Classical Mechanics in depth introduction, Lecture notes of Physics

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Classical Mechanics

(2nd^ Edition)

Cenalo Vaz

University of Cincinnati

Contents

• 1 Vectors
  • 1.1 Displacements
  • 1.2 Linear Coordinate Transformations
  • 1.3 Vectors and Scalars
  • 1.4 Rotations in two dimensions
  • 1.5 Rotations in three dimensions
  • 1.6 Algebraic Operations on Vectors
    • 1.6.1 The scalar product
    • 1.6.2 The vector product
  • 1.7 Vector Spaces
  • 1.8 Some Algebraic Identities
  • 1.9 Differentiation of Vectors
    • 1.9.1 Time derivatives
    • 1.9.2 The Gradient Operator
  • 1.10 Some Differential Identities
  • 1.11 Vector Integration
    • 1.11.1 Line Integrals
    • 1.11.2 Surface integrals
    • 1.11.3 Volume Integrals
  • 1.12 Integral Theorems
    • 1.12.1 Corollaries of Stokes’ Theorem
    • 1.12.2 Corollaries of Gauss’ theorem
• 2 Newton’s Laws and Simple Applications
  • 2.1 Introduction
  • 2.2 The Serret-Frenet description of curves
  • 2.3 Galilean Transformations
  • 2.4 Newton’s Laws
  • 2.5 Newton’s Laws and the Serret Frenet Formulæ
  • 2.6 One dimensional motion CONTENTS iii
  • 2.7 Motion in a resisting medium
    • 2.7.1 Drag and the projectile
    • 2.7.2 Perturbative expansions: an example
  • 2.8 Harmonic motion
    • 2.8.1 Harmonic motion in one dimension
    • 2.8.2 One dimensional oscillations with damping
    • 2.8.3 Two dimensional oscillations
    • 2.8.4 Trajectories in the plane
    • 2.8.5 Lissajou’s figures
  • 2.9 One dimensional free fall
  • 2.10 Systems with variable mass: the rocket
• 3 Conservation Theorems
  • 3.1 Single Particle Conservation Theorems
    • 3.1.1 Conservation of momentum
    • 3.1.2 Conservation of angular momentum
    • 3.1.3 Work and the conservation of energy
  • 3.2 Frictional forces and mechanical energy
  • 3.3 Examples of conservative forces
  • 3.4 The damped and driven oscillator
    • 3.4.1 Fourier Expansion
    • 3.4.2 Green’s Function
  • 3.5 Systems of many particles
    • 3.5.1 Conservation of momentum.
    • 3.5.2 Conservation of angular momentum.
    • 3.5.3 The Work-Energy theorem
  • 3.6 Collisions
    • 3.6.1 One Dimensional Collisions
    • 3.6.2 Two Dimensional Collisions
  • 3.7 The Virial Theorem
• 4 Newtonian Gravity
  • 4.1 The force law
  • 4.2 Two properties of the gravitational field
  • 4.3 Simple Applications of Gauss’ Law
    • 4.3.1 Point mass.
    • 4.3.2 Spherical charge distribution.
    • 4.3.3 Spherical shell.
    • 4.3.4 Infinite line of constant linear mass density (cosmic string).
    • 4.3.5 Infinite sheet of constant areal mass density: (domain wall) iv CONTENTS
  • 4.4 The Poisson and Laplace Equations
• 5 Motion under a Central Force
  • 5.1 Symmetries
    • 5.1.1 Spherical Coordinates
    • 5.1.2 Cylindrical coordinates
  • 5.2 Central Forces
  • 5.3 Inverse square force
    • 5.3.1 Conic sections
    • 5.3.2 Analysis of solutions
    • 5.3.3 Kepler’s laws
  • 5.4 Other examples of central forces
  • 5.5 Stability of Circular Orbits
  • 5.6 Scattering by a Central Force
• 6 Motion in Non-Inertial Reference Frames
  • 6.1 Newton’s second law in an accelerating frame
  • 6.2 Rotating Frames
  • 6.3 Motion near the surface of the earth.
    • 6.3.1 Deflection of a freely falling particle
    • 6.3.2 Motion of a projectile
    • 6.3.3 The Foucault Pendulum
• 7 Rigid Bodies
  • 7.1 Equations of motion
  • 7.2 The Inertia Tensor
  • 7.3 Computing the Inertia Tensor: examples
    • 7.3.1 Homogeneous sphere
    • 7.3.2 Homogeneous cube
  • 7.4 The parallel axis theorem
  • 7.5 Dynamics
• 8 Mechanical Waves
  • 8.1 The Wave Equation
  • 8.2 The Wave Equation from Dynamics
    • 8.2.1 Waves in Strings
    • 8.2.2 Sound Waves in Media
