analytic dynamics

1

ANALYTIC DYNAMICSSUMMARY

SOLO HERMELIN

2

Analytic DynamicsSOLO

1. Background

Table of Content

1.1 Newton’s Laws of Motion

1.2 Work and Energy1.3 The Principal Laws of Analytic Dynamics

1.4 Basic Definitions

1.5 Constraints

1.6 Generalized Coordinates

1.7 The Stationary Value of a Function and of a Definite Integral

1.8 The Principle of Virtual Work

2. D’Alembert Principle

3. Hamilton’s Principle

4. Lagrange’s Equations of Motion 5. Hamilton’s Equations

6. Kane’s Equations

7. Gibbs-Appel’s Equations

References

3


1.1 Newton’s Laws of Motion

“The Mathematical Principles of Natural Philosophy” 1687

First Law Every body continues in its state of rest or of uniform motion in

straight line unless it is compelled to change that state by forcesimpressed upon it.

Second Law The rate of change of momentum is proportional to the forceimpressed and in the same direction as that force.

Third Law To every action there is always opposed an equal reaction.

→=→= constvF

0

( )vmtd

dp

td

dF

==

2112 FF

−=

vmp = td

pdF

=

12F

1 2

21F

4


1.2 Work and Energy

The work W of a force acting on a particle m that moves as a result of this alonga curve s from to is defined by:

F

1r

2r

∫∫ ⋅

=⋅=

⋅∆ 2

1

2

1

12

r

r

r

r

rdrmdt

drdFW

r

1r

2r

rd

rdr+

1

2

F

m

s

rd

is the displacement on a real path.

⋅⋅∆⋅= rrmT

2

1

The kinetic energy T is defined as:

1212

2

1

2

1

2

12

TTrrdm

dtrrdt

dmrdrm

dt

dW

r

r

r

r

r

r

−=

⋅=⋅

=⋅

= ∫∫∫

⋅

⋅

⋅⋅⋅⋅⋅

For a constant mass m

5


Work and Energy (continue(

When the force depends on the position alone, i.e. , and the quantityis a perfect differential

( )rFF

= rdF

⋅

( ) ( )rdVrdrF

−=⋅

The force field is said to be conservative and the function is known as the Potential Energy. In this case:

( )rV

( ) ( ) ( ) 212112

2

1

2

1

VVrVrVrdVrdFWr

r

r

r

−=−=−=⋅= ∫∫∆

The work does not depend on the path from to . It is clear that in a conservativefield, the integral of over a closed path is zero.

12W 1r

2r

rdF

⋅

( ) ( ) 01221

21

1

2

2

1

=−+−=⋅+⋅=⋅ ∫∫∫ VVVVrdFrdFrdF

path

r

r

path

r

rC

Using Stoke’s Theorem it means that∫∫∫ ⋅×∇=⋅SC

sdFrdF

0=×∇= FFrot

Therefore is the gradient of some scalar functionF

( ) rdrVdVrdF

⋅−∇=−=⋅

( )rVF

−∇=

6


Work and Energy (continue(

and⋅

→∆→∆⋅−=⋅−=

∆∆= rF

dt

rdF

t

V

dt

dVtt

00limlim

But also for a constant mass system

⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅=⋅=

⋅+⋅=

⋅= rFrrmrrrr

mrrm

dt

d

dt

dT

22

1

Therefore for a constant mass in a conservative field

( ) .0 constEnergyTotalVTVTdt

d ==+⇒=+

7


1.3 The Principal Laws of Analytical Dynamics

The basic laws of dynamics can be formulated (expressed mathematically( in several waysother that that given by Newton’s Laws. The most important are:

(a( D’Alembert Principle

(b( Lagrange’s Equations

(c( Hamilton’s Equations

(d( Hamilton’s Principle

(e( Kane’s Equations

(f( Gibbs-Appell’s Equations

All are basically equivalent.

8


1.4 Basic Definitions

Given a system of N particles defined by their coordinates:

( ) ( ) ( ) ( ) Nkktzjtyitxzyxrr kkkkkkkk ,,2,1,, =++==

where are the unit vectors defining any Inertial Coordinate System kji,,

The real displacement of the particle : km

( ) ( ) ( ) Nkktdzjtdyitdxrd kkkk ,,2,1 =++=

is the infinitesimal change in the coordinates along real path caused by all theforces acting on the particle . km

The virtual displacements are infinitesimal changes in thecoordinates; they are not real changes because they are not caused by real forces.

The virtual displacements define a virtual path that coincides with the real one atthe end points.

( )tzyx kkk ∆∆∆∆ ,,,

9


Basic Definitions (continue(

( )trk

( )1trk

( )2trk

krd

1

2F

km

),,,( dttdzzdyydxxPrdr kkkkkkkk ++++=+

),,,( tzyxP kkk

),,,( ttzzyyxxP kkkkkk ∆+∆+∆+∆+

),,,( tzzyyxxP kkkkkk ∆+∆+∆+

tvrd kk ∆=

i j

k

True (Dynamical or Newton) Path

Virtual Path

10


1.5 Constraints

If the N particles are free the system has n = 3 N degrees of freedom. ( ) Nkzyxr kkkk ,,2,1,, =

The constraints on the system can be of the following types:

(1( Equality Constraints: The general form (the Pffafian form(

( ) ( ) ( )[ ] ( ) mldttradztradytradxtra lt

N

kk

lzkk

lykk

lxk ,,2,10,,,,

1

==+++∑

=

or

{ } maaarankmldtarda lzk

lyk

lxk

lt

N

kk

lk ===+⋅∑

=

,,,,2,101

We can classify the constraints as follows:

(a( Time Dependency

(a1( Catastatic mla lt ,,2,10 ==

(a2( Acatastatic mla lt ,,2,10 =≠

(1( Equality Constraints

(2( Inequality Constraints

11


Constraints (continue(

Equality Constraints: The general form (the Pffafian form( (continue(


lyk

lxk

lt

N

kk

lk ===+⋅∑

=

,,,,2,101

(b( Integrability

(b1( Holonomic if the Pffafian forms are integrable; i.e.:

mldtt

fzd

z

fyd

y

fxd

x

fdf

N

k

lk

k

lk

k

lk

k

ll ,,2,1

1

=∂

∂+

∂∂+

∂∂+

∂∂= ∑

=

or( ) mltzyxzyxf NNNl ,,2,10,,,,,,, 111 ==

(b2( Non-holonomic if the Pffafian forms are not integrable

(b2.1( Scleronomic:

