3v 8 ) +2Ω(v 2 v 3 v 5 + v 6 )) i (ω 24mω b Ω b(v 3v 2 v 3 + 3v 4 + v 5 3v 6 v 7 + 3v 8 ) +2Ω(v 2 v 3 v 5 + v 6 )) i (ω 2mω b Ω b( v + v 3 v 5 + v 7 ) +Ω(v 2 v 3 v 5 + v 6 )) i 24 (v + 3v 2 v 3 3v 4 v 5 3v 6 + v 7 + 3v 8 2v 2 + 2v 3 2v 5 + 2v 6 ) i 24 (v 3v 2 v 3 + 3v 4 v 5 + 3v 6 + v 7 3v 8 2v 2 + 2v 3 2v 5 + 2v 6 ) i (v 2 v 3 v 5 + v 7 + v 2 v 3 + v 5 v 6 ) p y 24 ( v 3v 2 v 3 3v 4 v 5 3v 6 v 7 3v 8 + 2v 2 + 2v 3 + 2v 5 + 2v 6 ) p y2 24 ( v + 3v 2 v 3 + 3v 4 v 5 + 3v 6 v 7 + 3v 8 + 2v 2 + 2v 3 + 2v 5 + 2v 6 ) p y3 2 (v + v 3 + v 5 + v 7 + v 2 + v 3 + v 5 + v 6 )
12 y y x x (a) (b) y y x x (c) (d) Figure : 2D parametric plots of possible xy motion generated by combinations of oscillatory elementary solutions, which determine three separate modes of the system. Paths (a) - (c) correspond to orbits generated by solutions v - v 8 (double arrows indicate direction of motion), while path (d) represents the mode generated by v 5 and v 6. 2
13 References [] A. M. Perelomov. Integrable Systems Of Classical Mechanics and Lie Algebras, volume I. Birkhäuser, Boston, MA, 990. [2] Herbert Goldstein. Classical Mechanics. Addison Wesley, San Francisco, CA, 3rd edition,
Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism
Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism Benjamin Hornberger 1/26/1 Phy 55, Classical Electrodynamics, Prof. Goldhaber Lecture notes from Oct. 26, 21 Lecture held by Prof. Weisberger
More informationPhysics 5153 Classical Mechanics. Canonical Transformations-1
1 Introduction Physics 5153 Classical Mechanics Canonical Transformations The choice of generalized coordinates used to describe a physical system is completely arbitrary, but the Lagrangian is invariant
More informationGeneral Exam Part II, Fall 1998 Quantum Mechanics Solutions
General Exam Part II, Fall 1998 Quantum Mechanics Solutions Leo C. Stein Problem 1 Consider a particle of charge q and mass m confined to the x-y plane and subject to a harmonic oscillator potential V
More informationAssignment 8. [η j, η k ] = J jk
Assignment 8 Goldstein 9.8 Prove directly that the transformation is canonical and find a generating function. Q 1 = q 1, P 1 = p 1 p Q = p, P = q 1 q We can establish that the transformation is canonical
More informationM2A2 Problem Sheet 3 - Hamiltonian Mechanics
MA Problem Sheet 3 - Hamiltonian Mechanics. The particle in a cone. A particle slides under gravity, inside a smooth circular cone with a vertical axis, z = k x + y. Write down its Lagrangian in a) Cartesian,
More information[#1] R 3 bracket for the spherical pendulum
.. Holm Tuesday 11 January 2011 Solutions to MSc Enhanced Coursework for MA16 1 M3/4A16 MSc Enhanced Coursework arryl Holm Solutions Tuesday 11 January 2011 [#1] R 3 bracket for the spherical pendulum
More informationHomework 3. 1 Goldstein Part (a) Theoretical Dynamics September 24, The Hamiltonian is given by
Theoretical Dynamics September 4, 010 Instructor: Dr. Thomas Cohen Homework 3 Submitted by: Vivek Saxena 1 Goldstein 8.1 1.1 Part (a) The Hamiltonian is given by H(q i, p i, t) = p i q i L(q i, q i, t)
More informationMATHEMATICAL PHYSICS
MATHEMATICAL PHYSICS Third Year SEMESTER 1 015 016 Classical Mechanics MP350 Prof. S. J. Hands, Prof. D. M. Heffernan, Dr. J.-I. Skullerud and Dr. M. Fremling Time allowed: 1 1 hours Answer two questions
More informationClassical Mechanics Comprehensive Exam Solution
Classical Mechanics Comprehensive Exam Solution January 31, 011, 1:00 pm 5:pm Solve the following six problems. In the following problems, e x, e y, and e z are unit vectors in the x, y, and z directions,
More informationPHY 5246: Theoretical Dynamics, Fall November 16 th, 2015 Assignment # 11, Solutions. p θ = L θ = mr2 θ, p φ = L θ = mr2 sin 2 θ φ.
