Mechanics , Lecture Notes - Physics, Study notes of Mechanics

Download Mechanics , Lecture Notes - Physics and more Study notes Mechanics in PDF only on Docsity! Mechanics Physics 151 Fall 2003 Masahiro Morii Teaching Staff ! Lecturer: Masahiro Morii ! Tuesday/Thursday 11:30 – 1:00. Jefferson 256 ! Section leaders: Srinivas Paruchuri and Abdol-Reza Mansouri ! Two or three 1-hour sections per week ! Date/time to be announced ! Please fill out the student survey ! Course assistant: Carol Davis ! She will have all course materials (problem sets, etc.) Textbook ! Classical Mechanics, Goldstein, Poole and Safko ! Required ! Classic (literally) textbook. Originally published in 1950 ! A must-read for serious physicists ! 3rd edition came out in 2001 ! 2nd edition still good (or better) – Get it if you can ! Will follow this textbook closely ! Except for skipping a few advanced materials ! It’s a 600-page book written for graduate students Grading ! Grades will be based on a weighted average of ! Homework 40% ! Mid-term exam 20% ! 1-hour exam. After 10 lectures ! Final exam 40% ! Exam period. 3 hours Homework ! Problem sets are distributed on Thursdays ! Reports are due at the next week’s Thursday lecture ! Typical format: ! 2–3 problems that will be discussed at sections ! 3–4 problems you solve and turn in report ! Work together in groups ! Groups will be assigned according to the Survey ! Each of you must turn in your own report, though Mechanics ! Mechanics concerns ! Motion of objects " Velocity and acceleration ! Cause of the motion " Force and energy ! The objects move, but do not change their properties ! Idealized particles and rigid bodies ! Mass and moment of inertia are all what matters ! Newton’s Three Laws of Motion ! You remember them, right? ! Principia (1687) pretty much wrapped it up Mechanics: A branch of physical science that deals with energy and forces and their effect on bodies (Webster’s) Classical vs. Modern ! “Modern” in physics means “20th century” ! Quantum Mechanics ! Relativity ! Classical Mechanics = pre-Quantum Mechanics ! We include special relativity as well as E&M ! What happened between the 17th and 20th centuries? Do We Care? ! We know Relativity and QM are the “right answers” ! Newtonian Mechanics is a human-scale approximation ! Isn’t that enough? ! Why should we learn the theory that has been superseded? (An advanced course in classical mechanics) introduces no new physical concepts to the graduate student. It does not lead him directly into current physics research. Nor does it aid him, to any appreciable extent, in solving the practical mechanics problems he encounters in the laboratory. Goldstein, Preface to the First Edition Generalizing Equation of Motion ! Newtonian Mechanics deals with the object’s position ! Goal: finding x = x(t), y = y(t), z = z(t) ! 3 coordinates for each object " 3N for N objects ! But there are infinite other ways to describe motion ! E.g. a more natural way for a pendulum ! Number of free variables may not be 3N ! Let’s call the new variables generalized coordinates ! What are the Equations of Motion for generalized coordinates? cos , sin , 0, ( )x L y L z tθ θ θ θ= = − = = Lagrangian Formulation ! Newton’s Equation is about force ! You start from F = F(x, t) for all particles ! 3N functions corresponding to 3N coordinates ! Forget the force. Introduce something else ! Lagrangian: ! Lagrange’s Equation ! Lagrangian does not depend on a coordinate system ! Switching to a different set of coordinates is a snap ( , )L L q q= ! Coordinate q and its time derivative 0d L L dt q q  ∂ ∂− = ∂ ∂ ! Everything about this system is embodied in a scalar function L m=F a Hamilton’s Principle ! Hamilton’s Principle derives Lagrange’s Equation from a simple rule: Rather weird statement… ! Newton’s Laws were found by induction ! “It is so because it agrees with many observations” ! Deriving them from a principle means knowing why it is so ! Not quite that dramatic, but it does suggest deeper reason ! Eventually connected to Feynman’s path integral ! Besides, calculus of variations is a useful technique The time integral of L is stationary for the path taken by an actual physical system 0 2 1 =δ∫ Ldt What We Will Study ! Lagrange’s Equations, Hamilton’s Principle ! Central force problem ! Rigid body motion ! Oscillation ! Extension to special relativity ! Hamilton Equation, Canonical transformations ! Hamilton-Jacobi Equation ! Advanced stuff ! Classical chaos? ! Perturbation theory? ! Field theory? Mechanics Physics 151 Lecture 1 Elementary Principles (Goldstein Chapter 1) Goals for Today ! Review basic principles of Newtonian Mechanics ! Very quickly so that you don’t fall asleep ! Discuss motion of a single particle ! Define standard notations and usages ! Momenta, conservation laws, kinetic & potential energies ! You (should) already know all this Inertial Systems ! Consider two inertial systems A and B ! A particle is at rA in A, rB in B ! Origin of A is at rB – rA in B ! Equivalence of such systems was pointed out by Galileo ! Hence the name Galilean system Ar Br AB rr − AO BOconst 0 =−→ =−→== AB ABBA mm rr rrrrF !! !!!!!!!! Any two inertial systems are moving relative to each other at a constant velocity Angular Momentum ! Define ! Angular momentum ! Moment of force (= torque) ! From one can deduce ! Too easy to write down here ! Subtlety: the definitions depend on the origin O ! Because r is defined from O ! The equation holds for any origin prL ×= FrN ×= The order matters! LN !=pF != LN != Momentum Conservation ! Two conservation theorems follow ! From ! From LN != pF != If the total force F is zero, the linear momentum p is conserved If the total torque N is zero, the angular momentum L is conserved This is really getting too easy… Potential Energy ! F is conservative ↔ F is expressed by ! V is the potential energy ! Work W12 is then expressed by ! Which was equal to T2 – T1 )(rF V−∇= 21 2 112 VVdW −=⋅= ∫ sF 2211 VTVT +=+ If the force is conservative, the total energy T + V is conserved Energy Conservation Theorem Summary ! Reviewed basic principles of Newtonian Mechanics ! Define standard notations and usages ! Momenta, conservation laws, kinetic & potential energies ! I hope everything looked familiar, if boring ! It will get better from here ☺ ! Next: multi-particle system & constraints