(b2.2( Rheonomic:

ml

l

t

f

,,2,1

0=

=∂

∂or

( ) mlzyxzyxf NNNl ,,2,10,,,,,, 111 ==

ml

l

t

f

,,2,1

0=

≠∂

∂

12



(2( Inequality Constraints:

(a( Stationary Boundaries (time independent(:

(b( Non-stationary Boundaries (time dependent(:

( ) mlzyxzyxf NNNl ,,2,10,,,,,, 111 =≥

( ) mltzyxzyxf NNNl ,,2,10,,,,,,, 111 =≥

13



Displacements Consistent with the Constraints:

The real displacement consistent with theGeneral Equality Constraints (Pffafian form) is:

The virtual displacement consistent with theGeneral Equality Constraints (Pffafian form) is:

dtkdzjdyidxrd kkkk , ++=

[ ] mldtardadtadzadyadxa lt

N

kk

lk

lt

N

kk

lzkk

lykk

lxk ,,2,10

11

==+⋅=+++ ∑∑

==

tkzjyixr kkkk ∆∆+∆+∆=∆ ,

[ ] mltaratazayaxa lt

N

kk

lk

lt

N

kk

lzkk

lykk

lxk ,,2,10

11

==∆+∆⋅=∆+∆+∆+∆ ∑∑

==

Dividing the Pffafian equation by dt and taking the limit, we obtain:

mlraaN

kk

lk

lt ,,2,1

1

=⋅−= ∑

=

⋅

Now replace in the virtual displacement equationlta

mltrraN

kkk

lk ,,2,10

1

==

∆−∆⋅∑

=

⋅

Define the δ variation as:

td

dt∆−∆=

∆δ

14



Displacements Consistent with the Constraints (continue(:

Define the δ variation as:td

dt∆−∆=

∆δ

( )trk kr

δkm

),,,( dttdzzdyydxxPrdr kkkkkkkk ++++=+

),,,( tzyxP kkk

),,,( ttzzyyxxP kkkkkk ∆+∆+∆+∆+

),,,( tzzyyxxP kkkkkk ∆+∆+∆+

dtrrd kk

⋅=

i j

k

True (Dynamical or Newton) Path

Virtual Path

kr∆ trr kk ∆=∆

⋅

Then: kkk rtd

dtrr

∆−∆=∆

δ

From the Figure we can see that δ variation corresponds to a virtual

displacement in which the time t is

held fixed and the coordinates varied

to the constraints imposed on the system.

mlraN

kk

lk ,,2,10

1

==⋅∑

=

δ

For the Holonomic Constraints: ( ) mltzyxzyxf NNNl ,,2,10,,,,,,, 111 ==

mlrfN

kklk ,,2,10

1

==⋅∇∑

=

δ

15


1.6 Generalized Coordinates

The motion of a mechanical system of N particles is completely defined by n = 3N coordinates . Quite frequently we may find it more

advantageous to express the motion of the system in terms of a different set of coordinates, say . If we take in consideration the m constraints wecan reduce the coordinates to n = 3N-m generalized coordinates.

( ) ( ) ( ) ( )Nktztytx kkk ,,2,1,, =

( ) Tnqqqq ,,, 21 =

( ) ( ) ( ) ( ) ( ) Nkktqzjtqyitqxtqqqrtqr kkknkk ,,2,1,,,,,,,, 21

=++==

Nkkdzjdyidxdtt

rdq

q

rrd kkk

kj

n

j j

kk ,,2,1

1

=++=∂∂

+∂∂

= ∑=

Nkt

rq

q

r

td

rdrv k

j

n

j j

kkkk ,,2,1

1

=

∂∂

+∂∂

=== ∑=

⋅

In the same way

Nkkzjyixtt

rq

q

rr kkk

kj

n

j j

kk ,,2,1

1

=∆+∆+∆=∆∂∂

+∆∂∂

=∆ ∑=

and

Nktt

rtq

q

rt

t

rq

q

rtrrr k

j

n

j j

kkj

n

j j

kkkk ,,2,1

11

==∆

∂∂

−∆∂∂

−∆∂∂

+∆∂∂

=∆−∆= ∑∑==

⋅δ

16


Generalized Coordinates (continue(

( ) Nkqq

rtqq

q

rr

n

jj

j

kjj

n

j j

kk ,,2,1

11

=

∂∂

=∆−∆∂∂

= ∑∑==

δδ

where tqqq jjj ∆−∆=∆

δ

The Generalized Equality Constraints in Generalized Coordinates will be:

mldtt

raadq

q

ra

dtt

raadq

q

radtarda

N

k

klk

lti

n

i i

kN

k

lk

N

k

N

k

klk

lti

n

i i

klk

lt

N

kk

lk

,,2,1011 1

1 111

==

∂∂

⋅++

∂∂

⋅=

=

∂∂

⋅++∂∂

⋅=+⋅

∑∑ ∑

∑ ∑∑∑

== =

= ===

If we define

∑ ∑∑= =

∆

=

∆

∂∂

⋅+=∂∂

⋅=N

k

N

k

klk

lt

lt

n

i i

klk

li t

raac

q

rac

1 11

&

we obtain mldtcdqc lti

n

i

li ,,2,10

1

==+∑=

and the virtual displacements compatible with the constraints are

mlqc i

n

i

li ,,2,10

1

==∑=

δ

17


Generalized Coordinates (continue(

The number of degrees of freedom of the system is n = 3N-m. However, when thesystem is nonholonomic, it is possible to solve the m constraint equations for thecorresponding coordinates so that we are forced to work with a number of coordinates exceeding the degrees of freedom of the system. This is permissibleprovided the surplus number of coordinates matches the number of constraintequations. Although in the case of a holonomic system it may be possible to solvefor the excess coordinates, thus eliminating them, this is not always necessary ordesirable. If surplus coordinates are used, the corresponding constraint equationsmust be retained.

