STiCM
Select / Special Topics in Classical Mechanics
P. C. Deshmukh
Department of Physics Indian Institute of Technology Madras Chennai 600036
STiCM Lecture 13: Unit 4 Symmetry and Conservation Laws Dynamical Symmetry in the Kepler Problem
PCD_STiCM 1 Unit 1 : Equations of Motion (Newton / Lagrange / Hamilton )
Unit 2: Oscillators (Free/damped/driven), Resonances, Waves
Unit 3: Polar Coordinate Systems
Symmetry Conservation laws (Noether)
Unit 4 : Dynamical Symmetry in the Kepler Problem
PCD_STiCM 2 Unit 4: Learning goals: Recapitulate: Conservation of energy that is well-known in the Kepler-Bohr problem stems from the symmetry with regard to temporal translations (displacements on the time axis).
GETTING CONSERVATION LAWS FROM THE EQUATION OF MOTION Conservation of angular momentum, likewise, stems from the central field symmetry in the Kepler problem.
Neither of these accounts for the fact that the Kepler ellipse remains fixed; that the ellipse does not undergo a ‘rosette’ motion.
PCD_STiCM 3 This unit will discover the ‘dynamical’ symmetry of the Kepler problem and its relation with the constancy (conservation) of the LRL vector, which keeps the orbit ‘fixed’.
Motivation: Connection between symmetry & conservation laws has important consequences on issues at the very frontiers of physics and technology. Mechanics of
Flights into SpacePCD_STiCM 4 Mechanics of Flights into Space
Konstantin Tsiolkovsky Robert H. Goddard (1857-1935) (1882-1945) Dr. Vikram Sarabhai
Father of India’s Hermann Oberth Space Program (1894-1989) PCD_STiCM 5 Indian Space Research Organisation (ISRO)
[1] Indian National Satellites (INSAT) - for communication services [2] Indian Remote Sensing (IRS) Satellites - for management of natural resources [3] Polar Satellite Launch Vehicle (PSLV) - for launching IRS type of satellites [4] Geostationary Satellite Launch Vehicle (GSLV)
- for launching INSAT type of satellites. PCD_STiCM 6 Gravity plays the most important role in designing satellite trajectories, of course, and hence we study the Kepler TWO-BODY problem
We must then adapt the formalism to understand the models, methods and applications of satellite orbits,
etc. PCD_STiCM 7 Other than ‘energy’ and ‘angular momentum’, what else is conserved, and what is the associated symmetry?
For given physical laws of nature, what quantities are conserved?
Rather, if you can observe what physical quantities are conserved, can you discover the physical laws of nature?
PCD_STiCM 8 How did Kepler deduce that planetary orbits are ellipses around the sun ?
How would you solve this problem?
Kepler had no knowledge of : (a) differential equations
Johannes Kepler (b) inverse square force 1571- 1630 (gravity).
PCD_STiCM 9 How did Kepler deduce that planetary orbits are ellipses around the sun ?
Kepler got his elliptic orbits not by solving differential equations for gravitational force, but by doing clever curve fitting of Tycho Brahe’s experimental data.
PCD_STiCM 10 Tycho Brahe (1546-1601):
Danish astronomer appointed as the imperial astronomer under Rudolf II in Prague, which then came under the Roman empire.
Brahe had his own “Tychonian” model of planetary motion:
Brahe had planets revolve around the sun (like the Copernican system), and the sun and the moon going around the earth (like the system of Ptolemy).
What was Brahe’s nose was made of ?
PCD_STiCM 11 Tycho Brahe (1546-1601):
He discovered the supernova in 1572, in “Cassiopeia’.
Kepler and Brahe never could collaborate successfully.