  • 8.3 Energy Transfer
    • 8.3.1 Waves in Strings
    • 8.3.2 Sound Waves CONTENTS v
  • 8.4 Solutions of the Wave Equation
  • 8.5 Boundary Conditions and Particular Solutions
    • 8.5.1 Standing Waves
    • 8.5.2 Traveling Wave at an Interface
  • 8.6 The Doppler Effect
    • 8.6.1 Stationary Source, Moving Observer
    • 8.6.2 Moving Source, Stationary Observer
    • 8.6.3 Generalizations
  • 8.7 Superposition
    • 8.7.1 Interference
    • 8.7.2 Beats
    • 8.7.3 Wave Packets
• 9 The Calculus of Variations
  • 9.1 Functionals
  • 9.2 Euler’s equation for extrema
  • 9.3 Examples
    • 9.3.1 Geodesics
    • 9.3.2 Minimum surface of revolution
    • 9.3.3 The rotating bucket
    • 9.3.4 The Brachistochrone
  • 9.4 Functional Derivatives
  • 9.5 An alternate form of Euler’s equation
  • 9.6 Functionals involving several functions
  • 9.7 Constraints
• 10 The Lagrangian
  • 10.1 Fermat’s least time principle
  • 10.2 The variational principle of mechanics
  • 10.3 Examples
  • 10.4 Symmetries and Noether’s theorems
• 11 The Hamiltonian
  • 11.1 Legendre Transformations
  • 11.2 The Canonical equations of motion
  • 11.3 Poisson Brackets
  • 11.4 Examples
  • 11.5 The Dirac-Bergmann Algorithm for Singular Systems
    • 11.5.1 Dirac Bracket
    • 11.5.2 Examples vi CONTENTS
• 12 Canonical Transformations
  • 12.1 Hamilton’s equations from a Variational Principle
  • 12.2 The Generating Function
  • 12.3 Examples
  • 12.4 The Symplectic Approach
  • 12.5 Infinitesimal Transformations
  • 12.6 Hamiltonian as the generator of time translations
• 13 Hamilton-Jacobi Theory
  • 13.1 The Hamilton-Jacobi equation
  • 13.2 Two examples
  • 13.3 Hamilton’s Characteristic Function
  • 13.4 Separability
  • 13.5 Periodic motion and Action-Angle Variables
  • 13.6 Further Examples
• 14 Special Relativity
  • 14.1 The Principle of Covariance
    • 14.1.1 Galilean tranformations
    • 14.1.2 Lorentz Transformations
  • 14.2 Elementary consequences of Lorentz transformations
    • 14.2.1 Simultaneity
    • 14.2.2 Length Contraction
    • 14.2.3 Time Dilation
    • 14.2.4 Velocity Addition
    • 14.2.5 Aberration
  • 14.3 Tensors on the fly
  • 14.4 Waves and the Relativistic Doppler Effect
  • 14.5 Dynamics in Special Relativity
  • 14.6 Conservation Laws
  • 14.7 Relativistic Collisions
  • 14.8 Accelerated Observers
• 15 More general coordinate systems*
  • 15.1 Introduction
  • 15.2 Vectors and Tensors
  • 15.3 Differentiation
    • 15.3.1 Lie Derivative
    • 15.3.2 Covariant Derivative: the Connection CONTENTS vii
    • 15.3.3 Absolute Derivative: parallel transport
    • 15.3.4 The Laplacian
  • 15.4 Examples
  • 15.5 Integration: The Volume Element
• 16 Ideal Fluids
  • 16.1 Introduction
  • 16.2 Equation of Continuity
  • 16.3 Ideal Fluids
  • 16.4 Euler’s equation for an Ideal Fluid
  • 16.5 Waves in Fluids
  • 16.6 Special Flows
    • 16.6.1 Hydrostatics
    • 16.6.2 Steady Flows
    • 16.6.3 Irrotational or Potential Flows
    • 16.6.4 Incompressible Flows
  • 16.7 Elementary Applications
    • 16.7.1 Hydrostatics
    • 16.7.2 Steady Flows
    • 16.7.3 Potential flows of Incompressible fluids
  • 16.8 The Circulation
• 17 Energy and Momentum in Fluids
  • 17.1 The Energy Flux Density Vector
  • 17.2 Momentum Flux Density Tensor
  • 17.3 The Stress Tensor
  • 17.4 Energy Dissipation
  • 17.5 Boundary Conditions
  • 17.6 Reynolds and Froude Numbers
  • 17.7 Applications of the Navier-Stokes equation
    • 17.7.1 Plane Poiseuille Flow
    • 17.7.2 Couette Flow
    • 17.7.3 Hagen-Poiseuille Flow
  • 17.8 Relativistic Fluids
    • 17.8.1 Perfect Fluids
    • 17.8.2 Conserved Currents
    • 17.8.3 Imperfect Fluids
  • 17.9 Scaling behavior of fluid flows
  • 17.10An Example