18


1.7 The Stationary Value of a Function and of a Definite Integral

In problems of dynamics is often sufficient to find the stationary value of functionsinstead of the extremum (minimum or maximum(.

Definition: A function is said to have a stationary value at a certain point if the rate of change in

every direction of the point is zero.

Examples:

(1( ( ) niu

fdu

u

fdfuuuf

i

n

ii

in ,,2,100,,,

121 ==

∂∂→=

∂∂=→ ∑

=

By solving those n equations we obtain for which f is stationary

( )nuuu ,,, 21

19


The Stationary Value of a Function and of a Definite Integral(continue(

Examples (continue(:

(2( ( )nuuuf ,,, 21 with the constraints { } marankmldua lk

N

kk

lk ===∑

=

,,2,101

Lagrange’s multipliers solution gives:

01 1

=

+

∂∂= ∑ ∑

= =i

n

i

m

l

lil

i

duau

fdf λ

By choosing the m Lagrange’s multipliers to annihilate the coefficients of them dependent differentials we have

lλidu

equationsmn

mldua

niau

f

n

li

li

m

l

lil

i +

==

==+∂∂

∑

∑

=

=

,,2,10

,,2,10

1

1

λ

20




(3( The functional ( ) ( )∫

=

2

1

,,x

x

dxxd

xydxyxFI

We want to find such that I is stationary, when the end points and are given.

( )xy ( )1xy ( )2xy

( )xy

( ) ( ) ( ) ( )xxyxyxy ηεδ +=+

( )11, yx

( )22 , yx

x

y

The variation of is( )xy

( ) ( ) ( ) ( ) ( ) ( ) ( ) 021 ==+=+= xxxxyxyxyxy ηηηεδ

and

( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫

++=

=

2

1

2

1

,,,,x

x

x

x

dxxd

xd

xd

xydxxyxFdx

xd

xydxyxFI

ηεηεε

( ) ( ) +++====

2

0

2

2

02

10 ε

εε

εεε

εε

dd

Idd

d

IdII

21




Continue: The functional ( ) ( )∫

=

2

1

,,x

x

dxxd

xydxyxFI

The necessary condition for a stationary value is

( ) ( )

( ) ( ) ( )[ ] 00

12

0

2

1

2

1

=−

∂

∂+

∂

∂−∂∂=

∂

∂+∂∂=

∫

∫=

xx

xdyd

Fdxx

xdyd

F

xd

d

y

F

dxxd

xd

xdyd

Fx

y

F

d

Id

x

x

nintegratio

partsby

x

x

ηηη

ηηε ε

Since this must be true for every continuous function we have( )xη

210 xxx

xd

yd

F

xd

d

y

F ≤≤=

∂

∂−∂∂

Euler-Lagrange Differential Equation

By solving this differential equation, ,for which I is stationary is found. ( )xy

JOSEPH-LOUISLAGRANGE

1736-1813

LEONHARD EULER1707-1783

22


1.8 The Principle of Virtual Work

This is a statement of the Static Equation of a mechanical system.

If the system of N particles is in dynamic equilibrium the resultant force on each

particle is zero; i.e.: 0=iR

01

=⋅= ∑=

N

iii rRW

δδ

Because of this, for a virtual displacement the Virtual Work of the system isirδ

If the system is subjected to the constraints:


lyk

lxk

lt

N

kk

lk ===+⋅∑

=

,,,,2,101

Then we denote the external forces on particle i by and the constraint’s forces

by . The resultant force on i is:

iF

iF '

0' =+= iii FFR

23


The Principle of Virtual Work (continue(

We want to find the Virtual Work of the constraint forces.

There are two kind of constraints:

(1( The particle i is constrained to move on a definite surface. We assume that the

motion is without friction and therefore the constraint forces must be

normal to the surface. The virtual variation compatible with the constraint

must be on the surface, therefore .

iF

irδ

0' =⋅ ii rF

δ

irδ

iF '

(2( The particle i is acting on the particle j and the distance between them is l(t(. .

iF '

i j

jF '

ir

jr

( )tl

24



( ) ( ) ( )tlrrrr jiji2=−⋅−

( ) ( ) tllrrrr jiji ∆=∆−∆⋅−

( ) llrrrr jiji =

−⋅−

⋅⋅( )

( ) ( ) ji

rr

jiji

jjiiji

rrrrrr

trrtrrrr

ji

δδδδ =→=−⋅−→

→=

∆−∆−

∆−∆⋅−

≠

⋅⋅

0

0

ji FF ''

−=

If we compute the virtual variation and differential and we multiply the secondequation by and add to the first we obtaint∆−

Because is a real (not a generalized( force we can use Newton’s Third Law: i.e.: iF '

and the virtual work of the constraint forces of this system is:

( ) 0''''' =⋅−+⋅=⋅+⋅= rFrFrFrFW iijjii

δδδδδ

We can generalized this by saying that:

0'1

=⋅∑=

N

iii rF

δ

The work done by the constraint forces in virtual displacements compatible withthe constraints (without dissipation( is zero.

