3.7 Generalised dynamics

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3.7 Generalised dynamics	79

3.7 Generalised dynamics Lagrangian dynamics

action (δS = 0 for the motion)

Action

S = t1

L(q,q˙,t) dt

(3.213)

q˙

generalised velocities

generalised coordinates

Lagrangian

Euler–Lagrange

∂L

(3.214)

mass

equation

∂q˙i

− ∂qi

= 0

time

velocity

Lagrangian of

− U(r,t)

(3.215)

L =

position vector

particle in

external ﬁeld

= T − U

(3.216)

potential energy

kinetic energy

(rest) mass

Relativistic

m0c2

Lorentz factor

Lagrangian of a

L = −

− e(φ− A · v)

(3.217)

positive charge

charged particle

electric potential

magnetic vector potential

Generalised

pi =

∂L

(3.218)

generalised momenta

momenta

∂q˙i

Hamiltonian dynamics

Lagrangian

Hamiltonian

H =

piq˙i

−

(3.219)

generalised momenta

q˙ii

generalised velocities

Hamilton’s

∂H

Hamiltonian

equations

q˙i =

;

p˙i = −

(3.220)

∂pi

∂qi

generalised coordinates

Hamiltonian

H =

+ U(r,t)

(3.221)

particle speed

position vector

of particle in

potential energy

external ﬁeld

= T + U

(3.222)

kinetic energy

(rest) mass

Relativistic

speed of light

Hamiltonian

2 2

1/2

of a charged

H = (m0c

|p − eA|

c )

+ eφ

(3.223)

positive charge

particle

electric potential

vector potential

∂f ∂g

[f,g] =

−

(3.224)

particle momentum

∂qi

∂pi

∂qi

Poisson

∂g

time

∂g

f,g

arbitrary functions

brackets

[qi,g] = ∂pi ,

[pi,g] = − ∂qi

(3.225)

[·,·]

Poisson bracket (also see

∂g

[H,g] = 0

= 0,

= 0

(3.226)

Commutators on page 26)

∂t

Hamilton–

∂S

Jacobi

action

∂t + H qi, ∂qi ,t = 0

(3.227)

equation

80	Dynamics and mechanics

3.8Elasticity

Elasticity deﬁnitions (simple)a						F
			τ	stress			A
			τ	stress			A
Stress	τ = F/A	(3.228)	F	applied force
Stress	τ = F/A	(3.228)	A	cross-sectional
			A	cross-sectional	l
				area	l
				area
			e	strain
			e	strain
Strain	e = δl/l	(3.229)	δl	change in length
Strain	e = δl/l	(3.229)	δl	change in length
			l	length
			l	length		w
						w
Young modulus	E = τ/e = constant	(3.230)	E	Young modulus
(Hooke’s law)	E = τ/e = constant	(3.230)	E	Young modulus
(Hooke’s law)
			σ	Poisson ratio
	δw/w		σ	Poisson ratio
Poisson ratiob	δw/w		δw	change in width

	σ = − δl/l	(3.231)
	σ = − δl/l	(3.231)	w	width
			w	width

aThese apply to a thin wire under longitudinal stress.

bSolids obeying Hooke’s law are restricted by thermodynamics to −1 ≤ σ ≤ 1/2, but none are known with σ < 0. Non-Hookean materials can show σ > 1/2.

Elasticity deﬁnitions (general)

Stress tensora

τij =

force i direction

(3.232)

τij

stress tensor (τij = τji)

area j direction

∂uk

∂ul

ekl

strain tensor (ekl = elk)

Strain tensor

ekl =

(3.233)

displacement to xk

∂xl

∂xk

coordinate system

Elastic modulus

τij = λijklekl

(3.234)

λijkl

elastic modulus

Elastic energyb

U =

λijkleij ekl

(3.235)

potential energy

Volume strain

δV

volume strain

ev =

+ e33

(3.236)

(dilatation)

= e11 + e22

δV

change in volume

volume

Shear strain

ekl = (ekl −

evδkl)+

evδkl

(3.237)

δkl

Kronecker delta

Hydrostatic

)

pure shear

dilatation

compression

τij = −pδij

(3.238)

hydrostatic pressure

aτii are normal stresses, τij (i = j) are torsional stresses.

bAs usual, products are implicitly summed over repeated indices.