PHY 5246: Theoretical Dynamics, Fall 215 November 16 th, 215 Assignment # 11, Solutions 1 Graded problems Problem 1 1.a) The Lagrangian is L = 1 2 m(ṙ2 +r 2 θ2 +r 2 sin 2 θ φ 2 ) V(r), (1) and the conjugate
More informationThe Particle-Field Hamiltonian
The Particle-Field Hamiltonian For a fundamental understanding of the interaction of a particle with the electromagnetic field we need to know the total energy of the system consisting of particle and
More informationREVIEW. Hamilton s principle. based on FW-18. Variational statement of mechanics: (for conservative forces) action Equivalent to Newton s laws!
Hamilton s principle Variational statement of mechanics: (for conservative forces) action Equivalent to Newton s laws! based on FW-18 REVIEW the particle takes the path that minimizes the integrated difference
More informationCOMPLETE ALL ROUGH WORKINGS IN THE ANSWER BOOK AND CROSS THROUGH ANY WORK WHICH IS NOT TO BE ASSESSED.
BSc/MSci EXAMINATION PHY-304 Time Allowed: Physical Dynamics 2 hours 30 minutes Date: 28 th May 2009 Time: 10:00 Instructions: Answer ALL questions in section A. Answer ONLY TWO questions from section
More informationCurves in the configuration space Q or in the velocity phase space Ω satisfying the Euler-Lagrange (EL) equations,
Physics 6010, Fall 2010 Hamiltonian Formalism: Hamilton s equations. Conservation laws. Reduction. Poisson Brackets. Relevant Sections in Text: 8.1 8.3, 9.5 The Hamiltonian Formalism We now return to formal
More informationQuantum Mechanics I Physics 5701
Quantum Mechanics I Physics 5701 Z. E. Meziani 02/24//2017 Physics 5701 Lecture Commutation of Observables and First Consequences of the Postulates Outline 1 Commutation Relations 2 Uncertainty Relations
More informationfor changing independent variables. Most simply for a function f(x) the Legendre transformation f(x) B(s) takes the form B(s) = xs f(x) with s = df
Physics 106a, Caltech 1 November, 2018 Lecture 10: Hamiltonian Mechanics I The Hamiltonian In the Hamiltonian formulation of dynamics each second order ODE given by the Euler- Lagrange equation in terms
More informationHamiltonian Dynamics from Lie Poisson Brackets
1 Hamiltonian Dynamics from Lie Poisson Brackets Jean-Luc Thiffeault Department of Applied Physics and Applied Mathematics Columbia University http://plasma.ap.columbia.edu/~jeanluc 12 February 2002 2
More informationComplex Numbers. The set of complex numbers can be defined as the set of pairs of real numbers, {(x, y)}, with two operations: (i) addition,
Complex Numbers Complex Algebra The set of complex numbers can be defined as the set of pairs of real numbers, {(x, y)}, with two operations: (i) addition, and (ii) complex multiplication, (x 1, y 1 )
More informationCanonical 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
More informationQuantization of scalar fields
Quantization of scalar fields March 8, 06 We have introduced several distinct types of fields, with actions that give their field equations. These include scalar fields, S α ϕ α ϕ m ϕ d 4 x and complex
More informationLecture I: Constrained Hamiltonian systems
Lecture I: Constrained Hamiltonian systems (Courses in canonical gravity) Yaser Tavakoli December 15, 2014 1 Introduction In canonical formulation of general relativity, geometry of space-time is given
More information= 0. = q i., q i = E
Summary of the Above Newton s second law: d 2 r dt 2 = Φ( r) Complicated vector arithmetic & coordinate system dependence Lagrangian Formalism: L q i d dt ( L q i ) = 0 n second-order differential equations
More informationPhysical Dynamics (PHY-304)
Physical Dynamics (PHY-304) Gabriele Travaglini March 31, 2012 1 Review of Newtonian Mechanics 1.1 One particle Lectures 1-2. Frame, velocity, acceleration, number of degrees of freedom, generalised coordinates.
