CLASSICAL MECHANICS 1 - astro3.sci.hokudai.ac.jpalex/teaching_materials/CM1_P1.pdf · • 40%...

CLASSICAL MECHANICS 1

ISP LV2

Advanced Mechanics in a pre-analytical worldASSISTANT

PROF. PETTITT

COURSE OVERVIEW

• x14, 90minute lectures, with an exam in 15th lecture

OUTLINE

• About 7 homework assignments.

• Breakdown:

• 50% Exam

• 40% Homework

• 10% Attendance/attitude/seminar

• A final 90min exam in the 15th lecture.

MATERIAL

COURSE OVERVIEW

• As with Physics 1, all content can be found on my personal website: https://astro3.sci.hokudai.ac.jp/~alex/

• Course can be supplemented by a few textbooks:

• Kibble & Berkshire, Classical Mechanics

• Taylor, Classical Mechanics

• Marion & Thornton, Classical Dynamics

https://astro3.sci.hokudai.ac.jp/~alex/

• There are many classical mechanics texts on the market. Some are quite high level. The following would “future-proof” yourselves for CMII, but would be a bit hard to use for this course.

• Hand & Finch Analytical Mechanics

• Landau Lifshitz Mechanics

• Goldstein Classical Mechanics

MATERIAL

COURSE OVERVIEW

CONTENT

COURSE OVERVIEW

• This course is meant to act to fill out the basics you learned in the mechanics portion of Physics I.

• Now include much more in-depth problems, mathematics, and complicated systems.

• e.g., last year we looked at the solutions of a damped harmonic oscillator. BUT we did not really have the mathematical tools to understand where it came from.

CONTENT

COURSE OVERVIEW

• We will also devoted some time to topics we did not have time for last year (~50%), such as celestial mechanics, frames of reference, and rigid body motion.

(e.g. Fourier Analysis, Complex Numbers in Fall 2018)

• There will be some advanced mathematics needed for the course. Some will be brought up here, others in dedicated courses.

@H@q

= �dp

dt

@L@q

� d

dt

@L@q

= 0

• We will stop before we get to true “analytical mechanics”. This is the subject of Classical Mechanics II, in the Fall semester. Lagrangians, Hamiltonians, Green’s functions, oh my etc. will not be covered in this course.

SYLLABUS

COURSE OVERVIEW

2. “Realistic” equations of motion

1. Coordinate systems

0. Review of Physics I and Newton’s laws

We will spend ~1-2 lectures per topic, some will be longer than others.

SYLLABUS

COURSE OVERVIEW

3. Momentum of many bodied systems and centre of mass

4. Energy and equilibrium positions

5. Damped and driven oscillators

StarTrek IV (aka the one with the whales)

6. Gravity and central force problems

Is Thor’s hammer just made of aluminium?

SYLLABUS

COURSE OVERVIEW

We didn’t last year, but there is a chance we will finish early, and there are some extra things we can cover

A. Numerical methods

B. Introduction to relativity

7. Motion in different reference frames

8. Rigid body rotation and motion

C. Collision theory

SYLLABUS

COURSE OVERVIEW

SYLLABUS

COURSE OVERVIEW

Wait, where’s the rest?

Hydro(dynamics)now too hard, has to wait until MSc course

Opticscovered in quantum optics

and electromagnetism [see QM1+2, E&M1+2]

WHAT IS CLASSICAL MECHANICS?

PART 1: REVIEW AND NEWTON’S LAWS

KINEMATICS: describes motion

• Coordinate systems

• Vector nature of motion (+rotations)

DYNAMICS: explains the motion

• Newton’s laws

• Equations of motion, conservation laws

Galileo (16C)

Newton (17C)

Lagrange (18C)

Hamilton (19C)

Einstein (20C)

Phys

ics I

CM I

CM II

QMMSc

NEWTON’S LAWS

• Law 1: Object at rest of uniform motion remains that way if not acting on by some external force,


• Law 3: For every action there is an equal and opposite reaction.~FAB = �~FBA

• Law 2: The rate of change of the momentum of a system is equal to the net external force acting,

~Fnet,ext =

d~psys

dt~⌧net,ext =

d~Lsys

dt

~Fnet,ext = 0 (Law of inertia)

NEWTON’S LAWS


• Conservation of linear and angular momentum:

~Fnet,ext = 0

~⌧net,ext = 0

~psys = conserved

~Lsys = conserved

• Conservation of total energy of system: if forces are all conservative then total energy is conserved (e.g. potential and kinetic).