25



From equation we obtain: 0' =+= iii FFR

∑∑∑∑====

⋅=⋅+⋅=⋅=N

iii

N

iii

N

iii

N

iii rFrFrFrR

1

0

111

'0

δδδδ

or

01

=⋅= ∑=

N

iii rFW

δδ

The Principle of Virtual Work

The work done by the applied forces in infinitesimal virtual displacements compatible with the constraints (without dissipation( is zero

26



{ } mjNimaaarank

mjra

jzi

jyi

jxi

N

kk

jk

,,2,1&,,1,,

,,2,101

===

==⋅∑=

δ

We found that the General Equality Constraint the virtual displacement compatible with the constraint must be:

irδ

Let adjoin the m constraint equations by the m Lagrange’s multipliers λ j and add to thevirtual work equation:

01 11 11

=⋅

+=

⋅+⋅= ∑ ∑∑ ∑∑

= == ==

N

ii

m

j

jiji

m

j

N

ii

jij

N

iii raFrarFW

δλδλδδ

There are 3N virtual displacements from which m are dependent of the constraint λj relations and 3N-m are independent. We will choose the m Lagrange’s multipliers

to annihilate the coefficients of the m dependent variables:

−+=

==+ ∑

= iationsmNtindependenNmi

mtheofbecausemiaF

jm

j

jiji

var33,,1

,,2,10

1

λλ

27



From we obtain: 0' =+= iii FFR ∑

==

m

j

jiji aF

1

'

λ

where are chosen such that jλ mjforaFm

j

jiji ,,2,10

1

==+ ∑=

λ

Since , we obtain: k

n

k k

ii q

q

rr δδ ∑

= ∂∂=

1

01 1 11

1 111 1

=

∂∂

+

∂∂

=

=∂∂

⋅

+=⋅

+=

∑ ∑ ∑∑

∑ ∑∑∑ ∑

= = ==

= === =

n

kk

m

j

N

i k

ijij

N

i k

ii

N

i

n

kk

k

im

j

jiji

N

ii

m

j

jiji

qq

ra

q

rF

qq

raFraFW

δλ

δλδλδ

We define:

nkq

rFQ

N

i k

iik ,,2,1

1

=

∂∂

=∑=

∆

nkcq

raQ

m

j

jkj

m

j

N

i k

ijijk ,,2,1'

11 1

==

∂∂

= ∑∑ ∑== =

∆λλ nk

q

rac

N

i k

iji

jk ,,2,1

1

=

∂∂

=∑=

∆

Generalized Forces

Generalized Constraint Forces

28


2. D’Alembert Principle

Newton’s Second Law for a particle of mass and a linear momentumVector can be written as

im

iii vmp =

D’Alembert Principle: 0' =−+⋅

iii pFF

where and are the applied and constraint forces, respectively. iF

iF '

D’Alembert Principle enables us to trait dynamical problems as if they were statical.Let extend the Principle of Virtual Work to dynamic systems:

0'1

=⋅

−+∑

=

⋅N

iiiii rpFF

δ

Assuming that the constraints are without friction the virtual work of the constraintforce is zero . Then we have

Generalized D’Alembert Principle: 01

=⋅

−∑

=

⋅N

iiii rpF

δ

0'1

=⋅∑=

N

iii rF

δ

The Generalized D’Alembert Principle The total Virtual Work performed by the effective forces through infinitesimal

Virtual Displacement, compatible with the system constraints are zero.

0=−⋅

ii pF

is the effective force.

Jean Le Rondd’ Alembert1717-1783

“Traité de Dynamique”

1743

29


3. Hamilton’s Principle

William RowanHamilton1805-1865

Let write the D’Alembert Principle: in integral form01

=⋅

−∑

=

⋅N

iiii rpF

δ

But

( )∑∑∑===

⋅⋅+

⋅−=⋅−

N

iiii

N

iiii

N

iiii r

td

dvmrvm

td

drvm

111

δδδ

Let find ( )irtd

d δ

iiiii rtd

dttvrr

∆−∆=∆−∆=δ are the virtual displacements compatible with the

constraints mjraN

ii

ji ,,2,10

1

==⋅∑

=δ

( )tri

irδ

tvi∆

( )tPi( )tP i'

( )ttP i ∆+'ir∆

Virtual Path True Path (P)Newtonian orDynamic Path

The ConstraintSpace at t

mjraN

ii

ji ,,10

1

==⋅∑

=δ

( ) ( )( ) ( )( ) ( )

=∆=∆=∆=∆==

0

0

0

21

21

21

tttt

trtr

trtr

ii

ii

δδ

1t

2t

02

11

=⋅

−∫∑

=

⋅t

t

N

iiii dtrpF

δ

30


Hamilton’s Principle (continue(Since

td

rdvv iPi i

==

( )( )

( )

ttd

dvr

td

dvt

td

dr

td

dv

ttd

d

rtd

d

td

rd

tdtd

rdrd

ttd

rrdvvv

iiiii

ii

iiiittPii i

∆−∆+≈

∆−

∆+≈

∆+

∆+=

=∆+∆+

=∆+∆+

==∆+ ∆+

11

'

( ) ( ) tartd

dtatvr

td

dt

td

dvr

td

dv iiiiiiii ∆+=∆+∆−∆=∆−∆=∆ δ

Therefore

( ) ( )

ecommutativaretd

dand

rtd

dvv

td

dttavr

td

diiiiii

δ

δδδ

→

→==

∆−∆=∆−∆=

31


Hamilton’s Principle (continue(Now we can develop the expression:

tavmvvmrvmtd

dram

N

iiii

N

iiii

N

iiii

N

iiii ∆⋅−

∆⋅+

⋅−=⋅− ∑∑∑∑

==== 1111

δδ

But the Kinetic Energy T of the system is:

∑=

⋅=N

iiii vvmT

12

1

∑=

∆⋅=∆N

iiii vvmT

1

∑∑∑===

⋅⋅=⋅=⋅=

N

iiii

N

iiii

N

i

iii vFmavmvvmT111

Therefore

Trvmtd

d

tTTrvmtd

dram

N

iiii

N

iiii

N

iiii

δδ

δδ

+

⋅−=

=∆−∆+

⋅−=⋅−

∑

∑∑

=

==

1

11

32


Hamilton’s Principle (continue(From the integral form of D’Alembert Principle we have:

( )

∫ ∑∫ ∑∑

∫ ∑∫ ∑

∫∑

⋅+=

⋅++⋅−=

=

⋅++

⋅−=

=⋅+−=

===

==

=

2

1

2

1

2

1

2

1

2

1

2

1

1110

11

1

0

t

t

N

iii

t

t

N

iii

N

i

t

tiii

t

t

N

iii

t

t

N

iiii

t

t

N

iiiii

dtrFTdtrFTrvm

dtrFTdtrvmtd

d

dtrFam

δδδδδ

δδδ

δ

We obtained

( ) 02

1

2

11

=+=

⋅+ ∫∫ ∑

=

dtWTdtrFTt

t

t

t

N

iii δδδ

Extended Hamilton’s Principle

33


Hamilton’s Principle (continue(If we develop and we can writetTTT ∆−∆= δ tvrr iii ∆−∆= δ