3.8 Elasticity	81

Isotropic elastic solids

µ =

(3.239)

Lame´ coe cients

2(1 + σ)

Eσ

λ =

(3.240)

(1 + σ)(1 − 2σ)

Longitudinal

Ml =

E(1

−

σ)

= λ+ 2µ

(3.241)

modulusa

(1 + σ)(1 − 2σ)

[τii −

eii =

σ(τjj + τkk)]

(3.242)

Diagonalised

τii = Ml eii +

(ejj + ekk)

(3.243)

equationsb

1 − σ

t = 2µe + λ1tr(e)

(3.244)

K =

= λ+

(3.245)

Bulk modulus

3(1 − 2σ)

(compression

1 ∂V

= − V ∂p

(3.246)

modulus)

−

p = Kev

(3.247)

Shear modulus

µ =

(3.248)

2(1 + σ)

(rigidity modulus)

τT = µθsh

(3.249)

Young modulus

E =

9µK

(3.250)

µ+ 3K

Poisson ratio

σ =

3K − 2µ

(3.251)

2(3K + µ)

µ,λ Lame´ coe cients

EYoung modulus

σPoisson ratio

longitudinal elastic modulus

eii	strain in i direction	3
τii	stress in i direction

estrain tensor

tstress tensor

1 unit matrix tr(·) trace

Kbulk modulus

KT isothermal bulk modulus

Vvolume

ppressure

Ttemperature

ev volume strain

µshear modulus

τT	transverse stress
θsh	shear strain

τT

θsh

aIn an extended medium.

bAxes aligned along eigenvectors of the stress and strain tensors.

Torsion

Torsional rigidity

twisting couple

torsional rigidity

(for a

G = C

(3.252)

rod length

homogeneous

rod)

twist angle in

length l

radius

Thin circular

C = 2πa3µt

(3.253)

wall thickness

cylinder

shear modulus

Thick circular

C =

µπ(a24 − a14)

(3.254)

inner radius

cylinder

outer radius

cross-sectional

Arbitrary

4A2µt

C =

(3.255)

area

thin-walled tube

perimeter

Long ﬂat ribbon

C =

µwt3

(3.256)

cross-sectional

width

82	Dynamics and mechanics

Bending beamsa

bending moment

ξ2 ds

Young modulus

Gb =

(3.257)

radius of curvature

Bending

area element

moment

(3.258)

distance to neutral

surface from ds

moment of area

displacement from

Light beam,

horizontal

horizontal at

end-weight

x = 0, weight

y = 2EI

l − 3

(3.259)

beam length

at x = l

distance along beam

Heavy beam

d4y

beam weight per

EI dx4 = w(x)

(3.260)

unit length

π2EI/l2

(free ends)

Euler strut

Fc = 4π2EI/l2

(ﬁxed ends)

critical compression

force

failure

EI/(4l

)

(1 free end)

strut length

(3.261)

aThe radius of curvature is approximated by 1/Rc d2y/dx2.

neutral surface

(cross section)

free

Fc Fc

ﬁxed

Elastic wave velocitiesa

	vt = (µ/ρ)1/2					(3.262)
In an inﬁnite	vl = (Ml/ρ)1/2					(3.263)
isotropic solidb		vl	2		2σ	1/2
		vl	2		2σ
			=	1 −	2σ	(3.264)
		vt	=	1 −	2σ	(3.264)
				−
In a ﬂuid	vl = (K/ρ)1/2					(3.265)

On a thin plate (wave travelling along x, plate thin in z)

vl(x) =

1/2

(3.266)

ρ(1

σ2)

v(y)

−

= (µ/ρ)1/2

(3.267)

= k

Et2

1/2

vt(z)

(3.268)

12ρ(1 − σ2)

vl = (E/ρ)1/2

(3.269)