=3X

i=1

riei

COORDINATE SYSTEMS


• Laws of physics are independent of our choice of coordinate system.

• We can use vector notation to make this clearer…• Cartesian coordinates are what

you are familiar with:~r = (x, y, z)

~r = xx+ yy + zz

~r = xi+ yj + zk

~r

position vector from origin

~r = x1e1 + x2e2 + x3e3 useful for >3D

naff

• A useful vector operation:~A · ~B =

X

i

AiBi = |A||B| cos ✓AB

COORDINATE SYSTEMS


• Each of these unit vectors (i,j,k) has no units, and simply “points” somewhere: |r| =

pr · r = 1

• This measures the alignment of two independent vector quantities.

E.g. force and displacement give work only if they are not perpendicular.

[E.g. the cosine rule!]

• Another few useful vector operations:

COORDINATE SYSTEMS


Cx

= Ay

Bz �Az

ByC

y

= A

z

Bx�A

x

Bz

C

z

= A

x

By �A

y

Bx

• This measures the alignment of two independent vector quantities, and their degree of linear dependence.

~A⇥ ~B = det

2

4i j kA

x

Ay

Az

Bx

By

Bz

3

5C =| ~A⇥ ~B| = | ~A|| ~B| sin ✓AB

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additionally:

you maybe saw in linear algebra?

~F = q~v ⇥ ~B<latexit sha1_base64="IkNXnR9cRygXx91mH+L6JAESSm0=">AAACCXicbZDLSsNAFIZP6q3WW9Slm8EiuCqJCLoRSgVxWcFeoAllMp20QycXZyaFErJ146u4caGIW9/AnW/jNM1CWw8MfPz/OZw5vxdzJpVlfRulldW19Y3yZmVre2d3z9w/aMsoEYS2SMQj0fWwpJyFtKWY4rQbC4oDj9OON76e+Z0JFZJF4b2axtQN8DBkPiNYaalvImdCSXqToSv0kOMkcxQLqJwbjaxvVq2alRdaBruAKhTV7JtfziAiSUBDRTiWsmdbsXJTLBQjnGYVJ5E0xmSMh7SnMcR6mZvml2ToRCsD5EdCv1ChXP09keJAymng6c4Aq5Fc9Gbif14vUf6lm7IwThQNyXyRn3CkIjSLBQ2YoETxqQZMBNN/RWSEBSZKh1fRIdiLJy9D+6xmWzX77rxabxRxlOEIjuEUbLiAOtxCE1pA4BGe4RXejCfjxXg3PuatJaOYOYQ/ZXz+AHIqmis=</latexit><latexit sha1_base64="IkNXnR9cRygXx91mH+L6JAESSm0=">AAACCXicbZDLSsNAFIZP6q3WW9Slm8EiuCqJCLoRSgVxWcFeoAllMp20QycXZyaFErJ146u4caGIW9/AnW/jNM1CWw8MfPz/OZw5vxdzJpVlfRulldW19Y3yZmVre2d3z9w/aMsoEYS2SMQj0fWwpJyFtKWY4rQbC4oDj9OON76e+Z0JFZJF4b2axtQN8DBkPiNYaalvImdCSXqToSv0kOMkcxQLqJwbjaxvVq2alRdaBruAKhTV7JtfziAiSUBDRTiWsmdbsXJTLBQjnGYVJ5E0xmSMh7SnMcR6mZvml2ToRCsD5EdCv1ChXP09keJAymng6c4Aq5Fc9Gbif14vUf6lm7IwThQNyXyRn3CkIjSLBQ2YoETxqQZMBNN/RWSEBSZKh1fRIdiLJy9D+6xmWzX77rxabxRxlOEIjuEUbLiAOtxCE1pA4BGe4RXejCfjxXg3PuatJaOYOYQ/ZXz+AHIqmis=</latexit><latexit sha1_base64="IkNXnR9cRygXx91mH+L6JAESSm0=">AAACCXicbZDLSsNAFIZP6q3WW9Slm8EiuCqJCLoRSgVxWcFeoAllMp20QycXZyaFErJ146u4caGIW9/AnW/jNM1CWw8MfPz/OZw5vxdzJpVlfRulldW19Y3yZmVre2d3z9w/aMsoEYS2SMQj0fWwpJyFtKWY4rQbC4oDj9OON76e+Z0JFZJF4b2axtQN8DBkPiNYaalvImdCSXqToSv0kOMkcxQLqJwbjaxvVq2alRdaBruAKhTV7JtfziAiSUBDRTiWsmdbsXJTLBQjnGYVJ5E0xmSMh7SnMcR6mZvml2ToRCsD5EdCv1ChXP09keJAymng6c4Aq5Fc9Gbif14vUf6lm7IwThQNyXyRn3CkIjSLBQ2YoETxqQZMBNN/RWSEBSZKh1fRIdiLJy9D+6xmWzX77rxabxRxlOEIjuEUbLiAOtxCE1pA4BGe4RXejCfjxXg3PuatJaOYOYQ/ZXz+AHIqmis=</latexit><latexit sha1_base64="IkNXnR9cRygXx91mH+L6JAESSm0=">AAACCXicbZDLSsNAFIZP6q3WW9Slm8EiuCqJCLoRSgVxWcFeoAllMp20QycXZyaFErJ146u4caGIW9/AnW/jNM1CWw8MfPz/OZw5vxdzJpVlfRulldW19Y3yZmVre2d3z9w/aMsoEYS2SMQj0fWwpJyFtKWY4rQbC4oDj9OON76e+Z0JFZJF4b2axtQN8DBkPiNYaalvImdCSXqToSv0kOMkcxQLqJwbjaxvVq2alRdaBruAKhTV7JtfziAiSUBDRTiWsmdbsXJTLBQjnGYVJ5E0xmSMh7SnMcR6mZvml2ToRCsD5EdCv1ChXP09keJAymng6c4Aq5Fc9Gbif14vUf6lm7IwThQNyXyRn3CkIjSLBQ2YoETxqQZMBNN/RWSEBSZKh1fRIdiLJy9D+6xmWzX77rxabxRxlOEIjuEUbLiAOtxCE1pA4BGe4RXejCfjxXg3PuatJaOYOYQ/ZXz+AHIqmis=</latexit>