They quarreled, and Tycho did not provide Kepler any access to the high precision observational data he (Brahe) had complied. It was only on his deathbed, saying “……let me not seem to have lived in vain…….” that Brahe handed over his observational data to Kepler [Ref.:Sagan]. PCD_STiCM 12 Johannes Kepler Galileo Galilei Isaac Newton 1571- 1630 1564 - 1642 (1642-1727)
Causality, Determinism, Equation of Motion ‘Dynamics’ came well AFTER KEPLER! PCD_STiCM 13 Two-body problem m 1 r R R 21 r R uˆ 1 r m2 R2
Fby_1_ on _ 2 Fby_ 2_ on _1
mm12 mm12 m22 R G2 uˆ m11 R G2 uˆ RR21 RR21
PCD_STiCM 14 Two-body problem: Centre of Mass
m1 R 1 m 2 R 2 RCM r11 R RCM m1 mm12 r22 R RCM R1 RCM m2 mm12 R2
m1 r 1 m 2 r 2 m 1()() R 1 RCM m 2 R 2 R CM
m1 R 1 m 2 R 2 m 1 m 2 RCM 0.
r R2 R 1 r 2 r 1
PCD_STiCM 15 Fby_1_ on _ 2 Fby_ 2_ on _1 r mm uˆ mm m R G12 uˆ 12 22 2 r m11 R G2 uˆ RR21 RR21 r r R2 R 1 G m 1 m 2 r R R r r r 3 2 1 2 1 r ...where G m1 m21 Gm ....for m12m r 3 This equation of motion describes the r r 0. ‘relative motion’ of the smaller mass r 3 relative to the larger mass, assuming that 32 LT the differencePCD_STiCM in the masses is huge. 16 r We now take the dot product of the r 0. velocity with the ‘Eq. of motion’: r 3 r r r r 0 r 3 dd r r ruˆ ru ˆ ru ˆ dt dt
r r v v vv r r ruˆ ruˆ ruˆ rr
vvrr 0 r3 r lim v t0 v lim t t0 2 PCD_STiCM tr17 r We took the dot product of the ‘Eq. of r 0. motion’ with velocity: r 3 r lim v t0 v lim t Integration t0 tr2 w.r.t. time
Differentiation Total ‘SPECIFIC’ 2 w.r.t. time (i.e. per unit mass) v MECHANICAL E DIMENSIONS ENERGY: Constant 2 r of integral / 22 2 ELT Constant of Motion. v 32 LT INVARIANCE, E SYMMETRY. 2 r PCD_STiCM 18 r r 0. r 3 Equation of Motion We took the dot product of the ‘Eq. of motion’ with velocity: r r r r 0 r 3
…….. and discovered a CONSERVED QUANTITY! v2 E 2 r E: constant symmetry with respect to translations in time.
(E,t): canonicallyPCD_STiCM conjugate pair of variables 19 We now take the cross product of the position vector with the ‘Eq. of motion’: r r r r 0. r 3 Therefore: r r 0 Force: RADIAL central force symmetry d Now, ( rr ) rrrrrr 0 dt Therefore H r r = r v is also a constant of motion. SPECIFIC ANGULAR MOMENTUM
INVARIANCE, SYMMETRY
PCD_STiCM 20 Homogeneity of time Conservation of energy Homogeneity of space Conservation of linear Translational Symmetry momentum Isotropy of space Conservation of angular Rotational Symmetry momentum
Emmy Noether 1882 to 1935
SYMMETRY CONSERVATION LAWS
Her entry to the Senate of the University of Gottingen, Germany, was resisted. David Hilbert argued in favor of admitting her to the University PCD_STiCM 21 Senate. L 0 Time is homogeneous: t Lagrangian of a closed system does not depend explicitly on time. L q - L is a CONSTANT. q
Hamiltonian Conservation of Energy is thus connected with the “ENERGY” symmetry principle regarding invariance with respect to temporal translations. Hamiltonian / Hamilton’s Principal Function
PCD_STiCM 22 Space is homogenous and isotropic
the condition for homogeneity of space : L ( x , y , z ) 0 LLL i. e ., L x y z 0 x y z L which implies 0 where q x , y , z q
L d L L since 0, this means i . e . p is conserved. q dt q q ie. ., is independent of time, is a constant of motion
Law of conservation of momentum, arises from the homogeneity of space. Symmetry Conservation Laws PCD_STiCM 23 N ()i 0 FSk k 1
N ()i Displacement is arbitrary, hence, Fk 0; k 1 N dd()i i. e . 0 pk = P dtk 1 dt Thus P is conserved in the absence of external forces.