viii CONTENTS

A The δ−function i A.1 Introduction.................................... i A.1.1 An example................................ i A.1.2 Another example............................. ii A.1.3 Properties................................. iv A.2 The δ−function in curviliear coordinates.................... v

2 CHAPTER 1. VECTORS

x 1

x 3

x 2

displacement

i

f

Figure 1.1: Displacement vector

regardless of the initial and final points. So also, what defines a vector is its magnitude and direction and not its location in space. We must also consider how displacements in particular and vectors in general may be represented algebraically. In a two dimensional plane, we introduce two mutually perpendicular axes intersecting at some point O, the origin, order them in some way calling one the x axis and the other the y axis, and label points by an ordered pair, the coordinates (x, y), where x represents the projection of the point on the x axis and y its projection on the y axis. A more fruitful way to think about this Cartesian coordinate system is to imagine that we have two mutually perpendicular and space filling one parameter families of parallel straight lines in the plane (see figure (1.2). Because the families are space filling, every point will lie on the intersection of one “vertical” and one “horizontal” line. Label a point by the parameter values of the straight lines it lies on. Why is this a better way to think about coordinates? Because it is now easily generalized. Straight lines are not the only curves possible. We could also consider circles of radius r about an origin together with radial lines from the origin, each making an angle θ with some chosen direction [see figure (1.3)]. Every point in the plane lies on the intersection of some circle with some radial line and could therefore be labeled by the pair (r, θ). These, of course, are the familiar polar coordinates. The system is ill defined at the origin because θ cannot be defined there. The situation is similar in three dimensions, except that the curves are now replaced by surfaces. A coordinate system in three dimensions is a set of three independent, space filling, one parameter families of surfaces relative to which points are labeled. In the Cartesian system this set consists of three mutually perpendicular one parameter families of parallel planes. All points in R^3 will lie on the intersection of a unique set of three

1.1. DISPLACEMENTS 3

y=-

y=-

y=-

y=

y=

y=

y=

y=

y=

x=-2x=-1x=0x=1x=2x=3x=4x=5x=6x=7x=

Figure 1.2: Cartesian coordinates in the plane.

