Transcript Lecture1x
Theoretical Mechanics: Lecture 1 Martikainen Jani-Petri Practicalities • [email protected] (N201b) • In 2nd period [email protected] • 5cp, 2 h lectures and 2 h exercises • Return exercises before your exercise session on the following week (ask Antti/Miika how last week in PII handled) • Book: Fetter & Walecka: “ Theoretical mechanics of particles and Continua” • Recommend you to get it! Practicalities Practicalities • 1st Midterm wed 28.10.2016 9 am-12 (at B/C?) • Language: english (also in the exam. If you are not OK with this, you need a special permit under current rules. Not up to teachers.) • Grading (OK?): o midterms 4 problems max 24 points each o Max 6+6 points bonus from exercises and activities o Exercise bonus=min(midterm points,Exercise bonus) so that all have an incentive to work in both periods o Max points=24+6+24+6=60 • Last year somebody got through with 25 points • Some changes from last year… Practicalities • We will be using MyCourses • Additional material etc. will be there. • If you registered in Oodi you should have access to the course in mycourses.aalto.fi • Assistant: Miika Mäkelä ([email protected]) Overview: Period I 1. 2. 3. 4. 5. 6. Recap/warm up on classical mechanics Generalized coordinates, constraints etc… Lagrangian dynamics Variational calculation, Hamilton’s principle Hamilton’s dynamics Small oscillations In 2nd period: Fluid mechanics, elastic materials… General content: st 1 period • You have learned about Newton’s • Here we will formulate alternative and easier ways to deal with classical mechanics • Lagrangian formulation, Hamiltonian formulation… • Generalize more naturally to field theories, quantum theories • Makes many computations easier: less vector quantities, coordinate choices more natural, symmetries more transparent…. • Same toolbox for all of physics • Mechanics “done right” Recap:concepts from classical mechanics Newton’s laws • Understand how everyday objects behave • Accurate within reason. (Quantum mechanics, relativity…) • Machines, atmosphere, satellites, students… Newton’s I law • Define primary inertial coordinate system at rest with respect to fixed stars o “In this primary inertial frame, every body remains at rest or in uniform motion unless acted on by a force F , The condition F=0 thus implies a constant velocity v and a constant momentum p=mv.”F&W • Aristotle would have disagreed! • Galileo Galilei laid the groundwork for N1 and Rene Descartes stated it clearly in 1644 (Newton was born 1642!) • This is actually quite counterintuitive based on real world experience • Experimental verification always approximate (gravity is always there) Galileo: inertia • If friction could be eliminated… …ball reaches the same height Ball continues forever! Need force to make it stop. Galileo’s thought experiment • Fall with same acceleration Newton’s II law • “In the primary inertial frame, application of a force alters the momentum, in an amount specified by the quantitative relation…” • If mass is conserved • …and Newton’s III law • “To each action, there is an equal and opposite reaction. Thus if F12 is the force exerted on particle 1 by particle 2, then • and these forces act along the line separating the particles”: F&W Newtons’s laws • Note that 3rd law together with F=m a can be used to define “mass” in terms of fundamental unit m* • This is independent of specific force law • Galilean relativity: Any frame moving with constant velocity relative to inertial frame is also inertial. • Equations of motion are the same (at least if F12=F12(|r1-r2|), as is usually the case…why?) What if mass is not conserved? • Examples? • Rockets, droplets, ablation pressure (rockets/inertial confinement fusion/thermonuclear weapons?) Conservation of momentum • See notes… • Linear momentum: p conserved if no forces Conservation of energy • See notes… • Energy (conservative forces) • What goes on here? Tic-Tac bounce? Remember also rotational degrees of freedom! Conservation of angular momentum • See notes… • Angular momentum conserved if no torque Why are conservation laws useful? Conservation laws • …are in fact related to continuous symmetries of the underlying laws (we will get back to this) • Translational invariance momentum conservation • Time invariance -> energy • Rotational invariance angular momentum • Magnetic field trickier for angular momentum conservation. Many particles • N particles…define center of mass R (M=total mass): • External forces and forces from other particles • For the CM Many particles • Last term vanishes. Why? • So we have • CM moves as if external force acted on M concentrated at R • When is the total momentum constant? • Total angular momentum similarly Many particles • We will get (if forces Fij along ri-rj)… (always true?) Relating change in total angular momentum to external torque. • Often useful to move to a coordinate system centered at center of mass • We then have Many particles • The primed one was the internal angular momentum about the CM • L is independent of R ONLY if V=0. • Change in L’ about CM is equal to external torque about CM • This holds even in a non-inertial frame! • L’ constant of motion if no torque about center of mass Many particles • For energy we can do similar things. Total kinetic energy • Potential energy (conservative forces and no selfforces V(rii)=0): Many particles • Similarly as for 1-particle system we get a conservation law for energy • Rigid body means what here? How does it simplify things? • For rigid body rij are fixed so sum over the interparticle potentials just adds a constant. Then only external potential can change the kinetic energy. Warm up: double pendulum • Newton’s approach • Let us do this together Non-inertial frames Accelerating coordinate systems: HOWTO • Newton’s laws simple in inertial frames • Move to non-inertial ones and surprising terms may appear in equations of motion • These are known as fictitious forces • Examples? centrifugal force, Coriolis force Centripetal force is a real one Accelerating coordinate systems • Newton’s law in accelerated/rotated coordinate system 10.2 • See notes/book… V is any vector, ω angular velocity vector in the inertial frame Accelerating coordinate systems • Examples of effects… Central forces and other stuff • • • • • • 2-body problem, reduced mass Kepler’s laws Scattering Accelerating coordinate systems Foucault’s pendulum (study the material yourself from the book) Next week • Lagrangian dynamics! • Beginning of Chapter 3 from the book…until calculus of variations. Earth: case study • • • • • • Inertial frame with sun in the center Non-inertial fixed in rotating earth with origin in earths center Earth around sun in 32x10^7 s Earth around itself in 8.62x10^4 s Earths gravity relative to sun (at earth) about 1660 bigger Motion of the particle on the rotating earth… • Examples… • Particle on scale --> geoid idea (tangential component towards equator) • Falling particle