May the force be with you!

Fundamental vs. non-fundamental force

TosolveNewton’smechanicsproblems,oneneedtoknowforces.Therearefourfundamentalforces:Thegravita=onalforcelawfoundbyNewtonisfundamental.Bynow,allfundamentallawsofotherthreeforcesareknownaswell:electromagne=c,weakandstrong.

Friction

The E&M force: Coulomb’s law

Strong force

• Strongforceistheonethatbindstogethertheneutronsandprotonsinnuclei.• Thisfundamentalforcewasfinallyunderstoodin1970swhencolorchargeandatheoryofcolorinteractionsareproposedandverified.(Chromodynamics)• However,solvingthe“NewtonEquation”isveryverydifficulty.Theonlywaytodothisisusingalargescalecomputer!

Weak force

• Weakforceistheonethatchangestheprotonintoaneutronandhenceamongdifferentelements.• Weakforceisthefirststepinthesunburning.• Thefundamentaltheoryofweakforcewasproposedin1967,wasverifiedovertheyears.• Theweakforcehasthesameoriginastheelectromagneticforce.• Theweakforcealsogivethemasstotheelectron!

Pseudo force due to non-inertial frame• Ininertialframes,alltheforcesare“physical”.Alllawsininertialframesarethesame.• However,whenworkinginnon-inertialframe,thereareextraforcesinNewton’sequation.Forexamplewhenaframeisacceleration,oneneedstotakeintoaccountthisspurious“force”• Sometimeunavoidable.LiketheCoriolisforceontheEarth->Foucaultpendulum.

Question

Question

Braess' paradox

Coordinates transformation

Coordinate translation

Translation

Translational invariance

• Thenewton’slawofphysicsisvariantundertranslation

• Theforcedoesnotchangeundertranslation

• Theaccelerationdoesnotchangeundertranslation

• Thisistheoriginofmomentumconservation

• Webelievealllawsofphysicsareinvariantunder

translation

Time translational symmetry

•

Rotations

11 12 13

21 22 23

31 32 33

'''

x a a a xy a a a yz a a a z

⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟=⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

Rotational matrix

•

Rotational symmetry

•

Galileo transformation

• Twocoordinatesarerelatedbyconstantspeed,sowehavetwocoordinates

ThisiscalledGalileantransformation.

0

0

0

'''

'

x

y

z

t tx x v ty y v t

z z v t

=

= −

= −

= −

0

''t tr r v t=

= −! ! !

Newton’s equations are invariant under Galileo transformation

• Galileanprincipleofrelativity.Nottruewhenvelocityislarge

0

''t tr r v t=

= −! ! ! 0 0dv

dt=

!'a a=! !

'F F=! !

F ma=! ! ' 'F ma=

! !