In a thin circular

vφ = (µ/ρ)1/2

(3.270)

rod

1/2

vt =

(3.271)

vt	speed of transverse wave
vl	speed of longitudinal wave

µshear modulus

ρdensity

Ml	longitudinal modulus
		!
	=	E(1−σ)
		(1+σ)(1−2σ)

Kbulk modulus

vl(i)	speed of longitudinal
	wave (displacement i)
vt(i)	speed of transverse wave
	(displacement i)

EYoung modulus

σPoisson ratio

kwavenumber (= 2π/λ)

tplate thickness (in z, t λ)

vφ torsional wave velocity a rod radius ( λ)

aWaves that produce “bending” are generally dispersive. Wave (phase) speeds are quoted throughout.

bTransverse waves are also known as shear waves, or S-waves. Longitudinal waves are also known as pressure waves, or P-waves.

3.8 Elasticity	83

Waves in strings and springsa

			vl	speed of longitudinal wave
In a spring	vl = (κl/ρl)1/2	(3.272)	κ	spring constantb
			l	spring length
			ρl	mass per unit lengthc
On a stretched	vt = (T /ρl)1/2	(3.273)	vt	speed of transverse wave
string			T	tension

On a stretched	vt = (τ/ρA)1/2	(3.274)	τ	tension per unit width	3
sheet			ρA	mass per unit area	3

aWave amplitude assumed wavelength. bIn the sense κ = force/extension.

cMeasured along the axis of the spring.

Propagation of elastic waves

Acoustic

force

(3.275)

Z = response velocity

u˙

impedance

= (E ρ)1/2

stress force

(3.276)

strain displacement

v =

1/2

Wave velocity/

elastic modulus

impedance

(3.277)

density

relation

then

Z = (E ρ)1/2 = ρv

(3.278)

wave phase velocity

Mean energy

E k2u2

(3.279)

energy density

wavenumber

density

ρω2u02

angular frequency

(nondispersive

(3.280)

waves)

maximum displacement

P = Uv

(3.281)

mean energy ﬂux

r =

−

τr

Z1 − Z2

(3.282)

reﬂection coe cient

Normal

τi

Z1 + Z2

transmission coe cient

coe cientsa

2Z1

(3.283)

t = Z1 + Z2

stress

θi

angle of incidence

sinθi

sinθr

sinθt

Snell’s lawb

(3.284)

θr

angle of reﬂection

θt

angle of refraction

aFor stress and strain amplitudes. Because these reﬂection and transmission coe cients are usually deﬁned in terms of displacement, u, rather than stress, there are di erences between these coe cients and their equivalents deﬁned in electromagnetism [see Equation (7.179) and page 154].

bAngles deﬁned from the normal to the interface. An incident plane pressure wave will generally excite both shear and pressure waves in reﬂection and transmission. Use the velocity appropriate for the wave type.

84 Dynamics and mechanics

3.9 Fluid dynamics Ideal ﬂuidsa

∂ρ

density

Continuityb

ﬂuid velocity ﬁeld

∂t + · (ρv) = 0

(3.285)

time

circulation

Γ =

dl = constant

(3.286)

loop element

Kelvin circulation

element of surface

= S ω · ds

(3.287)

bounded by loop

vorticity (= ×v)

∂v

+ (v

)v =

−

+ g

(3.288)

pressure

Euler’s equationc

∂t

∂

gravitational ﬁeld

( ×v) = ×[v×( ×v)]

(3.289)

strength

(v · )

advective operator

∂t

Bernoulli’s equation

ρv2 + p+ ρgz = constant

(3.290)

altitude

(incompressible ﬂow)

ratio of speciﬁc heat

Bernoulli’s equation

+ gz = constant

(3.291)

capacities (cp/cV )

− 1

(compressible

speciﬁc heat capacity

adiabatic ﬂow)d

at constant pressure

= 2 v

+ cpT + gz

(3.292)

temperature

Hydrostatics

p = ρg

(3.293)

Adiabatic lapse rate

= −

(3.294)

(ideal gas)

aNo thermal conductivity or viscosity. bTrue generally.

cThe second form of Euler’s equation applies to incompressible ﬂow only. dEquation (3.292) is true only for an ideal gas.