02

1

2

1111

=

∆

⋅+−

∆⋅+∆=

⋅+ ∫ ∑∑∫ ∑

===

dttvFTrFTdtrFTt

t

N

iii

N

iii

t

t

N

iii

δδ

and because ∑=

⋅=N

iii vFT

1

022

11

=

∆−

∆⋅+∆∫ ∑

=

dttTrFTt

t

N

iii

The pair and is arbitrary but compatible with the constraints: ir∆ t∆

mjtara jt

N

ii

ji ,,2,10

1

==∆+∆⋅∑

=

34


Hamilton’s Principle (continue(For a Conservative System VF ii −∇=

VrVrFWN

iii

N

iii δδδδ −=⋅∇−=⋅= ∑∑

== 11

We have ( ) ( ) 02

1

2

1

2

1

==−=+ ∫∫∫t

t

t

t

t

t

dtLdtVTdtWT δδδ

where VTL −=∆

=−∇=−==

∆

∫ NiVFVTLdtL ii

t

t

,,2,1;02

1

δHamilton’s Principle

forConservative Systems

Hamilton’s Principle for Conservative Systems: The actual path of a conservative system in the configuration space renders

the value of the integral stationary with respect to all arbitraryvariations (compatible with the constraints) of the path between the twoinstants and provided that the path variations vanish at those two points.

∫=2

1

t

t

dtLI

1t 2t

35


4. Lagrange’s Equations of Motion

Joseph LouisLagrange1736-1813

“Mecanique Analitique”

1788

The Extended Hamilton’s Principle states: 02

11

=

⋅+∫ ∑

=

dtrFTt

t

N

iii

δδ

where are the virtual displacements compatible with theconstraints:

irδ

mjqcqq

rara

n

kk

ki

n

kk

N

i k

iji

N

ii

ji ,,2,10

11 11

===

∂∂

=⋅ ∑∑ ∑∑== ==

δδδ

T the kinetic energy of the system is given by:

∑ ∑∑∑=

⋅

===

⋅⋅

=

∂∂

+∂∂

⋅

∂∂

+∂∂

=⋅=N

j

n

i

ji

i

jn

i

ji

i

jj

N

j

jjj tqqTt

rq

q

r

t

rq

q

rmrrmT

1 111

,,2

1

2

1

where is the vector of generalized coordinates. ( )nqqqq ,,, 21 =

−∆

∂∂+

∆

∂∂+∆

∂∂+

=

−

∆+∆+∆+=∆

⋅

=

⋅⋅⋅⋅

∑ tqqTtt

Tq

q

Tq

q

TtqqTtqqTttqqqqTT

n

ii

ii

i

,,,,,,,,1

t

Tq

q

Tq

q

TT

n

ii

ii

i ∂∂+

∂∂+

∂∂= ∑

=1

( ) ( ) ∑∑==

∂∂+

∂∂=

∆−∆

∂∂+∆−∆

∂∂=∆−∆=

n

ii

ii

i

n

iii

iii

i

qq

Tq

q

Ttqq

q

Ttqq

q

TtTTT

11

δδδ

36


Lagrange’s Equations of Motion (continue(

( ) ( ) ∑∑==

∂∂+

∂∂=

∆−∆

∂∂+∆−∆

∂∂=∆−∆=

n

ii

ii

i

n

iii

iii

i

qq

Tq

q

Ttqq

q

Ttqq

q

TtTTT

11

δδδ

But because δ and are commutative and: td

d ( )ii qdt

dq δδ =

∑=

∂∂+

∂∂=

n

ii

ii

i

qdt

d

q

Tq

q

TT

1

δδδ

This is an expected result because the variation δ keeps the time t constant.

We found that , therefore∑= ∂

∂=

n

ii

i

jj q

q

rr

1

δδ

∫∑∫∑ ∑∫ ∑ ∑∫ ∑== == ==

=

∂∂

⋅=

∂∂

⋅=

⋅

2

1

2

1

2

1

2

111 11 11

t

t

n

iii

t

t

n

ii

N

j i

jj

t

t

N

j

n

ii

i

jj

t

t

N

jjj dtqQdtq

q

rFdtq

q

rFdtrF δδδδ

where ForcesdGeneralizeniq

rFQ

N

j i

jji ,,2,1

1

=∂∂

⋅=∑=

∆

Now( )

∫∑∑

∫∑∫ ∑

==

==

−

∂∂−

∂∂−

∂∂=

+

∂∂+

∂∂=

⋅+=

2

1

2

1

2

1

2

1

110

.int

11

0

t

t

n

iiii

ii

i

n

i

t

tii

partsby

t

t

n

iiii

ii

i

t

t

N

jjj

dtqQqq

Tq

q

T

td

dq

q

T

dtqQqq

Tq

td

d

q

TdtrFT

δδδδ

δδδδδ

37



02

11

=

−

∂∂−

∂∂

∫∑=

t

t

i

n

ii

ii

dtqQq

T

q

T

td

d δ

where the virtual displacements must be consistent with the constraints . Let adjoin the previous equations by the constraints multiplied

by the Lagrange’s multipliers

iqδmkqc

n

ii

ki ,,2,10

1

==∑=

δ

( )mkk ,,2,1 =λ

01 11 1

=

=

∑ ∑∑ ∑

= == =

n

ii

m

k

kik

m

k

n

ii

kik qcqc δλδλ

to obtain

02

11 1

=

−−

∂∂−

∂∂

∫∑ ∑= =

t

t

i

n

i

m

k

kiki

ii

dtqcQq

T

q

T

td

d δλ

While the virtual displacements are still not independent, we can chose the Lagrangian’s multipliers so as to render the bracketed coefficients of

equal to zero. The remaining being independent can be chosenarbitrarily, which leads to the conclusion that the coefficients ofare zero. It follows

iqδ

iqδ

( )mkk ,,2,1 =λ

( )nmiqi ,,2,1 +=δ

( )miqi ,,2,1 =δ

nicQq

T

q

T

dt

d m

k

kiki

ii

,,2,11

=+=∂∂−

∂∂ ∑

=

λ

38



nicQq

T

q

T

dt

d m

k

kiki

ii

,,2,11

=+=∂∂−

∂∂ ∑

=

λ

We have here n equations with n+m unknowns . To find all theunknowns we must add the m equations defined by the constraints, to obtain