More informationTranslation versus Momentum and Rotation versus Angular Momentum
APPENDIX F Translation versus Momentum and Rotation versus Angular Momentum In Chapter 2, it was shown that the Hamiltonian Ĥ commutes with any translation p. 68) or rotation p. 69) operator, denoted as
More informationIsotropic harmonic oscillator
Isotropic harmonic oscillator 1 Isotropic harmonic oscillator The hamiltonian of the isotropic harmonic oscillator is H = h m + 1 mω r (1) = [ h d m dρ + 1 ] m ω ρ, () ρ=x,y,z a sum of three one-dimensional
More informationQuantization of the E-M field
Quantization of the E-M field 0.1 Classical E&M First we will wor in the transverse gauge where there are no sources. Then A = 0, nabla A = B, and E = 1 A and Maxwell s equations are B = 1 E E = 1 B E
More informationAn introduction to Birkhoff normal form
An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an
More informationSolutions to Problems in Goldstein, Classical Mechanics, Second Edition. Chapter 9
Solutions to Problems in Goldstein, Classical Mechanics, Second Edition Homer Reid October 29, 2002 Chater 9 Problem 9. One of the attemts at combining the two sets of Hamilton s equations into one tries
More information15. Hamiltonian Mechanics
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 15. Hamiltonian Mechanics Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationHamilton s principle and Symmetries
Hamilton s principle and Symmetries Sourendu Gupta TIFR, Mumbai, India Classical Mechanics 2011 August 18, 2011 The Hamiltonian The change in the Lagrangian due to a virtual change of coordinates is dl
More informationPhysical Dynamics (SPA5304) Lecture Plan 2018
Physical Dynamics (SPA5304) Lecture Plan 2018 The numbers on the left margin are approximate lecture numbers. Items in gray are not covered this year 1 Advanced Review of Newtonian Mechanics 1.1 One Particle
More informationPhysics 106b: Lecture 7 25 January, 2018
Physics 106b: Lecture 7 25 January, 2018 Hamiltonian Chaos: Introduction Integrable Systems We start with systems that do not exhibit chaos, but instead have simple periodic motion (like the SHO) with
More informationIntroduction to Modern Quantum Field Theory
Department of Mathematics University of Texas at Arlington Arlington, TX USA Febuary, 2016 Recall Einstein s famous equation, E 2 = (Mc 2 ) 2 + (c p) 2, where c is the speed of light, M is the classical
More informationHamilton-Jacobi theory
Hamilton-Jacobi theory November 9, 04 We conclude with the crowning theorem of Hamiltonian dynamics: a proof that for any Hamiltonian dynamical system there exists a canonical transformation to a set of
More information4. Complex Oscillations
4. Complex Oscillations The most common use of complex numbers in physics is for analyzing oscillations and waves. We will illustrate this with a simple but crucially important model, the damped harmonic
More information2 Canonical quantization
Phys540.nb 7 Canonical quantization.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system?.1.1.lagrangian Lagrangian mechanics is a reformulation of classical mechanics.
More informationQuantization of Scalar Field
Quantization of Scalar Field Wei Wang 2017.10.12 Wei Wang(SJTU) Lectures on QFT 2017.10.12 1 / 41 Contents 1 From classical theory to quantum theory 2 Quantization of real scalar field 3 Quantization of
More informationDifferential Equations Grinshpan Two-Dimensional Homogeneous Linear Systems with Constant Coefficients. Purely Imaginary Eigenvalues. Recall the equation mẍ+k = of a simple harmonic oscillator with frequency
More information4.3 Lecture 18: Quantum Mechanics
CHAPTER 4. QUANTUM SYSTEMS 73 4.3 Lecture 18: Quantum Mechanics 4.3.1 Basics Now that we have mathematical tools of linear algebra we are ready to develop a framework of quantum mechanics. The framework
More informationSecond quantization: where quantization and particles come from?