Potential ﬂowa

Velocity potential	v = φ							(3.295)	v	velocity
	2	φ = 0						(3.296)
		φ = 0						(3.296)	φ	velocity potential
									φ	velocity potential
									ω	vorticity
	ω = ×v = 0								ω	vorticity
Vorticity condition	ω = ×v = 0							(3.297)	F	drag force on moving
										sphere
									a	sphere radius
									a	sphere radius
Drag force on a		2		πρa3u˙	= −	1			u˙	sphere acceleration
sphereb	F = −						Mdu˙	(3.298)
sphereb	F = −		3			2	Mdu˙	(3.298)	ρ	ﬂuid density
									Md	displaced ﬂuid mass

aFor incompressible ﬂuids.

bThe e ect of this drag force is to give the sphere an additional e ective mass equal to half the mass of ﬂuid displaced.

3.9 Fluid dynamics	85

Viscous ﬂow (incompressible)a

τij

ﬂuid stress tensor

∂vi

∂vj

hydrostatic pressure

Fluid stress

(3.299)

velocity along i axis

τij = −pδij + η ∂xj

∂xi

shear viscosity

δij

Kronecker delta

∂v

+ (v

)v =

−

ω + g

(3.300)

ﬂuid velocity ﬁeld

Navier–Stokes

∂t

vorticity

equationb

−

2v + g

(3.301)

gravitational acceleration

density

Kinematic

ν = η/ρ

(3.302)

kinematic viscosity

viscosity

aI.e., · v = 0, η = 0.

bNeglecting bulk (second) viscosity.

Laminar viscous ﬂow

ﬂow velocity

Between

∂p

direction of ﬂow

y(h− y)

(3.303)

distance from

parallel plates

vz(y) =

2η

∂z

plate

shear viscosity

pressure

∂p

distance from

vz(r) =

(a − r

)

(3.304)

Along a

4η

∂z

pipe axis

circular pipea

πa

∂p

pipe radius

Q =

(3.305)

volume

8η

∂z

Circulating

axial couple

between

4πηa12a22

between cylinders

concentric

Gz =

(ω2 − ω1)

per unit length

rotating

2 −

(3.306)

ωi

angular velocity

cylindersb

of ith cylinder

ω1

Along an

Q =

π ∂p

a24

−

a14

(a22

− a12)2

inner radius

outer radius

ω2

annular pipe

8η ∂z

− ln(a2/a1)

volume discharge

(3.307)

rate

aPoiseuille ﬂow.

bCouette ﬂow.

Draga

On a sphere (Stokes’s law)	F = 6πaηv	(3.308)	F	drag force
On a sphere (Stokes’s law)	F = 6πaηv	(3.308)	a	radius
			a	radius
			v	velocity
On a disk, broadside to ﬂow	F = 16aηv	(3.309)	v	velocity
On a disk, broadside to ﬂow	F = 16aηv	(3.309)	η	shear viscosity
			η	shear viscosity

On a disk, edge on to ﬂow	F = 32aηv/3	(3.310)

aFor Reynolds numbers 1.

86 Dynamics and mechanics

Characteristic numbers

Reynolds number

Reynolds

ρUL

inertial force

density

Re =

number

(3.311)

characteristic velocity

viscous force

characteristic scale-length

shear viscosity

Froude

F =

inertial force

Froude number

numbera

(3.312)

gravitational acceleration

gravitational force

Strouhal

S =

Uτ

evolution scale

(3.313)

Strouhal number

numberb

physical scale

characteristic timescale

Prandtl number

Prandtl

P =

ηcp

momentum transport

(3.314)

Speciﬁc heat capacity at

number

heat transport

constant pressure

thermal conductivity

Mach

M =

speed

(3.315)

Mach number

number

sound speed

Rossby

Ro =

inertial force

Rossby number

number

(3.316)

ΩL

Coriolis force

Ω

angular velocity

aSometimes the square root of this expression. L is usually the ﬂuid depth. bSometimes the reciprocal of this expression.