Lines of constant angle

Circles of constant radius

q

^ r

^ r

^ r

^q ^ q

^ q

x

y

r

Figure 1.3: Polar coordinates in the plane.

1.1. DISPLACEMENTS 5

x 1 x 3

x 2

i ds = (dx ,dx ,dx ) 1 2 3

f

(x ,x ,x ) 1 2 3

(x +dx ,x +dx ,x +dx ) 1 1 2 2 3 3

0

Figure 1.4: Representation of the displacement vector

the constraint (^) ∑

i

cos^2 αi = 1, (1.1.4)

showing that one of the three angles is determined by the other two. We will sometimes denote the ith^ component, dxi, of d~r by [d~r]i. The following defini- tions are natural:

• Two displacements are equal if they have the same magnitude and direction:

d~r 1 = d~r 2 ⇔ [d~r 1 ]i = [d~r 2 ]i (1.1.5)

• If a is a real number, [ad~r]i = a[d~r]i (1.1.6) In particular, with a = −1, [−d~r]i = −[d~r]i.
• If d~r 1 and d~r 2 are two displacements then their sum is also a displacement given by

[d~r 1 + d~r 2 ]i = [d~r] 1 ,i + [d~r] 2 ,i (1.1.7)

(This definition can be understood as the algebraic equivalent of the familiar geo- metric parallelogram law of vector addition.)

Our implicit choice of coordinate system can be made explicit by assigning directions to the coordinate axes as follows: since every straight line is determined by two distinct points, on each axis choose two points one unit away from each other, in the direction of increasing coordinate value. There are only three corresponding displacements, which can be written as^2

x̂ 1 = ̂x = (1, 0 , 0), x̂ 2 = ̂y = (0, 1 , 0) and ̂x 3 = ̂z = (0, 0 , 1) (1.1.8) (^2) Carets, as opposed to arrows, are used to represent any displacement of unit magnitude.

6 CHAPTER 1. VECTORS

and it is straightforward that, using the scalar multiplication rule (1.1.6) and the sum rule (1.1.7), any displacement d~r could also be represented as

d~r = dx 1 x̂ 1 + dx 2 x̂ 2 + dx 3 x̂ 3 =

i

dxi ̂xi. (1.1.9)

The ̂xi represent unit displacements along the of our chosen Cartesian system and the set {̂xi} is called a basis. In R^3 , we could use the Cartesian coordinates of any point to represent its displacement from the origin. Displacements in R^3 from the origin

~r = (x 1 , x 2 , x 3 ) =

i

xi ̂xi. (1.1.10)

are called position vectors. It is extremely important to recognize that the representation of a displacement de- pends sensitively on the choice of coordinate system whereas the displacement itself does not. Therefore, we must distinguish between displacements (and, vectors, in general) and their representations. To see why this is important, we first examine how different Cartesian systems transform into one another.

1.2 Linear Coordinate Transformations

Two types of transformations exist between Cartesian frames, viz., translations of the origin of coordinates and rotations of the axes. Translations are just constant shifts of the coordinate origin. If the origin, O, is shifted to the point O′^ whose coordinates are (xO, yO, zO), measured from O, the coordinates get likewise shifted, each by the corre- sponding constant, x′^ = x − xO, y′^ = y − yO, z′^ = z − zO (1.2.1)

But since xO, yO and zO are all constants, such a transformation does not change the representation of a displacement vector,

d~r = (dx, dy, dz) = (dx′, dy′, dz′). (1.2.2)

Representations of displacement vectors are, however, affected by a rotation of the coordi- nate axes. Let us first consider rotations in two spatial dimensions [see figure (1.5)], where the primed axes are obtained from the original system by a rotation through some angle, θ. The coordinates (x 1 , x 2 ) of a point P in the original system would be (x′ 1 , x′ 2 ) in the rotated system. In particular, in terms of the length l of the hypotenuse OP of triangle AOP [figure (1.5)], we have

x 1 = l cos(α + θ) = (l cos α) cos θ − (l sin α) sin θ = x′ 1 cos θ − x′ 2 sin θ

8 CHAPTER 1. VECTORS

= (dx′ 1 cos θ − dx′ 2 sin θ)x̂ 1 + (dx′ 1 sin θ + dx′ 2 cos θ)̂x 2 (1.2.8)