( ) ( ) mn tqtq λλ ,,,,, 11

nicQq

T

q

T

dt

d m

k

kiki

ii

,,2,11

=+=∂∂−

∂∂ ∑

=

λLagrange’s Equations:

mkcqc kt

n

ii

ki ,,2,10

1

==+∑=

Let define

Generalized Constraint Forces: nicQm

k

kiki ,,2,1'

1

== ∑=

λ

39


Lagrange’s Equations of Motion (continue( If the system is acted upon by some forces which are derivable from a potential

Function and some forces which are not, we can write: ( ) ( )nn qqqVrrrV ,,,,,, 2121

−=− n

jF

njjj FVF

+−∇=

( ) ∑∑ ∑∑ ∑∑== == ==

=

∂∂

⋅+∂∂

⋅∇−=

∂∂

⋅+∇−=⋅n

iii

n

ii

N

j i

jnj

i

jj

N

j

n

ii

i

jnjj

N

jjj qQq

q

rF

q

rVq

q

rFVrF

11 11 11

δδδδ

But where∑= ∂

∂⋅∇=

∂∂ N

j i

jj

i q

rV

q

V

1

k

z

Vj

y

Vi

x

VV

jjjj

∂∂+

∂∂+

∂∂=∇

Therefore:

niQq

V

q

rF

q

rVQ in

i

N

j i

jnj

N

j i

jji ,,2,1

11

=+∂∂−=

∂∂

⋅+∂∂

⋅∇−= ∑∑==

Generalized External Forces:

Generalized External Nonconservative Forces:

niq

rFQ

N

j i

jnjin ,,2,1

1

=∂∂

⋅=∑=

∆

40


Lagrange’s Equations of Motion (continue(The Lagrange’s Equations nicQ

q

T

q

T

dt

d m

k

kiki

ii

,,2,11

=+=∂∂−

∂∂ ∑

=

λ

Define: ( ) ( ) ( )qVtqqTtqqL −=

∆,,,,

Because we assume that , we have ( )i

q

qV

i

∀=∂

∂0

Lagrange’s Equations: nicQq

L

q

L

dt

d m

k

kikin

ii

,,2,11

=+=∂∂−

∂∂ ∑

=

λ

mkcqc kt

n

ii

ki ,,2,10

1

==+∑=

We proved

=−∇=−==

∆

∫ NiVFVTLdtL ii

t

t

,,2,1;02

1

δ

Hamilton’s Principle for Conservative Systems

Lagrange’s Equations for a Conservative System without Constraints:

( )0,,,,2,10 =−=−∇===∂∂−

∂∂ k

iiiii

cVTLVFniq

L

q

L

dt

d

If they are no constraints, from the Lagrange’s Equations, or from Euler-Lagrange Equation for a stationary solution of , we obtain: ∫=

2

1

t

t

dtLI

41


5. Hamilton’s Equations

The Lagrange’s Equations nicQq

T

q

T

dt

d m

k

kiki

ii

,,2,11

=+=∂∂−

∂∂ ∑

=

λ

can be rewritten as:

nicQq

T

tq

Tq

qq

Tq

qq

T

q

T

dt

d m

k

kiki

i

n

i ij

jij

jii

,,2,111

222

=++∂∂=

∂∂

∂+∂∂

∂+∂∂

∂=

∂∂ ∑∑

==

λ

therefore consist of a set of n simultaneous second-order differential equations.

They must be solved tacking in consideration the m constraint equations.

mkcqc kt

n

ii

ki ,,2,10

1

==+∑=

A procedure for the replacement of the n second-order differential equations by2n first-order differential equations consists of formulating the problem in terms of2n Hamilton’s Equations.

We define first:General Momentum: ni

q

Tp

ii ,,2,1

=

∂∂=

∆

We want to find the transformation from the set of variables to the set by the Legendre’s Dual Transformation.

( )tqq ,,

( )tpq ,,

42


Hamilton’s Equations (continue(

Legendre’s Dual Transformation.

Adrien-MarieLegendre1752-1833

Let consider a function of n variables , m variables and time t. ix iy

( )tyyxxF mn ,,,,,, 11 and introduce a new set of variables ui defined by the transformation:

nix

Fu

ii ,,2,1 =

∂∂=

∆

We can see that:

∂∂∂

∂∂∂

∂∂∂

∂∂∂

+

∂∂

∂∂∂

∂∂∂

∂∂

=

mmnn

m

nnn

n

n dy

dy

dy

yx

F

yx

F

yx

F

yx

F

dx

dx

dx

x

F

xx

F

xx

F

x

F

du

du

du

2

1

2

1

2

1

2

11

2

2

1

2

2

1

2

1

2

21

2

2

1

We want to replace the variables by the new variables .We can see that the new n variables are independent if the Hessian Matrix

is nonsingular.