Fluid waves

vp =

wave (phase) speed

1/2

bulk modulus

Sound waves

(3.317)

pressure

dρ

density

In an ideal gas

ratio of heat capacities

γRT

1/2

γp 1/2

molar gas constant

(adiabatic

vp =

= ρ

(3.318)

(absolute) temperature

conditions)a

mean molecular mass

group speed of wave

ω2 = gktanhkh

(3.319)

liquid depth

Gravity waves on

g 1/2

(h λ)

wavelength

a liquid surfaceb

wavenumber

(3.320)

(gh)1/!2

λ)

gravitational acceleration

angular frequency

Capillary waves

σk

(3.321)

surface tension

(ripples)c

Capillary–gravity

ω2 = gk +

σk3

(3.322)

waves (h λ)

aIf the waves are isothermal rather than adiabatic then vp = (p/ρ)1/2. bAmplitude wavelength.

cIn the limit k2 gρ/σ.

3.9 Fluid dynamics	87

Doppler e ecta

Source at rest,

|u|

ν ,ν observed frequency

emitted frequency

observer

= 1

cosθ

(3.323)

moving at u

− vp

wave (phase) speed

in ﬂuid

Observer at

(3.324)

velocity

rest, source

|u|

angle between

cosθ

moving at u

− vp

wavevector, k, and u

aFor plane waves in a stationary ﬂuid.

Wave speeds

						vp	phase speed
		ω				ν	frequency
		ω
Phase speed	vp =		= νλ		(3.325)	ω	angular frequency (= 2πν)
Phase speed	vp =	k	= νλ		(3.325)	ω	angular frequency (= 2πν)
						λ	wavelength
						k	wavenumber (= 2π/λ)

	vg =	dω			(3.326)

Group speed		dk				vg	group speed
Group speed				dvp		vg	group speed
	= vp − λ			dvp	(3.327)
	= vp − λ			dλ	(3.327)

Shocks

θw

wedge semi-angle

Mach wedgea

sinθw =

(3.328)

wave (phase) speed

body speed

Kelvin

λK =

4πvb2

(3.329)

λK

characteristic

wavelength

wedgeb

θw = arcsin(1/3) = 19◦.5

(3.330)

gravitational

acceleration

shock radius

Spherical

Et2

1/5

energy release

adiabatic

time

r ρ0

(3.331)

shockc

ρ0

density of undisturbed

medium

upstream values

2γM12 − (γ − 1)

(3.332)

downstream values

Rankine–

γ + 1

pressure

Hugoniot

ρ2

γ + 1

(3.333)

velocity

shock

ρ1

(γ −

1) + 2/M12

temperature

relations

density

[2γM1 − (γ − 1)][2 + (γ − 1)M1

]

(3.334)

ratio of speciﬁc heats

(γ + 1)2M2

Mach number

aApproximating the wake generated by supersonic motion of a body in a nondispersive medium.

bFor gravity waves, e.g., in the wake of a boat. Note that the wedge semi-angle is independent of vb. cSedov–Taylor relation.

dSolutions for a steady, normal shock, in the frame moving with the shock front. If γ = 5/3 then v1/v2 ≤ 4.

88	Dynamics and mechanics

Surface tension

σlv =

surface energy

(3.335)

Deﬁnition

area

surface tension

(3.336)

length

Laplace’s

formulaa

∆p = σlv

(3.337)

Capillary

1/2

cc =

2σlv

constant

(3.338)

gρ

Capillary rise

h =

2σlv cosθ

(3.339)

(circular tube)

ρga

Contact angle

cosθ =

σwv − σwl

(3.340)

σlv

σlv	surface tension
	(liquid/vapour
	interface)
∆p	pressure di erence
	over surface
Ri	principal radii of
	curvature
cc	capillary constant

ρliquid density

ggravitational acceleration

hrise height

θcontact angle

atube radius

σwv wall/vapour surface tension

σwl wall/liquid surface tension

aFor a spherical bubble in a liquid ∆p = 2σlv/R. For a soap bubble (two surfaces) ∆p = 4σlv/R.

surface

ha θ

σwv

σwl θ σlv

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