110 Phys460.nb 7 Second quantization: where quantization and particles come from? 7.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system? 7.1.1.Lagrangian Lagrangian
More informationSimple one-dimensional potentials
Simple one-dimensional potentials Sourendu Gupta TIFR, Mumbai, India Quantum Mechanics 1 Ninth lecture Outline 1 Outline 2 Energy bands in periodic potentials 3 The harmonic oscillator 4 A charged particle
More informationProblem 1: Lagrangians and Conserved Quantities. Consider the following action for a particle of mass m moving in one dimension
105A Practice Final Solutions March 13, 01 William Kelly Problem 1: Lagrangians and Conserved Quantities Consider the following action for a particle of mass m moving in one dimension S = dtl = mc dt 1
More informationHAMILTON S PRINCIPLE
HAMILTON S PRINCIPLE In our previous derivation of Lagrange s equations we started from the Newtonian vector equations of motion and via D Alembert s Principle changed coordinates to generalised coordinates
More informationChaotic motion. Phys 420/580 Lecture 10
Chaotic motion Phys 420/580 Lecture 10 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t
More informationPhysics 139B Solutions to Homework Set 4 Fall 2009
Physics 139B Solutions to Homework Set 4 Fall 9 1. Liboff, problem 1.16 on page 594 595. Consider an atom whose electrons are L S coupled so that the good quantum numbers are j l s m j and eigenstates
More informationLecture Notes 2: Review of Quantum Mechanics
Quantum Field Theory for Leg Spinners 18/10/10 Lecture Notes 2: Review of Quantum Mechanics Lecturer: Prakash Panangaden Scribe: Jakub Závodný This lecture will briefly review some of the basic concepts
More informationPhonons and lattice dynamics
Chapter Phonons and lattice dynamics. Vibration modes of a cluster Consider a cluster or a molecule formed of an assembly of atoms bound due to a specific potential. First, the structure must be relaxed
More informationBACKGROUND IN SYMPLECTIC GEOMETRY
BACKGROUND IN SYMPLECTIC GEOMETRY NILAY KUMAR Today I want to introduce some of the symplectic structure underlying classical mechanics. The key idea is actually quite old and in its various formulations
More information(a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron, but neglecting spin-orbit interactions.
1. Quantum Mechanics (Spring 2007) Consider a hydrogen atom in a weak uniform magnetic field B = Bê z. (a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron,
More informationLecture 4. Alexey Boyarsky. October 6, 2015
Lecture 4 Alexey Boyarsky October 6, 2015 1 Conservation laws and symmetries 1.1 Ignorable Coordinates During the motion of a mechanical system, the 2s quantities q i and q i, (i = 1, 2,..., s) which specify
More informationarxiv:hep-th/ v1 8 Mar 1995
GALILEAN INVARIANCE IN 2+1 DIMENSIONS arxiv:hep-th/9503046v1 8 Mar 1995 Yves Brihaye Dept. of Math. Phys. University of Mons Av. Maistriau, 7000 Mons, Belgium Cezary Gonera Dept. of Physics U.J.A Antwerpen,
More informationAnalytical Mechanics for Relativity and Quantum Mechanics
Analytical Mechanics for Relativity and Quantum Mechanics Oliver Davis Johns San Francisco State University OXPORD UNIVERSITY PRESS CONTENTS Dedication Preface Acknowledgments v vii ix PART I INTRODUCTION:
More informationPhysics 215 Quantum Mechanics 1 Assignment 5
Physics 15 Quantum Mechanics 1 Assignment 5 Logan A. Morrison February 10, 016 Problem 1 A particle of mass m is confined to a one-dimensional region 0 x a. At t 0 its normalized wave function is 8 πx
More informationApproximation Methods in QM
