Mechanics Elementary Principles , Lecture Notes - Physics, Study notes of Mechanics

Download Mechanics Elementary Principles , Lecture Notes - Physics and more Study notes Mechanics in PDF only on Docsity! Mechanics Physics 151 Lecture 2 Elementary Principles (Goldstein Chapter 1) Administravia ! First Problem Set ! 3 problems for the section ! Work on them before coming to your section! ! 3 for the report (due next week) ! If you haven’t filled the Survey, please do it ! We need it for sectioning and study-grouping ! Section time: Tue. 6 PM, 7 PM and Wed. 5 PM ! If none of these slots works for you, let me know ! Sectioning will be announced on Monday by email ! We will also assign you into study groups (~6 each) System of Particles ! More than one particles? " Just add indices! ! Subtlety: F may be working between particles ! Distinguish between internal and external forces ! Now add up over i to see the overall picture ii LN !=ii pF != ( )e i ji i j = +∑F F F Force acting on particle i Force from particle j Force from outside Mechanics Physics 151 Lecture 2 Elementary Principles (Goldstein Chapter 1) Administravia ! First Problem Set ! 3 problems for the section ! Work on them before coming to your section! ! 3 for the report (due next week) ! If you haven’t filled the Survey, please do it ! We need it for sectioning and study-grouping ! Section time: Tue. 6 PM, 7 PM and Wed. 5 PM ! If none of these slots works for you, let me know ! Sectioning will be announced on Monday by email ! We will also assign you into study groups (~6 each) System of Particles ! More than one particles? " Just add indices! ! Subtlety: F may be working between particles ! Distinguish between internal and external forces ! Now add up over i to see the overall picture ii LN !=ii pF != ( )e i ji i j = +∑F F F Force acting on particle i Force from particle j Force from outside Sum of Particles ! This term vanishes if ! Weak law of action and reaction ( )( ) ( ) , e e i ji i ji ij i i i j i i j i i j < ≠ = + = + +∑ ∑ ∑ ∑ ∑F F F F F F jiij FF −= Forces two particle exert on each other are equal and opposite Forces two particle exert on each other are equal, opposite, and along the line joining the particles ! C.f. the strong law of action and reaction ( )e i i i i =∑ ∑F F Sum of Particles ! Now consider the equations of motion ! Define center of mass ∑∑∑∑ === i ii i i i e i i i mdt d rpFF 2 2 )( ! Center of mass moves like a particle of mass M under total external force F(e) i i i i i m m m M ≡ =∑ ∑ ∑ r r R )()( e i e iM FFR ≡=∑!! Total Angular Momentum ! Assuming strong law of action and reaction " Conservation of total angular momentum )()()( e i e i i e ii NNFrL ==×= ∑∑! If the total external torque N(e) is zero, the total angular momentum L is conserved A multi-particle system (= extended object) can be treated as if it were a single particle if the internal forces obey the strong law of action and reaction Laws of Action and Reaction ! Most forces we know obey strong law of action and reaction ! Gravity, electrostatic force ! There are rare exceptions ! E.g. Lorenz force felt by moving charges ! Conservation of linear & angular momenta fails +Q +Q v1 v2 F21 = 0 F12 ! Take into account the EM field ! Particles exchange forces with the field ! The field itself has linear & angular momenta " Conservation laws restored Conservation Laws ! We will see (in 2 lectures) that P and L must be conserved if the laws of physics are isotropic in space ! No special origin ! No special orientation ! If we accept these symmetries as fundamental principles, all forces must satisfy the action-reaction laws " “Proof” of Newton’s 3rd Law Conservation of P Conservation of L Weak law of action-reaction Strong law of action-reaction Potential Energy ! Assume conservative external force ! Assume also conservative internal forces ! To satisfy strong law of action/reaction 2 2 2( ) 11 1 e i i i i i i i i i d V d V⋅ = − ∇ ⋅ = −∑ ∑ ∑∫ ∫F s s       ∂ ∂ ∂ ∂ ∂ ∂≡∇ iii i zyx ii e i V−∇= )(F ijiji V−∇=F |)(| jiijij VV rr −= Potential depends onlyon the distance ∑ ≠ − ji ji ijV , 2 12 12 2 1 1 , , ji i i ij i i j i j i j i j d V d ≠ ≠ ⋅ = − ∇ ⋅∑ ∑∫ ∫F s s Bit of work Energy Conservation ! If all forces are conservative, one can define total potential energy: ! Then the total energy T + V is conserved ! The second term is internal potential energy ! It depends on the distances between all pairs of particles ! Constant if particles’ relative configuration is fixed " Rigid bodies ∑∑ ≠ += ji ji ij i i VVV ,2 1 Constraints ! Equation of motion assumes that particles can move anywhere in space ! Not generally true ! In fact never true – Free space is an idealization ! Amusement-park ride constrained (hopefully) on a rail ! Billiard balls on a pool table ! How can we accommodate constraints in the equation of motion? ! Depends on the type of the constraint ∑+== j ji e iiiim FFFr )(!! Generalized Coordinates ! N particles have 3N degrees of freedom ! Introducing k holonomic constraints reduces it to 3N – k ! Using generalized coordinates q1, q2,…, q3N – k ! Example: ),,,,( 321 tqqqii …rr = Transformation equations from (rl) to (ql)      θ= φθ= φθ= cos sinsin cossin cz cy cx Transformation from (x, y, z) to (θ, φ) Now What? ! We know the equations of motion for (ri) ! We know how to include constraints by switching to generalized coordinates ! How can we transform the equation of motion to the generalized coordinates? ∑+== j ji e iiiim FFFr )(!! ),,,,( 321 tqqqii …rr = Lagrange’s Equations Why Constraints? ! Constraint is an idealized classical concept ! Nothing is perfectly constrained in QM ! How useful is it to switch between coordinates? Uncertainty More than it seems Lagrange’s Equations ! Express L = T – V in terms of generalized coordinates , their time-derivatives , and time t ! The potential V = V(q, t) must exist ! i.e. all forces must be conservative ! Let’s do a quick example to see how it works 0 j j d L L dt q q  ∂ ∂− =  ∂ ∂ ! ( , , )L q q t T V≡ −! Kinetic energy Potential energy Lagrangian Recipe { }jq { }jq! Ex: Particle on a Line ! A particle moving on the x-axis ! Kinetic and potential energies: ! Equivalent to Newton’s Eqn given that 0 j j d L L dt q q  ∂ ∂− =  ∂ ∂ ! ( ), 0, 0x x t y z= = = 2 2 mT x= ! ( )V V x= 2 ( ) 2 mL x V x= −! Lagrange’s Eqn 0 Vmx x ∂+ = ∂ !! x VF x ∂= − ∂ OK, it works Summary ! Discussed multi-particle systems ! Internal and external forces ! Laws of action and reaction ! Momenta, conservation laws, kinetic & potential energies ! Introduced constraints ! Holonomic and nonholonomic constraints ! Generalized coordinates ! Introduced Lagrange’s Equations ! Next: Prove that Lagrange’s and Newton’s Equations are equivalent