Classical Mechanics and Mathematical Methods Review

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Explore a review of basic mathematical tools, mathematical methods, physical concepts, and functional dependencies covered in a Classical Mechanics and Mathematical Methods course lecture.

  • Mechanics
  • Mathematics
  • Functional Dependencies
  • Review
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  1. PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF in Olin 103 Notes on Lecture 39 Review of topics covered in this course 1. Basic mathematical tools 2. Mathematical methods 3. Physical concepts 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 1

  2. 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 2 2

  3. Some comments on basic functional dependencies and partial and total derivatives -- ( , ) ( , li da da a -- consider ( , ) b f a + ) f f a da b f a b f m a 0 b f a f b = + df da d b b a Note that in the case, a and b are independent variables. a b 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 3

  4. Now, consider the case where and are not independent, but ( , ( )). Partial derivatives are still defined as before: ( ). b a a b b = f f a b a + ( , ) da db db ( , ) f a b f f a da b f lim a d a f a f 0 b + ( , f a b ) ( , ) f a b lim db b b 0 a 2 2 f f = a b b a 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 4

  5. We often have the situation where , , and all depend on another variable, . Now, the total time derivative is given by f f f b t dt a dt dt because ( ) implies dependence only. a b f = ( ( ), ( ); ). f a t b t t t f = df da db da a t + + = Here d t a t t da dt This becomes more complicated when ( ) ( ) t ( ) b t a t df dt f da f a da dt f t = + + a dt df dt df dt d dt d dt f a f a Compare and a Compare and a 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 5

  6. Changing functional dependencies Legendre transformations Suppose we have a function ( , ) f x y = + f x f y ( , ) f x y df dx dy y x f y f x = udx vdy + Define: and u v df y x Now suppose we want to construct a related function ( , ) = + g u y g u g y ( , ) g u y dg du dy y u = Legendre suggests we try : ( , ) = ( , ) f x y g u y ux g u g y + = + = dg du dy df u dx xdu xdu vdy y u 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 6

  7. Changing functional dependencies Legendre transformations f x f y = + Summary -- ( , ) f x y df dx dy y x f x f y = udx vdy + Define: and u v df y x = Related function (thanks to Legendre) : ( , ) = + = ( , ) f x y g u y ux g u g y = + = dg du dy df udx xdu xdu vdy y u g u g y f y = = x v y u ( , ) g u y 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 7

  8. Recipe for constructing the Hamiltonian and analyzing the equations of motion ( , ) ) = Construct 1. Lagrangian function : ( ( , ) L L q t q t t L Compute . 2 generalize momenta d : p q = Construct . 3 Hamiltonia expression n : H q p L t ( , ) ) = motion Hamiltonia Form . 4 function n : ( ( , ) H of H q p t t : Analyze . 5 canonical equations dq dp H H = = dt p dt q Note that the equations of motion should yield equivalent trajectories for the Lagrangian and Hamiltonian formulations. 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 8

  9. Review of mathematical methods 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 9

  10. 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 10

  11. https://dlmf.nist.gov/ 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 11

  12. Example special functions 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 12

  13. Complex numbers = + 2 1 Define 1 i i = z x iy ( )( ) 2 = = + = + 2 2 * z zz x iy x i y x y Polar representation sin cos i + ( ) = = i z e Functions of complex variables ( ) f z = ( ) ( ) ( ) f z ( ) f z + + ( , ) u x y ( , ) iv x y i Derivatives: Cauchy-Riemann equations ( ) f z x ( ) u z x ( ) v z x f z x ( ) f z i y ( ) u z i y u z x ( ) v z i y ( ) v z y ( ) u z y = + = + = i i i ( ) ( ) f z i y ( ) ( ) v z y ( ) v z x ( ) u z y df dz = = = Argue that = and 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 13

  14. Analytic function ( ) is analytic if it is: continuous single valued its first derivative satisfies Cauchy-Rieman conditions A closed integral of an analytic function is zero. f z However: 1 z = = Behavior of ( ) about the point 0: f z z n For an integer , consider = n = 2 2 0 1 1 n n i 1 z in e i d e = = 1 (1 ) n i n dz e i d n n 2 i 0 0 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 14

  15. Contour integration methods -- ( ) = ( ) z 2 Res ( ) f dz i f z p ( ) z = p y C No contribution z p ( ) ( ) f z z Res ( ) ( ) f z p ( ) f z 2 Res i z p z p p ( ) z = x z p z p C 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 15

