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Dynamics and Relativity Lecture notes from Cambridge University

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The best lecture note from the University of Cambridge by Professor G. I. Ogilvie. The lecture provides a complete and easy to understand notes on Dynamics and Relativity.

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  • March 17, 2023
  • 82
  • 2017/2018
  • Lecture notes
  • G. i. ogilvie
  • All classes

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Part IA — Dynamics and Relativity
Based on lectures by G. I. Ogilvie
Notes taken by Dexter Chua

Lent 2015

These notes are not endorsed by the lecturers, and I have modified them (often
significantly) after lectures. They are nowhere near accurate representations of what
was actually lectured, and in particular, all errors are almost surely mine.

Familiarity with the topics covered in the non-examinable Mechanics course is assumed.


Basic concepts
Space and time, frames of reference, Galilean transformations. Newton’s laws. Dimen-
sional analysis. Examples of forces, including gravity, friction and Lorentz. [4]

Newtonian dynamics of a single particle
Equation of motion in Cartesian and plane polar coordinates. Work, conservative forces
and potential energy, motion and the shape of the potential energy function; stable
equilibria and small oscillations; effect of damping.
Angular velocity, angular momentum, torque.
Orbits: the u(θ) equation; escape velocity; Kepler’s laws; stability of orbits; motion in
a repulsive potential (Rutherford scattering). Rotating frames: centrifugal and coriolis
forces. *Brief discussion of Foucault pendulum.* [8]

Newtonian dynamics of systems of particles
Momentum, angular momentum, energy. Motion relative to the centre of mass; the
two body problem. Variable mass problems; the rocket equation. [2]

Rigid bodies
Moments of inertia, angular momentum and energy of a rigid body. Parallel axes
theorem. Simple examples of motion involving both rotation and translation (e.g.
rolling). [3]

Special relativity
The principle of relativity. Relativity and simultaneity. The invariant interval. Lorentz
transformations in (1 + 1)-dimensional spacetime. Time dilation and length contraction.
The Minkowski metric for (1 + 1)-dimensional spacetime. Lorentz transformations
in (3 + 1) dimensions. 4-vectors and Lorentz invariants. Proper time. 4-velocity
and 4-momentum. Conservation of 4-momentum in particle decay. Collisions. The
Newtonian limit. [7]




1

,Contents IA Dynamics and Relativity


Contents
0 Introduction 4

1 Newtonian dynamics of particles 5
1.1 Newton’s laws of motion . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Galilean transformations . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Newton’s Second Law . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Dimensional analysis 9
2.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Forces 12
3.1 Force and potential energy in one dimension . . . . . . . . . . . . 12
3.2 Motion in a potential . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Central forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.6 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Orbits 27
4.1 Polar coordinates in the plane . . . . . . . . . . . . . . . . . . . . 27
4.2 Motion in a central force field . . . . . . . . . . . . . . . . . . . . 28
4.3 Equation of the shape of the orbit . . . . . . . . . . . . . . . . . 31
4.4 The Kepler problem . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.5 Rutherford scattering . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Rotating frames 38
5.1 Motion in rotating frames . . . . . . . . . . . . . . . . . . . . . . 38
5.2 The centrifugal force . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3 The Coriolis force . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6 Systems of particles 43
6.1 Motion of the center of mass . . . . . . . . . . . . . . . . . . . . 43
6.2 Motion relative to the center of mass . . . . . . . . . . . . . . . . 45
6.3 The two-body problem . . . . . . . . . . . . . . . . . . . . . . . . 46
6.4 Variable-mass problem . . . . . . . . . . . . . . . . . . . . . . . . 47

7 Rigid bodies 49
7.1 Angular velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.2 Moment of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.3 Calculating the moment of inertia . . . . . . . . . . . . . . . . . 51
7.4 Motion of a rigid body . . . . . . . . . . . . . . . . . . . . . . . . 54

8 Special relativity 61
8.1 The Lorentz transformation . . . . . . . . . . . . . . . . . . . . . 61
8.2 Spacetime diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 64
8.3 Relativistic physics . . . . . . . . . . . . . . . . . . . . . . . . . . 65
8.4 Geometry of spacetime . . . . . . . . . . . . . . . . . . . . . . . . 71
8.5 Relativistic kinematics . . . . . . . . . . . . . . . . . . . . . . . . 74


2

,Contents IA Dynamics and Relativity


8.6 Particle physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79




3

, 0 Introduction IA Dynamics and Relativity


0 Introduction
You’ve been lied to. You thought you applied for mathematics. And here you
have a course on physics. No, this course is not created for students taking the
“Maths with Physics” option. They don’t have to take this course (don’t ask
why).
Ever since Newton invented calculus, mathematics is becoming more and more
important in physics. Physicists seek to describe the universe in a few equations,
and derive everyday (physical) phenomena as mathematical consequences of
these equations.
In this course, we will start with Newton’s laws of motion and use it to derive
a lot of physical phenomena, including planetary orbits, centrifugal forces1 and
the motion of rotating bodies.
The important thing to note is that we can “prove” all these phenomena just
under the assumption that Newton’s laws are correct (plus the formulas for, say,
the strength of the gravitational force). We are just doing mathematics here.
We don’t need to do any experiments to obtain the results (of course, we need
experiments to verify that Newton’s laws are indeed the equations that describe
this universe).
However, it turns out that Newton was wrong. While his theories were
accurate for most everyday phenomena, they weren’t able to adequately describe
electromagnetism. This lead to Einstein discovering special relativity. Special
relativity is also required to describe motion that is very fast. We will have a
brief introduction to special relativity at the end of the course.




1 Yes, they exist.


4

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