A simple comparison now gives

dx 1 = dx′ 1 cos θ − dx′ 2 sin θ

dx 2 = dx′ 1 sin θ + dx′ 2 cos θ (1.2.9)

or, upon inverting the relations,

dx′ 1 = dx 1 cos θ + dx 2 sin θ

dx′ 2 = −dx 1 sin θ + dx 2 cos θ. (1.2.10)

It is easy to see that these transformations can also be written in matrix form as ( dx′ 1 dx′ 2

cos θ sin θ − sin θ cos θ

dx 1 dx 2

and (^) ( dx 1 dx 2

cos θ − sin θ sin θ cos θ

dx′ 1 dx′ 2

Other, more complicated but rigid transformations of the coordinate system can always be represented as combinations of rotations and translations.

1.3 Vectors and Scalars

Definition: A vector is a quantity that can be represented in a Cartesian system by an ordered triplet (A 1 , A 2 , A 3 ) of components, which transform as the components of an infinitesimal displacement under a rotation of the reference coordinate system. Any vector can always be expressed as a linear combination of basis vectors, A~ = Ai xˆi.

In two dimensions, a vector may be represented by two Cartesian components A~ = (A 1 , A 2 ), which transform under a rotation of the Cartesian reference system as (A 1 , A 2 ) → (A′ 1 , A′ 2 ) such that (^) ( A′ 1 A′ 2

cos θ sin θ − sin θ cos θ

A 1

A 2

Definition: A scalar is any physical quantity that does not transform (stays invariant) under a rotation of the reference coordinate system.

1.4. ROTATIONS IN TWO DIMENSIONS 9

A typical scalar quantity in Newtonian mechanics would be the mass of a particle. The magnitude of a vector is also a scalar quantity, as we shall soon see. It is of great interest to determine scalar quantities in physics because these quantities are not sensitive to particular choices of coordinate systems and are therefore the same for all observers. Other examples of scalars within the context of Newtonian mechanics are temperature and density. In the Newtonian conception of space and time, time is also a scalar. Because time is a scalar all quantities constructed from the position vector of a particle moving in space by taking derivatives with respect to time are also vectors, therefore

• the velocity: ~v = d~dtr
• the acceleration: ~a = d~dtv
• the momentum: ~p = m~v and
• the force F~ = d~dtp

are all examples of vectors that arise naturally in mechanics. In electromagnetism, the electric and magnetic fields are vectors. As an example of a quantity that has the ap- pearance of a vector but is not a vector, consider A = (x, −y). Under a rotation of the coordinate system by an angle θ,

A′ 1 = A 1 cos θ − A 2 sin θ

A′ 2 = A 1 sin θ + A 2 cos θ (1.3.2)

which are not consistent with (1.3.1). The lesson is that the transformation properties must always be checked.

1.4 Rotations in two dimensions

Equation (1.3.1) can also be written as follows

A′ i =

j

R̂ ij Aj (1.4.1)

where

R̂ ij (θ) =

cos θ sin θ − sin θ cos θ

is just the two dimensional “rotation” matrix. We easily verify that it satisfies the following very interesting properties:

1.5. ROTATIONS IN THREE DIMENSIONS 11

1.5 Rotations in three dimensions

In two dimensions there is just one way to rotate the axes which, if we introduce a “x 3 ” axis, amounts to a rotation of the x 1 −x 2 axes about it. In three dimensions there are three such rotations possible: the rotation of the x 1 − x 2 axes about the x 3 axis, the rotation of the x 2 − x 3 axes about the x 1 axis and the rotation of the x 1 − x 3 axes about the x 2 axis. In each of these rotations the axis of rotation remains fixed, and each rotation is obviously independent of the others. Thus, we now need 3 × 3 matrices and may write