  16. General formula for determining residue: ( ) Res ( z ) f z ( ) g z p Suppose that in the neighborhood of , ( ) z f z ( ) p m z z z z z p p p Since ( ) is analytic near , we ca n make a Taylor exansi n about ( ( 1)! m dz o : g z z z p p ) 1 m z z 1 m ( ) ( ) dg z d g z ( ) p p p + + + + ( ) g z ( ) ... .... g z z z p p 1 m dz ) ( ( ) m 1 m ( ) d z z f z 1 ( ) p lim z = Res ( ) f z p 1 m ( 1 )! m dz z p ( ) = ( ) z 2 Res ( ) f dz i f z p p C 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 16

  17. Fourier transforms -- Definition of Fourier Transform function a for ( : ) t f = i t ( ) ( ) f t d F e Backward transform : 1 = i t ) F( dt f(t) e 2 Check : 1 = ' i t i t ( ) ' ' f t d dt f(t ) e e 2 1 ( ) = = ' i t t ( ) ' ' ' ' ' ( ) f t dt f(t ) d e dt f(t ) t t 2 Note: The location of the 2 factor varies among texts. 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 17

  18. Properties of Fourier transforms -- Parseval's theorem: ( ) ( ) * * = ( ) ( ) 2 dt f(t) f(t) d F F * ( ) ( ) ( ) * = ' i t i t Check: ' ' dt f(t) f(t ) dt d F e d F e ( ) ( ) ( ) ' i t * = ' ' d F d F dt e ( ) ( ) ( ) * = ' ' 2 d F d F ( ) ( ) F * = 2 d F 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 18

  19. Doubly discrete Fourier Transforms Doubly per 2 iodic func tions T + , and in ege ) ( , t r s t N 2 1 T N 1 N ( ) + 2 / 2 1 i N = f F e + 2 1 N = N N ( ) + 2 / 2 1 i N = F f e = N Fast Fourier Transforms (FFT) 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 19

  20. Notions of eigenvalues and eigenvectors In the context of linear algebra -- Eigenvalue properties of matrices = My y = = * Hy y Hermitian matrix: H H ij ji Theorem for Hermitian matrices: have real values and = H y y = y = H Uy y U U I Unitary matrix: = y = H 1 and In the context of Sturm-Liouville differential equations -- 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 20

  21. Notions of eigenvalues and eigenvectors -- continued Sturm Liouville differential equations, in terms of given functions ( ), ( ), and ( ) Eigenfunctions: d d x v x f x dx dx x v x x + = ( ) ( ) ( ) ( ) x f x ( ) n n n b ( ) x f x f = Orthogonality of eigenfunctions: ( ) ( ) x dx , N n m nm n a b 2 where ( )( x ( )) f x . N dx n n a 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 21

  22. Calculus of variation a method to find a function (y(x)) which optimizes a particular integral relationship. dy dx For ( ), y x , , f x x dy dx f a necessary condition to extremize : f y(x), ,x dx x i f y d dx f = 0 ( ) / dy dx dy dx , x , x y Euler-Lagrange equation 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 22

  23. Lagrangian in the presence of electromagnetic forces Lagrangian: (using cartesian coordinates) , , , , , , L L x y z x y z t = ( ) T U U mech EM q c ( ) ( ( ) ) ( ) = + + = 2 2 2 r r A r , , T m x y z U q t t 1 2 EM A r , t 1 c ( ) ( ) ( ) ( ) = = E r r B r A r where , , , , t t t t t q c ( ) ( ) ( ) = + + + r A r 2 2 2 r , , L m x y z U q t t 1 2 mech 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 23

  24. Digression on tools for solving ordinary differential equations Method of Frobenius https://mathshistory.st-andrews.ac.uk/Biographies/Frobenius/ Born: 26 October 1849 Berlin-Charlottenburg, Prussia (now Germany) Died: 3 August 1917 Berlin, Germany Summary: Georg Frobenius combined results from the theory of algebraic equations, geometry, and number theory, which led him to the study of abstract groups, the representation theory of groups and the character theory of groups. He also developed methods for solving linear differential equations. 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 24

  25. Simple example of ordinary differential equation: 2 1 r dr 1 r d dr d + = Solutions of the differential equation: ( ) f r 0 2 2 = Frobenius method for finding solutions near Guess series solution form: ( ) 0: r s n + = f r A r n = 0 n s n + s m + = = Evaluate: ( ) f r 0 for each power of to find O A Or r n = 0 n and the condition for non-trivial . Example ( thanks to F. B. Hildebrand): relationships between coefficients A m A 0 2 d dr r d dr A = + (1 2 ) 2 1 O r r 2 ( ) ( )( ) ( ) s n + s n + s n + = = + + + + 1 0 2 2 1 2 2 1 A O s n s n r s n r n n = = 0 0 n n ( ) = Condi ion for non-trivi l t a : 2 1 0 A s s 0 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 25