Chapter 3 Approximation Methods in QM Contents 3.1 Time independent PT (nondegenerate)............... 5 3. Degenerate perturbation theory (PT)................. 59 3.3 Time dependent PT and Fermi s golden
More informationRobotics, Geometry and Control - Rigid body motion and geometry
Robotics, Geometry and Control - Rigid body motion and geometry Ravi Banavar 1 1 Systems and Control Engineering IIT Bombay HYCON-EECI Graduate School - Spring 2008 The material for these slides is largely
More informationSTABILITY. Phase portraits and local stability
MAS271 Methods for differential equations Dr. R. Jain STABILITY Phase portraits and local stability We are interested in system of ordinary differential equations of the form ẋ = f(x, y), ẏ = g(x, y),
More informationProblem 1: Step Potential (10 points)
Problem 1: Step Potential (10 points) 1 Consider the potential V (x). V (x) = { 0, x 0 V, x > 0 A particle of mass m and kinetic energy E approaches the step from x < 0. a) Write the solution to Schrodinger
More informationPHYS2330 Intermediate Mechanics Fall Final Exam Tuesday, 21 Dec 2010
Name: PHYS2330 Intermediate Mechanics Fall 2010 Final Exam Tuesday, 21 Dec 2010 This exam has two parts. Part I has 20 multiple choice questions, worth two points each. Part II consists of six relatively
More informationLecture 4: Equations of motion and canonical quantization Read Sakurai Chapter 1.6 and 1.7
Lecture 4: Equations of motion and canonical quantization Read Sakurai Chapter 1.6 and 1.7 In Lecture 1 and 2, we have discussed how to represent the state of a quantum mechanical system based the superposition
More information(a) Sections 3.7 through (b) Sections 8.1 through 8.3. and Sections 8.5 through 8.6. Problem Set 4 due Monday, 10/14/02
Physics 601 Dr. Dragt Fall 2002 Reading Assignment #4: 1. Dragt (a) Sections 1.5 and 1.6 of Chapter 1 (Introductory Concepts). (b) Notes VI, Hamilton s Equations of Motion (to be found right after the
More informationLecture 5: Orbital angular momentum, spin and rotation
Lecture 5: Orbital angular momentum, spin and rotation 1 Orbital angular momentum operator According to the classic expression of orbital angular momentum L = r p, we define the quantum operator L x =
More informationNotes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers
Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop Eric Sommers 17 July 2009 2 Contents 1 Background 5 1.1 Linear algebra......................................... 5 1.1.1
More informationPhysics 342 Lecture 27. Spin. Lecture 27. Physics 342 Quantum Mechanics I
Physics 342 Lecture 27 Spin Lecture 27 Physics 342 Quantum Mechanics I Monday, April 5th, 2010 There is an intrinsic characteristic of point particles that has an analogue in but no direct derivation from
More informationChemistry 532 Practice Final Exam Fall 2012 Solutions
Chemistry 53 Practice Final Exam Fall Solutions x e ax dx π a 3/ ; π sin 3 xdx 4 3 π cos nx dx π; sin θ cos θ + K x n e ax dx n! a n+ ; r r r r ˆL h r ˆL z h i φ ˆL x i hsin φ + cot θ cos φ θ φ ) ˆLy i
More informationTheoretical physics. Deterministic chaos in classical physics. Martin Scholtz
Theoretical physics Deterministic chaos in classical physics Martin Scholtz scholtzzz@gmail.com Fundamental physical theories and role of classical mechanics. Intuitive characteristics of chaos. Newton
More informationHamilton-Jacobi theory on Lie algebroids: Applications to nonholonomic mechanics. Manuel de León Institute of Mathematical Sciences CSIC, Spain
Hamilton-Jacobi theory on Lie algebroids: Applications to nonholonomic mechanics Manuel de León Institute of Mathematical Sciences CSIC, Spain joint work with J.C. Marrero (University of La Laguna) D.