e.g.

~⌧ = ~r ⇥ ~F<latexit sha1_base64="AaGLY4P4J58ZYSRCeytw9mydLBI=">AAACCHicbVDLSsNAFJ34rPUVdenCwSK4KokIuhGKgrisYB/QhDKZTtuhk0mYuSmUkKUbf8WNC0Xc+gnu/BunaRbaemDg3HPu5c49QSy4Bsf5tpaWV1bX1ksb5c2t7Z1de2+/qaNEUdagkYhUOyCaCS5ZAzgI1o4VI2EgWCsY3Uz91pgpzSP5AJOY+SEZSN7nlICRuvaRN2Y09YAk2VVOVeYBD5nOi9usa1ecqpMDLxK3IBVUoN61v7xeRJOQSaCCaN1xnRj8lCjgVLCs7CWaxYSOyIB1DJXE7PLT/JAMnxilh/uRMk8CztXfEykJtZ6EgekMCQz1vDcV//M6CfQv/ZTLOAEm6WxRPxEYIjxNBfe4YhTExBBCFTd/xXRIFKFgsiubENz5kxdJ86zqOlX3/rxSuy7iKKFDdIxOkYsuUA3doTpqIIoe0TN6RW/Wk/VivVsfs9Ylq5g5QH9gff4AKtyasA==</latexit><latexit sha1_base64="AaGLY4P4J58ZYSRCeytw9mydLBI=">AAACCHicbVDLSsNAFJ34rPUVdenCwSK4KokIuhGKgrisYB/QhDKZTtuhk0mYuSmUkKUbf8WNC0Xc+gnu/BunaRbaemDg3HPu5c49QSy4Bsf5tpaWV1bX1ksb5c2t7Z1de2+/qaNEUdagkYhUOyCaCS5ZAzgI1o4VI2EgWCsY3Uz91pgpzSP5AJOY+SEZSN7nlICRuvaRN2Y09YAk2VVOVeYBD5nOi9usa1ecqpMDLxK3IBVUoN61v7xeRJOQSaCCaN1xnRj8lCjgVLCs7CWaxYSOyIB1DJXE7PLT/JAMnxilh/uRMk8CztXfEykJtZ6EgekMCQz1vDcV//M6CfQv/ZTLOAEm6WxRPxEYIjxNBfe4YhTExBBCFTd/xXRIFKFgsiubENz5kxdJ86zqOlX3/rxSuy7iKKFDdIxOkYsuUA3doTpqIIoe0TN6RW/Wk/VivVsfs9Ylq5g5QH9gff4AKtyasA==</latexit><latexit sha1_base64="AaGLY4P4J58ZYSRCeytw9mydLBI=">AAACCHicbVDLSsNAFJ34rPUVdenCwSK4KokIuhGKgrisYB/QhDKZTtuhk0mYuSmUkKUbf8WNC0Xc+gnu/BunaRbaemDg3HPu5c49QSy4Bsf5tpaWV1bX1ksb5c2t7Z1de2+/qaNEUdagkYhUOyCaCS5ZAzgI1o4VI2EgWCsY3Uz91pgpzSP5AJOY+SEZSN7nlICRuvaRN2Y09YAk2VVOVeYBD5nOi9usa1ecqpMDLxK3IBVUoN61v7xeRJOQSaCCaN1xnRj8lCjgVLCs7CWaxYSOyIB1DJXE7PLT/JAMnxilh/uRMk8CztXfEykJtZ6EgekMCQz1vDcV//M6CfQv/ZTLOAEm6WxRPxEYIjxNBfe4YhTExBBCFTd/xXRIFKFgsiubENz5kxdJ86zqOlX3/rxSuy7iKKFDdIxOkYsuUA3doTpqIIoe0TN6RW/Wk/VivVsfs9Ylq5g5QH9gff4AKtyasA==</latexit><latexit sha1_base64="AaGLY4P4J58ZYSRCeytw9mydLBI=">AAACCHicbVDLSsNAFJ34rPUVdenCwSK4KokIuhGKgrisYB/QhDKZTtuhk0mYuSmUkKUbf8WNC0Xc+gnu/BunaRbaemDg3HPu5c49QSy4Bsf5tpaWV1bX1ksb5c2t7Z1de2+/qaNEUdagkYhUOyCaCS5ZAzgI1o4VI2EgWCsY3Uz91pgpzSP5AJOY+SEZSN7nlICRuvaRN2Y09YAk2VVOVeYBD5nOi9usa1ecqpMDLxK3IBVUoN61v7xeRJOQSaCCaN1xnRj8lCjgVLCs7CWaxYSOyIB1DJXE7PLT/JAMnxilh/uRMk8CztXfEykJtZ6EgekMCQz1vDcV//M6CfQv/ZTLOAEm6WxRPxEYIjxNBfe4YhTExBBCFTd/xXRIFKFgsiubENz5kxdJ86zqOlX3/rxSuy7iKKFDdIxOkYsuUA3doTpqIIoe0TN6RW/Wk/VivVsfs9Ylq5g5QH9gff4AKtyasA==</latexit>