  26. Continued -- Example ( thanks to F. B. Hildebrand): 2 d dr r d dr A = + (1 2 ) 2 1 O r r 2 ( ) ( )( ) ( ) s n + s n + s n + = = + + + + 1 0 2 2 1 2 2 1 A O s n s n r s n r n n = = 0 0 n n ( ) = Condi ion for non-trivi l F irst solution: Coefficient of t a : 2 1 0 A s s 0 = 0 s + + + = m : (2 1)( 1) (2 1) 0 r A m m A m + 1 m m 2 3 r r = + + + + = r ( ) 1 ... f r A r A e 1 0 0 2 3! = S econd solution : s 1 2 ( ) + + + = m Coeffic ient of : (2 3 )( 1 ) 2 1 0 r A m m A m + 1 m m (infinite series, converges slowly) 2 3 2 3 2 3 5 2 5 = + + + 1 /2 2 3 ( ) f r 1 .... A r r r r 2 0 3 7 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 26

  27. Simple example of ordinary differential equation: 2 1 r dr 1 r d dr d + = Solutions of the differential equation: ( ) f r 0 2 2 We can use the Frobenius method for this example; in this case the series truncates. 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 27

  28. Mechanics topics Scattering theory Non-inertial reference frames Lagrangian mechanics Hamiltonian mechanics Liouville theorem Rigid body motion Normal modes of oscillation about equilibrium Wave motion Fluid mechanics (ideal or including viscosity; linear and nonlinear) Heat conduction Elasticity Note: The following review slides are necessarily brief. Please refer to the original lecture slides for details. Please email: natalie@wfu.edu with any corrections/suggestions 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 28

  29. Scattering theory Note: The notion of cross section is common to many areas of physics including classical mechanics, quantum mechanics, optics, etc. Only in the classical mechanics can we calculate it from a knowledge of the particle trajectory as it relates to the scattering geometry. d bdb b Figure from Marion & Thorton, Classical Dynamics d d d b db b db Note: We are assuming that the process is isotropic in = = d d d sin sin 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 29

  30. Lagrangian mechanics Given the Lagrangian function: ( ) , q = , , L L q t T U The physical trajectories of the generalized coordinates ( ) q t ( ) , q = Are those which minimize the action: , S L q t dt Euler-Lagrange equations: d dt L q L q d dt q L L q = = 0 for each : 0 q For the case that there both mechanical and electromagnetic contributions in terms of electric and magnetic fields: , 1 , , t t c t q L T U q t c ( ) A r t ( ) ( ) ( ) ( ) = = E r r B r A r , , t t ( ) ( ) = + r r A r , , t mech 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 30

  31. Example of solving coupled equations Lagrangian equations of motion for a Lorentz force ( ) q ( ) = + + + + 2 2 2 L m x y z B y x x y 1 0 2 2 c d q q q = = 0 0 m x B y B y m x B y 0 0 0 2 q 2 q dt c c c d q + + = + = 0 0 m y B x B x m y B x 0 0 0 2 2 dt c c c d = = 0 0 m z m z dt 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 31

  32. q ( ) ( ) = + + + + 2 2 2 L 1 2 m x y z B xy yx 0 2 c q c q c = + mx B y 0 = my B x 0 = 0 mz Need to find ( ), ( ), ( ). In this case, the initial conditions are (0) 0, (0) 0, (0) z x y = = z t x t y t = = = = 0 (0) 0, (0) x , (0) 0 z U y 0 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 32

  33. Recipe for constructing the Hamiltonian and analyzing the equations of motion ( ) = 1. Construct Lagrangian function: ( ) , ( ) , L L q t q t t L q 2. Compute generalized momenta: p = 3. Construct Hamiltonian expression: H q p L ( ) = 4. Form Hamiltonian function: ( ) , ( ) , H H q t p t t 5. A nalyze canonical equations of motion: dq H dt p dp dt H q = = 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 33

  34. Important tool for analyzing Lagrangian and/or Hamiltonian systems -- finding constants of the motion In Lagrangian formulation -- For independent generalized coordinates ( ): L q q t d dt q L L ( ) = = ( ) , ( ) , 0 L L q t q t t L q d dt q L q = = = Note that if 0, then 0 (constant) d dt L q L t = Additionally: L q L t L q = = For 0 (constant) L q E 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 34