The conservation of momentum was secured in ‘Unit 1’ on the basis of ‘translational invariance in homegenous space’, and not on the basis of Newton’s III law. PCD_STiCM 24 dP 0 dt d p d p 21 dt dt FF12 21 Given the symmetry related to translation in homogenous space, could you have discovered Newton’s 3rd law?
PCD_STiCM 25 Are the conservation principles consequences of the laws of nature? Or, are the laws of nature the consequences of the symmetry principles that govern them?
Until Einstein's special theory of relativity, it was believed that conservation principles are the result of the laws of nature.
Since Einstein's work, however, physicists began to analyze the conservation principles as consequences of certain underlying symmetry considerations, enabling the laws of nature to be revealed from this analysis.
PCD_STiCM 26 Instead of introducing Newton’s III law as a fundamental principle, we deduced it (in Unit 1) from symmetry / invariance.
This approach places SYMMETRY ahead of LAWS OF NATURE. It is this approach that is of greatest value to contemporary physics. This approach has its origins in the works of Albert Einstein, Emmily Noether and Eugene Wigner.
PCD_STiCM 27 (1879 – 1955) (1882 – 1935) (1902 – 1995) r Equation of Motion for r 3 0. r Kepler’s two-body problem H r r = r v, the SPECIFIC ANGULAR MOMENTUM
We shall get yet another constant of motion, a conserved quantity, by taking the cross product of the ‘SPECIFIC ANGULAR MOMENTUM H with the equation of motion: r H r H 0. 3 r PCD_STiCM 28 H r 3 r v0 r r r H r H 0. r 3 H r r r v - r v r 0 r 3
d dr Use:v r r reˆˆ r e rr dt dt
2 H r 3 rv - rrr 0 r
PCD_STiCM 29 2 H r 3 rv - rrr 0 r dd Now, H v H r H r .. since H 0 dt dt d Hv r22 v - r reˆ 0 3 d d r dt r H v0 dt dt r drv Hrv0 2 dt r r dr H v0 dt r PCD_STiCM 30 Observe how a dr H v0 dt r constant of Hv eˆ A , motion has constant emerged – - yet again Av H eˆ , constant -– by playing with LAPLACE – RUNGE – LENZ the equation of VECTOR motion! Physical LT32 Dimensions vH LT1 L 2 T 1 L 3 T 2 PCD_STiCM 31 We will take a Break… …… Any questions ? [email protected] References:
[1] Oliver Montenbruck and Eberhard Gill ‘Satellite orbits – Models, Methods, Applications’. (Springer, Berlin, 2000)
[2] Francis J. Hale ‘Introduction to Space Flight’ (Prentice Hall, Englewood Cliffs, 1994)
Next Lecture: Dynamical Symmetry of the Kepler Problem PCD_STiCM 32 STiCM
Select / Special Topics in Classical Mechanics
P. C. Deshmukh
Department of Physics Indian Institute of Technology Madras Chennai 600036
STiCM Lecture 14: Unit 4 Symmetry and Conservation Laws Dynamical Symmetry in the Kepler Problem
PCD_STiCM 33 vH eˆ A , constant
LAPLACE – RUNGE – LENZ VECTOR
Take dot product with r :
Equation to the orbit / trajectory
Without solving the differential equation of motion!