COORDINATE SYSTEMS


• Can easily define derivatives in terms of vector quantities:

~v =d~r

dt~a =

d2~r

dt2

• Where the derivative is the limit of some very small change in the quantity of interest:

d~r

dt= lim

�t!0

�~r

�t

�~r = ~r(t+�t)� ~r(t)

• The derivatives of a sum of vectors is separable.• The derivatives of a product of vector and scalar

is simply the product rule: d

dt(f~r) = f

d~r

dt+

df

dt~r

PART 1: REVIEW AND NEWTON’S LAWSCOORDINATE SYSTEMS

• When finding the rate of change of some vector, we can simply use the product and separable rules to show that:

makes sense?~v =d~r

dt

=dx

dt

i+dy

dt

j +dz

dt

k

~v = vx

i+ vy

j + vz

kif:

v

x

=dx

dt

, v

y

=dy

dt

, v

z

=dz

dt

then it simply follows that:

• May seem obvious, and we use it without really thinking about it. BUT it is only true for rectangular coordinates because the unit vectors (i,j,k) are constant.

• We shall see shortly that this is not the case in many coordinates systems…

COORDINATE SYSTEMS


• NL2 (and EoM’s in general) is in fact a differential equation for position, which are very common in all areas of physics. These will form a large portion of this course.

[e.g. position from constant force]

~a = (x, y, z)

F

x

= mx, F

y

= my, F

z

= mz

• Similar equations can be constructed for NL2 and acceleration.

• NL2 is an equation of motion (EoM), and is useful for its predictive power to determine the position and velocity at later times.

NEWTON NOTES


• These EoM’s hinge on some definition of mass, kinematic variables and force. e.g., we think of mass as an objects resistance to acceleration.

• Newton’s laws hold only in inertial frames, i.e. non-accelerating reference points.