  35. Constants of the motion in the Hamiltonian formulation ( ) H p = ( ) , ( ) , H H q t p t t dq dt dp dt H p H q = = constant if 0 q H q = = constant if 0 p dH dt dH dt H q H p H t = + + q p H t H t ( ) = + + = p q q p H t = constant if 0 H 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 35

  36. Question Why use this fancy formalism when simple conservation of energy or momentum intuitively apply? a. You should use your intuition whenever possible. b. You should never trust your intuition. c. The equations should be consistent with correct intuitive solutions and also reveal additional solutions (perhaps beyond intuition) 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 36

  37. Liouvilles Theorem (1838) The density of representative points in phase space corresponding to the motion of a system of particles remains constant during the motion. ( , ) t ) = Denote density the q of particles phase in space : ( ( , ) t D D q p t dD D D D = + + q p dt p t dD = According Liouville' to theorem s : 0 dt 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 37

  38. Rigid body motion Moment I In a reference frame attached to the object, there are 3 moments of inertia and 3 distinct principal axes of inertia p 1 tensor r : ( ) 2 r (dyad notation) m p r p p p Representation of rotational kinetic energy: ( 1 2 ) 1 1 1 ~ ~ ~ = + + 2 2 2 2 3 , , , , , T I I I 1 2 3 2 2 1 ( ) 2 = + sin cos sin I 1 2 1 ( ) 2 + + sin sin cos I 2 2 1 2 + + cos I 3 2 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 38

  39. Eulers transformation between body fixed and inertial reference frames ~ = + + 0 3 ' 2 e e e 3 0 3 e ( ) ) + = + e sin cos sin 1 ( + e sin sin cos 2 + + e cos 3 e 3 x y y x ' 2 e 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 39

  40. Normal modes of vibration -- potential in 2 and more dimensions 2 V x ( ) 2 + ( , ) V x y ( , ) V x y x x 1 2 eq eq eq 2 , x y eq eq 2 2 V y V ( ) ( )( ) 2 + + y y x x y y 1 2 eq eq eq 2 x y , , x y x y eq eq eq eq V(x,y) 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 40

  41. Example normal modes of a system with the symmetry of an equilateral triangle -- continued 3 u3 Potential contribution for spring 13: 1 2 ( ) 2 = + u u V k 13 13 3 1 13 13 2 ( ) u u 1 2 13 3 1 k u2 13 2 1 2 1 2 1 2 3 ( ) ( ) + k u u u u u1 3 1 3 1 x x y y 2 1 2 3 = + x y 13 13 2 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 41

  42. Example normal modes of a system with the symmetry of an equilateral triangle -- continued = + + Potential contributions: V V V V 12 13 23 2 2 ( ) ( ) u u u u 1 2 1 2 + 13 3 1 12 2 1 k k 12 13 2 ( ) u u 1 2 + 23 3 2 k 23 1 2 ( ) 2 k u u 2 1 x x 2 1 2 1 2 3 ( ) ( ) + + k u u u u 3 1 3 1 x x y y 2 2 1 2 1 2 3 ( ) ( ) + k u u u u 2 3 2 3 x x y y PHY 711 Fall 2024 -- Lecture 39 2 11/25/2024 42

  43. Example normal modes of a system with the symmetry of an equilateral triangle -- continued u u u u u u u u 1 x 1 x 2 x 2 x k m 3 x 3 x = 2 1 y 1 y u u 2 y 2 y u u 3 y 3 y 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 43

  44. Discrete particle interactions continuous media The wave equation Initial value solutions attributed to D'Alembert: to the wave equation; (x,t) 2 2 t = = ( ) and x = 2 0 where 0 0 ( ) x c (x, ) (x, ) 2 2 t x x ct + 1 2 1 2 ( ) ( , ) x t = x ct + + + ( ') x dx ( ) ( ) ' x ct c x ct 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 44

  45. Mechanical motion of fluids Newton s equations for fluids Use Euler formulation; following particles of fluid Density : Variables (x,y,z,t) Pressure p(x,y,z,t) (x,y,z,t) v Velocity Navier-Stokes equation + v 1 1 1 3 ( ) ( ) = f + + + 2 v v v v p t Continuity condition + ( ) = v 0 Viscosity contributions t 11/25/2024 PHY 711 Fall 2024 -- Lecture 39 45

  46. Fluid mechanics of incompressible fluid plus surface Non-linear effects in surface waves: p0 z h y z=0 x Dominant non-linear effects soliton solutions 3 x ct h ( , ) x t = = 2 sech constant 0 0 0 2 h gh = + where 1 0 h c gh PHY 711 Fall 2024 -- Lecture 39 0 1 / h 2 11/25/2024 46

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