PCD_STiCM 34 vH eˆ A where Hr v
Take dot product with r : vH r r A r
sign reversal : H v r r A r
Interchange ‘dot’ and ‘cross’: Hv r r A r
H2 r A r Ar cos
Ar, PCD_STiCM 35 H2 r Ar cos Ar, Ar, X r Acos H 2 H 2 Hp2 r Acos 1 A cos 1 cos
What is A ? PCD_STiCM 36 Ar,
F X
PCD_STiCM 37 H r v H , r 9000 & H , v 90 r vHA Hvv H uˆ r
2 2 Hvv H r 22 A r
Hv r v v v r v - v2 r
Hv r v r v r - v2 r r v r cos 2 v 2 r 2 v2r 2 sin 2 r,v 2 H2v 2 v 2 r 2 sin 2 2 A 2 r PCD_STiCM 38 2 H2v 2 v 2 r 2 sin 2 2 A 2 r
2 H2v 2 v 2 r 2 sin 2 2 A 2 rH 2 Hrv
2 HA2v 2 2 2 r HEA22 2 2
2 2 HE2 A A2 EH 2 221 2 1 PCD_STiCM 39 p r 1 cos with H22 A2 EH p & 1 2
PCD_STiCM 40 p r 1 cos 1: Hyperbola (open trajectory)
1: Parabola (open trajectory)
0 1: Ellipse (closed trajectory)
0 : Circle degenerate ellipse For satellite and ballistic missile trajectories, ellipses (inclusive of the circle) are of primary PCD_STiCM interest. 41 p r 1 cos 1: Hyperbola (open trajectory) 1 1: Parabola (open trajectory) 1 0 1: Ellipse (closed trajectory)
0 : Circle degenerate ellipse
Earth-orbiting 01 satellites are in elliptic motion.
Deep space probes leave earth’s gravity on hyperbolic orbits PCD_STiCM 42 b a Focus
a semi-major axis b semi-minor axis b2 e: eccentricity 1 e a2
b a e=0 e=0.5 e=0.75 e=0.95
PCD_STiCM Increasing eccentricity 43 p r ; 1 cos
apogee X 01 r perigee rmax min H22 A2 EH p & 1 2
p perigee r r when 0: r min min 1 p apogee r r when : r maxPCD_STiCM max 1 44 p H 222EH r ; p ; 1 1 cos 2 At apogee/perigee: apogee 2b r r r r2 a , constant perigee 2a p p2 p 2a r r 2 perigee apogee 2 a1 1 1 1 r pa1 2 1 cos
2 2 2 for circular 11 orbit E 2H2 2 p 2 a
PCD_STiCM 45 24 GPS satellites ---- Wikimedia Commons http://en.wikipedia.org/wiki/File:ConstellationGPS.gifPCD_STiCM 46 Hrv eˆ eˆ e ˆ e ˆ v eeˆˆ 2 eˆ Y z eˆ1 “Unit Circle” eˆ1 1
X O Av H eˆ eˆ,, e ˆ e ˆz Plane/Cylindrical Polar Coordinate System
PCD_STiCM 47 Laplace Runge Lenz Vector is constant for a strict 1 r The H potential. (specific) v H
eˆ angular eˆ v momentum A vector is S out of the plane of this figure, Av H eˆ toward us. PCD_STiCM 48 l v H H eˆ v eˆ S pl The Laplace - Runge - Lenz vector : A eˆ , or alternatively defined in terms of the specific angular momentum H as PCD_STiCM 49 Av H eˆ Av H eˆ
Central Field Symmetry dA dv dH deˆ H v Angular Momentum dt dt dt dt is Conserved
dA dv deˆ H dt dt dt deˆˆ e But eˆ dt dA dv He ˆ dt dt PCD_STiCM 50 Av H eˆ dA dv He ˆ dt dt
dv What is the form of the force per unit mass : ? dt We must know the form of the interaction! dv F ma m dt PCD_STiCM 51 v2 dv E The force per unit mass : - eˆ 2 r dt 2 dA dv H eˆ dt dt dA ˆ 2 ˆ ˆ - 2 e eez dt
eˆ, e ˆ , e ˆz : right handed basis set
eˆ e ˆz e ˆ dA 0 PCD_STiCM dt 52 Two-body central field Kepler-Bohr problem, Attractive force: inverse-square-law.
Rosette motion For this potential and for the associated field: no precession of orbit.
The constancy of the orbit Angular momentum is suggests a conserved quantity conserved, but major-axis not fixed. and one must look for an associated symmetry. PCD_STiCM 53 dA Conclusion: 0 dt H
Av H eˆ perigee
AH 0 LRL vector is in the plane of the orbit
Find the direction of A at the perigee. A Direction of the LRL vector is: focus to perigee. A : Must remain constant – no matter where the planet is!
This is preciselyPCD_STiCM what FIXES the orbit! 54 d v Force (per unit mass): eˆ dt 2 Newton Newton told us (but first, to Halley)!
