“On the Shoulders of Giants”

1. “On the Shoulders of Giants” An Introduction to Classical Mechanics

2. If I have seen further it is by standing on the shoulders of giants. • Isaac Newton, Letter to Robert Hooke, February 5, 1675English mathematician & physicist (1642 - 1727)

3. Quantum Mechanics (QM) is based on classical mechanics. It combines classical mechanics with statistics and statistical mechanics. For native English-speakers, it is somewhat unfortunate that it uses the word “quantum”. A better English word which describes the thrust of this approach would be “pixel”.

4. Lights! Camera! Action! • 2nd Century BC • Hero of Alexandria found that light, traveling from one point to another by a reflection from a plane mirror, always takes the shortest possible path. • 1657 • Pierre de Fermat reformulates the principle by postulating that the light travels in a path that takes the least time! • In hindsight, if c is constant then Hero and Fermat are in complete agreement. • Based on his reasoning, he is able to deduce both the law of reflection and Snell’s law (nsinQ = n’ sinQ’)

5. An Aside • Fermat is most famous for his last theorem: Xn +Yn = Zn where n=2 and … • On his deathbed, he wrote: And n= arrgh! I’m having a heartattack! • His last theorem was only solved by computer in the last 10 years…

6. Now we wait for the Math • 1686 • The calculus of variations is begun by Isaac Newton • 1696 • Johann and Jakob Bernoulli extend Newton’s ideas

7. Now we can get back • 1747 • Pierre-Louise-Moreau de Maupertuis asserts a “Principle of Least Action” • More Theological than Scientific • “Action is minimized through the Wisdom of God” • His idea of action is also kind of vague • Action (today’s definition)— • Has dimensions of length x momentum or energy x time • Hmm… p * x or E*t … seems familiar…

8. To the Physics • 1760 • Joseph Lagrange reformulates the principle of least action • The Lagrangian, L, is defined as L=T-V where T= kinetic energy of a system and V=potential energy of a system

9. Hamilton’s Principle • 1834-1835 • William Rowan Hamilton’s publishes two papers on which it is possible to base all of mechanics and most of classical physics. • Hamilton’s Principle is that a particle follows a path that minimizes L over a specific time interval (and consistent with any constraints). • A constraint, for example, may be that the particle is moving along a surface.

10. Lagrange’s Equations

11. Lagrange’s Equations And I can add zero to anything and not change the result

12. Expanding to 3 Dimensions Since x, y, and z are orthogonal and linearly independent, I can write a Lagrange’s EOM for each. In order to conserve space, I call x, y, and z to be dimensions 1, 2, and 3. So Amusingly enough, 1, 2, 3, could represent r, q, f (spherical coordinates) or r, q, z (cylindrical) or any other 3-dimensional coordinate system.

13. Example: Simple Harmonic Oscillator • Recall for SHO: V(x)= ½ kx2 and let T=1/2 mv2 • Hooke’s Law: F=-kx

14. Tip • The trick in the Lagrangian Formalism of mechanics is not the math but the proper choice of coordinate system. • The strength of this approach is that • Energy is a scalar and so is the Lagrangian • The Lagrangian is invariant with respect to coordinate transformations

15. Two Conditions Required for Lagrange’s Equations • The forces acting on the system (apart from the forces of constraint) must be derivable from a potential i.e. F=-dU/dx or some similar type of function • The equations of constraint must be relations that connect the coordinates of the particles and may be functions of time.

16. Your Turn • Projectile: • Go to the board and work a simple projectile problem in cartesian coordinates. Don’t worry about initial conditions yet. • Now do the same in polar coordinates. • Hint:

17. Introducing the Hamiltonian • First, any Lagrangian which describes a uniform force field is independent of time i.e. dL/dt=0.

18. Introducing the Hamiltonian • Hmmm… H for Hornblower or Hamilton?

19. Introducing the Hamiltonian

20. H is only E when • It is important to note that H is equal to E only if the following conditions are met: • The kinetic energy must be a homogeneous quadratic function of velocity • The potential energy must be velocity independent • While it is important to note that there is an association of H with E, it is equally important to note that these two are not necessarily the same value or even the same type of quantity!

21. Making Simple Problems Difficult with the Hamiltonian • Most students find that the Lagrangian formalism is much easier than the Hamiltonian formalism • So why bother?

22. Making Simple Problems Difficult with the Hamiltonian • First, we need to define one more quantity: generalized momenta, pj

23. SHO with the Hamiltonian • Big deal, right? • But look what we did • L=f(q,dq/dt,t) • H=f(q,p,t) • So our mechanics all depend on momentum but not velocity • Recall light has constant velocity, c, but a momentum which is p=hc/l !

24. The Big Deal • So if we are going to define mechanics for light, it does not make any sense to use the Lagrangian formulation, only the Hamiltonian!

25. That Feynman Guy! • Richard Feynman thought that Lagrangian mechanics was too powerful a tool to ignore. • Feynman developed the path integral formalism of quantum mechanics which is equivalent to the picture of Schroedinger and Dirac. • So which is better? Both and Neither • There seems to be no undergraduate treatment of path integral formalism.

26. Hamilton’s Equations of Motion • Just like Lagrangian formalism, the Hamiltonian formalism has equations of motion. There are two equations for every degree of freedom • They are

27. Finishing the SHO Hooke’s Law again!

28. Symmetry • Note that Hamilton’s EOM are symmetric in appearance i.e. that q and p can almost be interchanged! • Because of this symmetry, q and p are said to be conjugate

29. Definition of Cyclic • Consider a Hamiltonian of a free particle i.e. H=f(p)… then – dp/dt=0 i.e. momentum is a “constant of the motion” • Now in the projectile problem, U=-mgy and for x-component, H=f(px) only! • Thus, px= constant and the horizontal variable, x is said to “cyclic”! • A more practical definition of cyclic is “ignorable” and modern texts sometimes use this term.

30. Definition of canonical • Canonical is used to describe a simple, general set of something … such as equations or variables. • It was first introduced by Jacobi and rapidly gained common usuage but the reason for its introduction remained obscured even to contemporaries • Lord Kelvin was quoted as saying “Why it has been so called would be hard to say”

31. Poisson Brackets

32. Kronecker Delta • di,k=1 if i=k • di,k=0 if i≠k

33. Back to Fish • Consider two continuous functions g(q,p) and h(q,p) • If {g,h}=0 then h and g are said to commute In other words, the order of operations does not matter • If {g,h}=1 then quantities are canonically conjugate • A look ahead: we will find that canonically conjugate quantities obey the Uncertainty principle

34. Properties of Fish

35. Levi-Civita Notation

36. Levi-Civita Notation