R̂^3 (θ) =

cos θ sin θ 0 − sin θ cos θ 0 0 0 1

to represent the rotation of the x 1 − x 2 axes as before about the x 3 axis. Under such a rotation only the first and second component of a vector are transformed according to the rule A′ i =

j

R̂^3 ij (θ)Aj (1.5.2)

Rotations about the other two axes may be written likewise as follows:

R̂^1 (θ) =

0 cos θ sin θ 0 − sin θ cos θ

and^4

R̂^2 (θ) =

cos θ 0 − sin θ 0 1 0 sin θ 0 cos θ

The general rotation matrix in three dimensions may be constructed in many ways, one of which (originally due to Euler) is canonical:

• first rotate the (x 1 , x 2 ) about the x 3 axis through an angle θ. This gives the new axes (ξ, η, τ ) (τ ≡ z),
• existence of an inverse i.e., ∀ g ∈ G ∃ g−^1 ∈ G s.t. g ∗ g−^1 = g−^1 ∗ g = e, and
• associativity of ∗, i.e., ∀ g 1 , g 2 , g 3 ∈ G, g 1 ∗ (g 2 ∗ g 3 ) = (g 1 ∗ g 2 ) ∗ g 3

is called a group.

Definition: If ∀ g 1 , g 2 ∈ G, [g 1 , g 2 ] = g 1 ∗ g 2 − g 2 ∗ g 1 = 0 then the group (G, ∗) is called a “commutative” or “ Abelian” group. [g 1 , g 2 ] is called the commutator of the elements g 1 and g 2. (^4) Note the change in sign. It is because we are using a right-handed coordinate system. Convince yourself that it should be so.

12 CHAPTER 1. VECTORS

• then rotate (ξ, η, z) about the ξ axis through an angle φ. This gives the new axes (ξ′, η′, τ ′) (ξ′^ ≡ ξ),
• finally rotate (ξ, η′, τ ′) about the τ ′^ axis through an angle ψ to get (x′, y′, z′).

We get R̂ (θ, φ, ψ) = R̂^3 (ψ) · R̂^2 (φ) · R̂^3 (θ) (1.5.5)

The angles {θ, φ, ψ} are called the Euler angles after the the originator of this particular sequence of rotations.^5 The sequence is not unique however and there are many possible ways to make a general rotation. To count the number of ways, we need to keep in mind that three independent rotations are necessary:

• the first rotation can be performed in one of three ways, corresponding to the three independent rotations about the axes,
• the second rotation can be performed in one of two independent ways: we are not per- mitted to rotate about the axis around which the previous rotation was performed, and
• the third rotation can be performed in one of two independent ways: again we are not permitted to rotate about the axis around which the previous rotation was performed.

So in all there are 3 × 2 × 2 = 12 possible combinations of rotations that will give the desired general rotation matrix in three dimensions. Note that any scheme you choose will involve three and only three independent angles, whereas only one angle was needed to define the general rotation matrix in two dimensions. The general rotation matrix in n dimensions will require n(n − 1)/2 angles. Three dimensional rotation matrices satisfy some interesting properties that we will now outline:

• The product of any two rotation matrices is also a rotation matrix.
• The identity matrix is just the rotation matrix R̂ (0, 0 , 0).
• All three dimensional rotation matrices, like their two dimensional counterparts, obey the condition R̂ T^ · R̂ = 1 (1.5.6) (^5) Problem: Show that the general rotation matrix constructed with the Euler angles is

R̂(θ, φ, ψ) =

 

cos ψ cos θ − cos φ sin θ sin ψ cos ψ sin θ + cos φ cos θ sin ψ sin ψ sin φ − sin ψ cos θ − cos φ sin θ cos ψ − sin ψ sin θ + cos φ cos θ cos ψ cos ψ sin φ sin φ sin θ − sin φ cos θ cos φ

 