Given the constancy related to the conservation of the LRL vector, could you have discovered the law of gravity?
If you did that, wouldn’t you have discovered a law of nature? PCD_STiCM Halley55 Symmetry & Conservation Principles!
“It is now natural for us to try to derive the laws of nature and to test their validity by means of the laws of invariance, rather than to derive the laws of invariance from what we believe to be the laws of nature.” - Eugene Wigner
Eugene Paul Wigner Emmily (1902-1995) Noether PCD_STiCM 56 (1882 – 1935) Conservation law associated Av H eˆ Laplace Runge Lenz Vector : with ‘dynamical / accidental ’ 1 symmetry. constant for a strict potential. r
Wilhelm Carl David Tolmé Lenz 1888 -1957 Pierre-Simon Laplace Runge 1749 - 1827 1856 - 1927 Symmetry of the H atom: ‘old’ PCD_STiCM -572 quantum theory. En ~ n Further reading: • Symmetry & Conservation laws play an important role in understanding the very frontiers of Physics. • The implications go as far as testing the ‘standard’ model of physics, and exploring if there is any physics beyond the standards model. Do visit: Feynman's Messenger Lectures Online AKA Project Tuva http://www.fotuva.org/news/project_tuva.htmlPCD_STiCM 58 Useful references on ‘Symmetry & Conservation Laws’
P. C. Deshmukh and Shyamala Venkataraman Obtaining Conservation Principles from Laws of Nature -- and the other way around!
Bulletin of the Indian Association of Physics Teachers, Vol. 3, 143- 148 (2011)
P. C. Deshmukh and J. Libby (a) Symmetry Principles and Conservation Laws in Atomic and Subatomic Physics -1 Resonance, 15, 832 (2010) b) Symmetry Principles and Conservation Laws in Atomic and Subatomic Physics -2 Resonance, 15, 926 (2010)
PCD_STiCM 59 Continuous Symmetries - Translation, Rotation
Dynamical Symmetries - LRL, Fock Symmetry SO(4)
Discrete Symmetries - P : Parity - C : Charge Conjugation - T : Time Reversal
PCD_STiCM 60 Lorentz symmetry: associated with the PCT symmetry.
PCT theorem (Wolfgang Pauli) No experiment has revealed any violation of PCT symmetry.
This is predicted by the Standard Model of particle physics.
PCD_STiCM 61 The Standard Model today
The ‘standard model’ unifies all the fundamental building blocks of matter, and three of the four fundamental forces.
To complete the Model a new particle is needed – the Higgs Boson – that the physics community hopes to find in the new built accelerator LHC at CERN in Geneva. PCD_STiCM 62 2008 Nobel Prize in Physics "for the discovery of the "for the discovery of the origin of mechanism of spontaneous the broken symmetry which broken symmetry in predicts the existence of at least subatomic physics" three families of quarks in nature"
Makoto Kobayashi Toshihide Maskawa Yoichiro Nambu ¼ prize ¼ prize ½ prize High Energy Kyoto Sangyo Univ, Enrico Fermi Institute, Accelerator KyotoJapan Chicago, USA PCD_STiCMResearch Organization, 63 Tsukuba, Japan ‘every symmetry in nature yields a conservation law and conversely, every conservation law reveals an underlying symmetry’.
Noether’s theorem
PCD_STiCM 64 “In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.”
“In the realm of algebra, … , she discovered methods which have proved of enormous importance …. ”
“………. Her unselfish, significant work over a period of many years was rewarded by the new rulers of Germany with a dismissal, which cost her the means of maintaining her simple life and the opportunity to carry on her mathematical studies……”
ALBERT EINSTEIN. Princeton University, May 1, 1935.
New York Times May 5, 1935,PCD_STiCM Excerpts 65 http://www-history.mcs.st-and.ac.uk/history/Obits2/Noether_Emmy_Einstein.html We will take a Break… …… Any questions ?
Next: Unit 5: Inertial and non-inertial reference frames.
Moving coordinate systems. Pseudo forces. Inertial and non-inertial reference frames.
‘Deterministic’ cause-effect relations in inertial frame,
and their modificationsPCD_STiCM in a non -inertial frame. 66