More informationAssignment 6. Using the result for the Lagrangian for a double pendulum in Problem 1.22, we get
Assignment 6 Goldstein 6.4 Obtain the normal modes of vibration for the double pendulum shown in Figure.4, assuming equal lengths, but not equal masses. Show that when the lower mass is small compared
More informationColumbia University Department of Physics QUALIFYING EXAMINATION
Columbia University Department of Physics QUALIFYING EXAMINATION Monday, January 8, 2018 2:00PM to 4:00PM Classical Physics Section 2. Electricity, Magnetism & Electrodynamics Two hours are permitted for
More informationM3-4-5 A16 Notes for Geometric Mechanics: Oct Nov 2011
M3-4-5 A16 Notes for Geometric Mechanics: Oct Nov 2011 Text for the course: Professor Darryl D Holm 25 October 2011 Imperial College London d.holm@ic.ac.uk http://www.ma.ic.ac.uk/~dholm/ Geometric Mechanics
More informationSeminar 6: COUPLED HARMONIC OSCILLATORS
Seminar 6: COUPLED HARMONIC OSCILLATORS 1. Lagrangian Equations of Motion Let consider a system consisting of two harmonic oscillators that are coupled together. As a model, we will use two particles attached
More informationMath 1302, Week 8: Oscillations
Math 302, Week 8: Oscillations T y eq Y y = y eq + Y mg Figure : Simple harmonic motion. At equilibrium the string is of total length y eq. During the motion we let Y be the extension beyond equilibrium,
More informationThe Geometry of Euler s equation. Introduction
The Geometry of Euler s equation Introduction Part 1 Mechanical systems with constraints, symmetries flexible joint fixed length In principle can be dealt with by applying F=ma, but this can become complicated
More information7 Planar systems of linear ODE
7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution
More informationStatistical Mechanics Solution Set #1 Instructor: Rigoberto Hernandez MoSE 2100L, , (Dated: September 4, 2014)
CHEM 6481 TT 9:3-1:55 AM Fall 214 Statistical Mechanics Solution Set #1 Instructor: Rigoberto Hernandez MoSE 21L, 894-594, hernandez@gatech.edu (Dated: September 4, 214 1. Answered according to individual
More informationQuantum Mechanics: Vibration and Rotation of Molecules
Quantum Mechanics: Vibration and Rotation of Molecules 8th April 2008 I. 1-Dimensional Classical Harmonic Oscillator The classical picture for motion under a harmonic potential (mass attached to spring
More informationChaos in Hamiltonian systems
Chaos in Hamiltonian systems Teemu Laakso April 26, 2013 Course material: Chapter 7 from Ott 1993/2002, Chaos in Dynamical Systems, Cambridge http://matriisi.ee.tut.fi/courses/mat-35006 Useful reading:
More informationDamped harmonic oscillator with time-dependent frictional coefficient and time-dependent frequency. Abstract
Damped harmonic oscillator with time-dependent frictional coefficient and time-dependent frequency Eun Ji Jang, Jihun Cha, Young Kyu Lee, and Won Sang Chung Department of Physics and Research Institute
More informationDifferential Equations and Modeling
Differential Equations and Modeling Preliminary Lecture Notes Adolfo J. Rumbos c Draft date: March 22, 2018 March 22, 2018 2 Contents 1 Preface 5 2 Introduction to Modeling 7 2.1 Constructing Models.........................
More informationQuestion 1: A particle starts at rest and moves along a cycloid whose equation is. 2ay y a
Stephen Martin PHYS 10 Homework #1 Question 1: A particle starts at rest and moves along a cycloid whose equation is [ ( ) a y x = ± a cos 1 + ] ay y a There is a gravitational field of strength g in the
More informationTopics in Representation Theory: Cultural Background
Topics in Representation Theory: Cultural Background This semester we will be covering various topics in representation theory, see the separate syllabus for a detailed list of topics, including some that
More informationClassical mechanics of particles and fields
Classical mechanics of particles and fields D.V. Skryabin Department of Physics, University of Bath PACS numbers: The concise and transparent exposition of many topics covered in this unit can be found
More informationHomework 4. Goldstein 9.7. Part (a) Theoretical Dynamics October 01, 2010 (1) P i = F 1. Q i. p i = F 1 (3) q i (5) P i (6)
Theoretical Dynamics October 01, 2010 Instructor: Dr. Thomas Cohen Homework 4 Submitted by: Vivek Saxena Goldstein 9.7 Part (a) F 1 (q, Q, t) F 2 (q, P, t) P i F 1 Q i (1) F 2 (q, P, t) F 1 (q, Q, t) +
More informationPart III Symmetries, Fields and Particles
Part III Symmetries, Fields and Particles Theorems Based on lectures by N. Dorey Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often
More informationNonlinear Single-Particle Dynamics in High Energy Accelerators
Nonlinear Single-Particle Dynamics in High Energy Accelerators Part 2: Basic tools and concepts Nonlinear Single-Particle Dynamics in High Energy Accelerators This course consists of eight lectures: 1.