• Recall that circular motion is nonintertial, as there is always an acceleration (the Earth reference frame is noninertial!).

• Only valid for v<<c, and for the macroscopic world (non-QM).

• Special care also needs to be taken for EM fields… why?

NEWTON NOTES


• Lets look at NL’s in a 2-body system:

F2

= F21

+ F ext

2

= p2net force on 2:F1

= F12

+ F ext

1

net force on 1: = p1NL2

ptot

= p1

+ p2

= F12

+ F ext

1

+ F21

+ F ext

2

= F ext

1

+ F ext

2

⌘ F ext NL3

F ext = 0 p = const.so if:

then:

i.e. conservation of momentum

• This generalises to an N-body system as well. Prove this!

NEWTON NOTES


[e.g. P2 sliding block revision]

r

�

NON-CARTESIAN COORDINATE SYSTEMS


• Cartesian coordinates are great in some instances, but many real life problems are much harder to describe using 3 orthogonal basis vectors.

• Instead can describe via polar coordinates.

y

xx

y � r • These relate to Cartesian coords by:x = r cos�

y = r sin�

r =p

x

2 + y

2

� = arctan(y/x)

� = arctan 2(y, x)…technicallyNot the same as mathematicians…



• These are also orthogonal (90deg) basis vectors.

• What is a big difference between x/y and Φ/r?

y

xx

y

r

�

� r

• The polar frame changes orientation as the vector moves! This is going to make our lives a bit miserable.

y

xx

y

�

�r

� 6= 0



y

xx

y

r

�

� r

• We can see that our unit vector is a function of our vector, but we defined it with no units, so:

r ⌘ ~r

|r|~ri.e.

• The individual unit vectors are:

r = cos� x+ sin� y

ˆ

� = � sin� x+ cos� y

Prove this is true!



• So, we can expand forces in this orthogonal unit system like before: ~F = Fr r + F��

• Great, simple, it looks the same as before. So:

Fr = mr F� = m�• This is wrong, as the vector is not given by

~r = rr + ��• At the very least, it is not dimensionally correct.

• The real issue is that because r changes direction as the point moves, then it also has a derivative! So let’s differentiate our position vector: ~r = rr

= distance x direction



[to the board!]



• This gives us an expression for velocity in 2D polars:

~v = rr + r��

v� = r� = r!

Angular velocity

vr = r

Radial velocity

• Wizard… now we have to do it again.

~a ⌘ ~r =d

dt~r =

d

dt(rr + r��)

[to the board…]



• Exactly the same procedure as before, with the added faff of a derivative of Φ derivative.

• Eventually gives us:

~a =⇣r � r�2

⌘r +

⇣r�+ 2r�

⌘�

�r!2

Centripetal acceleration we have seen before!

Simple radial acceleration

r↵Angular acceleration!

errrr?

Check it makes sense in special cases…rock on string

no rotation



• So our equations for 2D Newton’s laws are:

Fr = m⇣r � r�2

⌘

F� = m⇣r�+ 2r�

⌘

2D polars

Fy = my

F

x

= mx

2D Cartesian

• Very useful for problems with “cylindrical symmetry”.



[e.g. a skateboarder in a halfpipe]



• What about in 3D?

• The simplest approach is to first generalise to cylindrical polars, whereby there is either no or only linear motion in z.

• Here we rename the radial coordinate as or where:$⇢

⇢ =px

2 + y

2

Fz = mz

• This gives the same versions of NL’s, with an additional z component:



• What about in 3D?

• For a spherically symmetric system, things get quite a bit uglier.

�

✓r

• Be EXTREMELY careful about definitions of angles and where they are measured form.

• We will derive NL’s for this coordinate system, but not right now.

r =p

x

2 + y

2 + z

2

• Radial position now includes the vertical:

z = r cos ✓

x = r sin ✓ cos�

y = r sin ✓ sin�

SUMMARY


• Looked at manipulation of vectors in multiple dimensions: dot and scalar products.

• Reviewed NL’s and the concept of an inertial frame.

• Unit vectors and other coordinate systems (2D polars).

• Constructed NL’s in different coordinate systems:

Fr = m(r � r�2)

F✓ = m(r�� 2r�)

F

x

= mx

Fy = my

NEXT TIME ON CMI…


Falling, with drag!

CLASSICAL MECHANICS 1 - astro3.sci.hokudai.ac.jpalex/teaching_materials/CM1_P1.pdf · • 40%...

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