( )nidxi ,,2,1 = ni

njji xx

F,,1

,,1

2

=

=

∂∂∂

43



Legendre’s Dual Transformation (continue(

Let define a new function G of the variables , and t. iu iy

( )tyyuuGFxuG mn

n

iii ,,,,,, 11

1

=−=∑=

∆

Then:( )

dtt

Fdy

y

Fdx

x

Fudux

dtt

Fdy

y

Fdx

x

FdxuduxdG

m

jj

j

n

ii

iiii

n

i

m

jj

ji

i

n

iiiii

∂∂−

∂∂−

∂∂−+=

=∂∂−

∂∂−

∂∂−+=

∑∑

∑ ∑∑

==

= ==

11

0

1 11

But because: ( )tyyuuGG mn ,,,,,, 11 =

dtt

Gdy

y

Gdu

u

GdG

n

i

m

jj

ji

i ∂∂+

∂∂+

∂∂= ∑ ∑

= =1 1

Because all the variations are independent we have:

t

F

t

Gmj

y

F

y

Gni

u

Gx

jjii ∂

∂−=∂∂=

∂∂−=

∂∂=

∂∂= ;,,1;,,1

44



Legendre’s Dual Transformation (continue(

Now we can define the Dual Legendre’s Transformation from

( )tyyxxF mn ,,,,,, 11 ( ) FxutyyuuGn

iiimn −= ∑

=111 ,,,,,, to

by using

nix

Fu

ii ,,2,1 =

∂∂=

niu

Gx

ii ,,2,1 =

∂∂=

End of Legendre’s Dual Transformation

45



Following the same pattern to find the transformation from the set of variables to the set , we introduce the Hamiltonian: ( )tqq ,,

( )tpq ,,

( )tqqTqpHn

iii ,,

1

−=∑

=

∆

whereni

q

Tp

ii ,,2,1

=

∂∂=

Then ( )tpqHH ,,=

dtt

Hdp

p

Hdq

q

Hdt

t

Tdq

q

Tdpq

dtt

Tdq

q

Tqd

q

TqdpdpqdH

n

ii

ii

i

n

ii

iii

n

ii

ii

iiiii

∂∂+

∂∂+

∂∂=

∂∂−

∂∂−=

=∂∂−

∂∂−

∂∂−+=

∑∑

∑

==

=

11

1

and

01

=

∂∂+

∂∂+

−

∂∂+

∂∂+

∂∂∑

=

dtt

T

t

Hdpq

p

Hdq

q

T

q

Hn

iii

ii

ii

46



If the Hessian Matrix is nonsingular, all the are independent, but not the that must be consistent with the constraints:

ni

njji qq

T,,1

,,1

2

=

=

∂∂∂

( )nidpi ,,2,1 =( )nidqi ,,2,1 =

mjdtcdqc jt

n

ii

ji ,,2,10

1

==+∑=

Let adjoin the previous equations by the constraint equations multiplied by the m

Lagrange’s multipliers :j'λ0'''

11 11 1

=+

=

+ ∑∑ ∑∑ ∑

== == =

m

j

jij

n

ii

m

j

jij

m

j

jt

n

ii

jij dtcdqcdtcdqc λλλ

We have

0''11 1

=

+

∂∂+

∂∂+

−

∂∂+

+

∂∂+

∂∂ ∑∑ ∑

== =

dtct

T

t

Hdpq

p

Hdqc

q

T

q

H m

j

jtj

n

iii

ii

m

j

jij

ii

λλ

47


Hamilton’s Equations (continue(By proper choosing the m Lagrange’s multipliers ,the remainder differentials and dt are independent and therefore we have:

j'λii dpdq ,

ni

ct

H

t

T

cq

H

q

T

p

Hq

m

j

jtj

m

j

jij

ii

ii

,,2,1

'

'

1

1

=

−∂

∂−=∂∂

−∂∂−=

∂∂

∂∂=

∑

∑

=

=

λ

λ Legendre’s Dual Transformation

By differentiating the General Momentum Equation and using Lagrange’s Equations we obtain:

( )∑∑==

−++∂∂−=++

∂∂=

∂∂=

m

j

jijji

i

m

j

jiji

iii cQ

q

HcQ

q

T

q

T

dt

dp

11

''''' λλλ

ni

cQq

Hp

p

Hq

m

j

jiji

ii

ii

,,2,1

1

=

++∂∂−=

∂∂=

∑=

λ

mkcqc kt

n

ii

ki ,,2,10

1

==+∑=

Extended Hamilton’s Equations

Constrained Differential Equations

48



For Holonomic Constraints (constraints of the form (we can (theoretically( reduce the number of generalized coordinates to n-m and wecan assume that and n represents the number of degrees of freedom of the system (this reduction is not possible for Nonholonomic Constraints(. Then:

( ) mjtqqf nj .,10,,,1 ==

0== jt

ji cc

ni

Qq

Hp

p

Hq

ii

i

ii

,,2,1

=

+∂∂−=

∂∂=

Extended Hamilton’s Equations for

Holonomic Constraints

niq

VQ

ii ,,2,1 =

∂∂−=

( ) ( ) ( )qVtqqTtqqL −=

∆,,,,

niq

Tp

ii ,,2,1

=

∂∂=

Extended Hamilton’s Equations for


and a

Conservative System

Conservative

System

49



Define:

Hamiltonian for

Conservative Systems( ) ( ) ( ) ( )qVtqqTtqqLqptqqH

n

iii

−=−=∑

=

∆,,,,,,

1

Hamilton’s Canonical Equations for

Conservative Systems

with


ni

q

Hp

p

Hq

ii

ii

,,2,1

=

∂∂−=

∂∂=

We have:

50


6. Kane’s Equations

In terms of generalized coordinates we can write:

Thomas R. Kane1924 -

Stanford University

( ) ( ) ( ) ( ) ( ) Niktqzjtqyitqxtqqrtqr iiinii .,1,,,,,,, 1

=++==

( ) Nikdzjdyidxdtt

rdq

q

rtqrd iii

in

jj

j

ii .,1,

1

=++=∂∂

+∂∂

= ∑=

( )Ni

t

rq

q

r

td

tqrdrv i

n

jj

j

iiii .,1

,

1

=∂∂

+∂∂

=== ∑=

→

6.1

6.2

6.3

Kane and Levinson have shown that with the n generalized coordinates , is usefulto define another n variables , which are linear functions of the n :

jq

iu jq

nrZqYu r

n

jjrjr .,1

1

=+=∑=

∆

6.4

where the matrix is invertible and[ ] { } nj

nrrjYY ,1

,1

==

= [ ] [ ] { } nj

nrrjWWY ,1

,1

1 ==

− ==

njXuWq j

n

rrrjj .,1

1

=+= ∑=

6.5

jrrjrj XandZWY ,, are functions of tandq

are called Generalized Speed (also Nonholonomic Velocities, Quasivelocities.etc.( and are not unique.iu