More informationQuantum Field Theory Notes. Ryan D. Reece
Quantum Field Theory Notes Ryan D. Reece November 27, 2007 Chapter 1 Preliminaries 1.1 Overview of Special Relativity 1.1.1 Lorentz Boosts Searches in the later part 19th century for the coordinate transformation
More informationSimple Harmonic Oscillator
Classical harmonic oscillator Linear force acting on a particle (Hooke s law): F =!kx From Newton s law: F = ma = m d x dt =!kx " d x dt + # x = 0, # = k / m Position and momentum solutions oscillate in
More informationHighest-weight Theory: Verma Modules
Highest-weight Theory: Verma Modules Math G4344, Spring 2012 We will now turn to the problem of classifying and constructing all finitedimensional representations of a complex semi-simple Lie algebra (or,
More informationPH 610/710-2A: Advanced Classical Mechanics I. Fall Semester 2007
PH 610/710-2A: Advanced Classical Mechanics I Fall Semester 2007 Time and location: Tuesdays & Thursdays 8:00am 9:15am (EB 144) Instructor and office hours: Dr. Renato Camata, camata@uab.edu CH 306, (205)
More informationFINAL EXAM GROUND RULES
PHYSICS 507 Fall 2011 FINAL EXAM Room: ARC-108 Time: Wednesday, December 21, 10am-1pm GROUND RULES There are four problems based on the above-listed material. Closed book Closed notes Partial credit will
More informationEigenvalues and Eigenvectors: An Introduction
Eigenvalues and Eigenvectors: An Introduction The eigenvalue problem is a problem of considerable theoretical interest and wide-ranging application. For example, this problem is crucial in solving systems
More informationL = 1 2 a(q) q2 V (q).
Physics 3550, Fall 2011 Motion near equilibrium - Small Oscillations Relevant Sections in Text: 5.1 5.6 Motion near equilibrium 1 degree of freedom One of the most important situations in physics is motion
More informationLie 2-algebras from 2-plectic geometry
Lie 2-algebras from 2-plectic geometry Chris Rogers joint with John Baez and Alex Hoffnung Department of Mathematics University of California, Riverside XVIIIth Oporto Meeting on Geometry, Topology and
More information221B Lecture Notes on Resonances in Classical Mechanics
1B Lecture Notes on Resonances in Classical Mechanics 1 Harmonic Oscillators Harmonic oscillators appear in many different contexts in classical mechanics. Examples include: spring, pendulum (with a small
More informationVariational principles and Hamiltonian Mechanics
A Primer on Geometric Mechanics Variational principles and Hamiltonian Mechanics Alex L. Castro, PUC Rio de Janeiro Henry O. Jacobs, CMS, Caltech Christian Lessig, CMS, Caltech Alex L. Castro (PUC-Rio)
More informationE = φ 1 A The dynamics of a particle with mass m and charge q is determined by the Hamiltonian
Lecture 9 Relevant sections in text: 2.6 Charged particle in an electromagnetic field We now turn to another extremely important example of quantum dynamics. Let us describe a non-relativistic particle
More informationQuantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours.
Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours. There are 10 problems, totalling 180 points. Do all problems. Answer all problems in the white books provided.
More informationP321(b), Assignement 1
P31(b), Assignement 1 1 Exercise 3.1 (Fetter and Walecka) a) The problem is that of a point mass rotating along a circle of radius a, rotating with a constant angular velocity Ω. Generally, 3 coordinates
More informationCanonical Quantization
Canonical Quantization March 6, 06 Canonical quantization of a particle. The Heisenberg picture One of the most direct ways to quantize a classical system is the method of canonical quantization introduced
More informationPart II. Classical Dynamics. Year
Part II Year 28 27 26 25 24 23 22 21 20 2009 2008 2007 2006 2005 28 Paper 1, Section I 8B Derive Hamilton s equations from an action principle. 22 Consider a two-dimensional phase space with the Hamiltonian
More informationTwo-Body Problem. Central Potential. 1D Motion
Two-Body Problem. Central Potential. D Motion The simplest non-trivial dynamical problem is the problem of two particles. The equations of motion read. m r = F 2, () We already know that the center of
More information