51


Kane’s Equations (continue(

Nonholonomic constraints are linear relations among either or the ; for mnonholonomic constraints:

6.6

6.7

where k may be n-m or n, depending on whether the nonholonomic constraints areincorporated.

iu jq

nmnsBuAu s

mn

rrsrs .,1

1

+−=+= ∑−

=

∆

If we substitute equations (6.6( in (6.5( we obtain a more general expression for :jq

njXuWq j

k

rrrjj .,1

1

=+= ∑=

Let substitute equation (6.6( in (6.3(:

( )

Nit

rX

q

ruW

q

r

t

rXuW

q

r

td

tqrdrv

ik

r

n

jj

j

ir

n

jrj

j

i

in

jj

k

rrrj

j

iiii

.,1

,

1 11

1 1

=∂∂+

∂∂+

∂∂=

=∂∂+

+

∂∂===

∑ ∑∑

∑ ∑

= ==

= =

→

From this equation we can see that ∑= ∂

∂=

∂∂ n

jrj

j

i

r

i Wq

r

u

v

1

52



6.8

6.9

Let use now the equation of differential work:

By defining we obtain:t

rX

q

rv i

n

jj

j

iit ∂

∂+

∂∂

=∑=

∆

1

Nivuu

vv t

i

k

rr

r

ii .,1

1

=+

∂∂

= ∑=

∑∑==

⋅=⋅=N

iiii

N

iii rdamrdFdW

11

Equation (6.9( is now rewritten using (6.8(. On the left side we obtain:

dtvuu

vFdtvFrdF

N

i

ti

k

rr

r

ii

N

iii

N

iii

+

∂∂⋅=⋅=⋅ ∑ ∑∑∑

= === 1 111

6.10

Similarly, the right side of (6.9( becomes:

dtvuu

vamdtvamrdam

N

i

ti

k

rr

r

iii

N

iiii

N

iiii

+

∂∂

⋅=⋅=⋅ ∑ ∑∑∑= === 1 111

6.11

Equations (6.10( and (6.11( are equated and terms re collected:

( ) ( ) 011 11

=

−⋅+

−⋅

∂∂+⋅

∂∂ ∑∑ ∑∑

== ==dtamFvdtuam

u

vF

u

v N

iiii

ti

k

rr

N

iii

r

iN

ii

r

i

6.12

53



( ) ( ) 011 11

=

−⋅+

−⋅

∂∂+⋅

∂∂ ∑∑ ∑∑

== ==dtamFvdtuam

u

vF

u

v N

iiii

ti

k

rr

N

iii

r

iN

ii

r

i

6.12

The and dt are nonzero and independent and so the coefficients of each of

them must be zero. Also using Newton’s Second Law: we have:

krur ,1=

0=− iii amF

nrZqYu r

n

jjrjr ,.,1

1

=+=∑=

∆

krFu

vQ

N

ii

r

ir ,,1

1

=⋅∂∂

=∑=

∆

( ) kramu

vQ

N

iii

r

ir ,,1'

1

=−⋅∂∂

=∑=

∆

krQQ rr ,,10' ==+

6.4

6.13

6.14

6.15

Generalized Speeds

Generalized Active Forces

Generalized Inertia Forces

Kane’s Equations

54


Gibbs-Appell Equations

7.1

Josiah Willard Gibbs

1839 - 1903

Paul Emile Appell

1855 - 1930

Differentiation of equation (6.8( gives: Nivuu

vv t

i

k

rr

r

ii .,1

1

=+

∂∂

= ∑=

About 100 years after Lagrange, Gibbs in 1879 and Appell in 1899;

independently devise what is known the Gibbs-Appell Equations.

Nivdt

du

u

v

dt

du

u

vva t

i

k

rr

r

ik

rr

r

iii .,1

11

=+

∂∂

+∂∂

== ∑∑==

From this equation we see that:

r

i

r

i

u

v

u

a

∂∂

=∂∂

7.2

If we substitute equation (7.2( in (6.14(

( ) ( ) krGu

aamu

amu

aam

u

vQ

r

N

iiii

r

N

iii

r

iN

iii

r

ir ,,1

2

1'

111

=∂∂−=⋅

∂∂−=−⋅

∂∂=−⋅

∂∂= ∑∑∑

===

∆

7.3

( ) kramu

vQ

N

iii

r

ir ,,1'

1

=−⋅∂∂

=∑=

∆

6.14

55


Gibbs-Appell Equations (continue(

( ) ( ) krGu

aamu

amu

aam

u

vQ

r

N

iiii

r

N

iii

r

iN

iii

r

ir ,,1

2

1'

111

=∂∂−=⋅

∂∂−=−⋅

∂∂=−⋅

∂∂= ∑∑∑

===

∆

7.3

From

and krQQ rr ,,10' ==+6.15

krQGu rr

,,1

==∂∂

∑=

∆⋅=

N

iiii aamG

1 2

1

nrZqYu r

n

jjrjr ,.,1

1

=+=∑=

∆

krFu

vQ

N

ii

r

ir ,,1

1

=⋅∂∂

=∑=

∆

Gibbs-Appell Equations

Gibbs Function:

Generalized Speed:

Generalized Active Forces:

56


References:

]1[ Goldstein, H., Classical Mechanics, 2nd ed., Addison-Wesley, 1981

]2[ Meirovitch, L., Methods of Analytical Dynamics, Mc Graw-Hill, 1970

]3[ Greenwood, D.T., Principle of Dynamics, 2nd ed., Prentice-Hall, 1977

]4[ Kane, T.R., Dynamics, 3th ed., Stanford University, 1972

]5[ Desloge, E.A., Relationship Between Kane’s Equations and the Gibbs-Appell Equations, J. Guidance, Vol. 10, No. 1, Jan.-Feb., 1987

August 12, 2015 57

SOLO

TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974– 2013

Stanford University1983 – 1986 PhD AA

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analytic dynamics

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