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1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J. Phys. 62 (9), 1994)

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Page 1: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

1

Lecture 1: Lagrange equations for N particles

Geometrical derivation

of Lagrangian Mechanics

(James Casey. Am. J. Phys. 62 (9), 1994)

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Newton’s second law in configuration space

1,2,3; 1,2,i n N

1( ) ,F n 2( ) ,F n3( )F n

( ) ( ) ( )i iF n M n x n

1 2 3

1 2 3

1 2 3

(1), (1), (1),

(2), (2), (2),

.............................

( ), ( ), ( )

x x x

x x x

x N x N x N

1 2 3 3 2 3 1 3, , , , , ,N N NP x x x x x x

1x

2x

3x

Physical Space (3-D) Cartesian Configuration Space (3N-D)

N Particles

P

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4

(1), (1), (1), , ( ), ( ), ( )M M M M N M N M N

1 2 3 3 2 3 1 3, , , , ,N N Nm m m m m m Cartesian mass components

1 2 3 1 2 3(1), (1), (1), , ( ), ( ), ( )F F F F N F N F N

1 2 3 3 2 3 1 3, , , , ,N N Nf f f f f f Cartesian force components

Components of Newton’s equation in configuration space

1, 2, , 3

,k

k k

k N

f m x

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5

Correspondence:

Physical space Configuration space

3 3

3 3

3 2 3 1 3

( 1, 2,3; 1, , )( ) ,

( ) ,

( ) ,

n i

i

i n i

n n n

i n Nx n x

F n f

M n m m m

2 2 2

1 2 3

1

1( ) ( ) ( ) ( )

2

N

i

T M i x i x i x i

Kinetic energy:

32

1

1( )

2

Nk

k

k

m x

Page 6: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

6

Configuration vectorial space

• New Cartesian coordinates of P: ,k k kmx x

m

where

1

( ),N

i

m M i

• The square of the distance from the origin to the position occupied by P is

defined as:

3 32 2 2

1 1

1( ) ( ) ,

N Nk k

OP k

k k

d x m xm

• We may represent P by its position vector in configuration space, and define

an inner product We choose a fixed orthonormal basis ,

and express as:

r2 .OPr r d 1 2 3, , , Ne e e

r

3

1

,N

k

k

k

r x e

1

2

3

1, 0, 0,0, ,0 ,

0,1, 0,0, ,0 ,

,

0, 0, 0,0, ,1 ,N

e

e

e

( 1, 2, , 3 ; 1, 2, , 3 )

1, ;

0, ;j k jk

j N k N

j ke e

j k

Page 7: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

7

• Besides the orthonormal basis it is useful to have a fixed pair of

reciprocal basis and , defined by ke ke

/ ,k k ke e m m / ,k

k ke e m m

( 1, 2, , 3 ; 1, 2, , 3 ),k k

j j j N k Ne e

3

1

Nk

k

k

r x e

3

1

,N

k

k

k

drv x e

dt

• Next, we construct a force vector in configuration space by setting f

3

1

,N

k

k

k

f f e

• We may write the position vector , and the velocity of P as: ( )r ( )v

3

1

,N

k

k

k

x e

Page 8: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

• The vectorial expression of Newton’s second law in configuration space is

• We observe that :

3 3

1 1

1 1,

2 2

N Nk j

k j

k j

mv v m x x e e T

,dv

f mdt

1 1 1( )2 2 2

dT d dv dv dvmv v m v mv m v f v

dt dt dt dt dt

(square of length of a line element ) 2 2 22,ds dr dr v v dt Tdt

m

,

1,2, ,3 ,

k k

dvf e m e

dt

k N

( ) ,

1,2, ,3 ,

kk

k k k k

dxf m e e m x

dt

k N

/km m

1, 2, , 3

,k

k k

k N

f m x

Page 9: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

Configuration manifold (M ) and its geometry

( 1, 2, , 3 ),( , ) ,j

j j Nx q t

Suppose we introduce a system of curvilinear coordinates,

called generalized coordinates, 1, 2, , 3 ,N ,q

1 2 3, , ,( , ), ,

Nq q q qr r q t

, 1, 2, , 3 ,r

a Nq

These 3N vector define an 3N-dimensional tangent space to M. denoted by

TPM. . Any vector ` TPM. can be resolved on the basis as w ,a

.w w a

P

Tangent to the

coordinates lines

• The velocity of P can be expressed as ( , )

,dr r q t

v q adt t

• We denote as generalized velocity. Notice that TPM.

q q a

M

,a

M

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10

2

21 12 1 02 2

( , ),

r q tT mv m q a T T T

t

3 3

2

1 1

1( ) 0,

2

N N

T m a a q q

• The kinetic energy of P can also be expressed as:

3

1

1

( ) ,N r

T m a qt

2

0

1.

2

rT m

t

3 3

2

1 1

( ) 0,N N

t cteds dr dr a a dq dq

• M is a manifold which moves through configuration space and whose

geometry is called Riemannian.

where

2220

Tdt

m

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11

Lagrange equations

,mv a f a

• Now, using the identities :

• Take a scalar product of both sides of Newton’s equation with each of the basis

vectors in the tangent space a

• Let (generalized force components) ,Q f a

; ,r v da v

aq q dt q

( ) , 1, ,3 .d T T

Q Ndt q q

Page 12: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

Geometrical constraints (holonomic)

• Suppose the system of N particles is subject to M

geometrical constraints (holonomic) of type:

( 1, 2, , 3 )( , ) 0 ,j j M Nr t

• Each of these constraints defines a hypersurface of

dimension 3N-1 . The intersection of the M hypersurfaces is

a subset M of dimension n=3N-M . The particle P must

remains on M and n (degrees of freedom) is the minimum

number of coordinates to locate P at time t on M . We may

describe M by n Gaussian coordinates called

generalized coordinates:

q

1 2, , , .( , ),

nq q q qr r q t

These n vectors define an n-dimensional tangent space to M denoted by

TPM. . Any vector TPM. can be resolved on the basis as

, 1, 2, ,r

a nq

w ,a .w w a

P

Subset M of

dimension n

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13

• The velocity of P can be expressed as

( , ),

r q tv q a

t

• We denote as generalized velocity. Notice that TPM. . q

2 1 0,T T T T

2

1 1

1( ) 0,

2

n n

T m a a q q

aq a

• The kinetic energy of P can also be expressed as:

1

1

( ) ,n r

T m a qt

2

0

1.

2

rT m

t

2 22

1 1

2( ) 0,

n n Tds a a dq dq dt

m

• M is a manifold which moves through configuration space and whose geometry

is called Riemannian.

where

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14

Lagrange equations

• Take a scalar product of both sides of Newton’s equation with each of the basis

vectors in the tangent space

,f a mv a

• Let (generalized force components).

• Now, using the identities :

a

,Q f a

;r v

aq q

,da v

dt q

( ) , 1, , .d T T

Q ndt q q

Page 15: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

15

Ideal constraints (ideal holonomic)

• The total force on the particle P in configuration space can be

written as

• We say a constraint is ideal if the corresponding constraint

force is orthogonal to the constraint :

where

* ,CHf f f

1, , ; 1, , 30, ( 0, )CH

j j M n N Mf a a

Given forces

Constraints forces 1

,M

CH

j

j

f

1, , ,,CH

j j j Mf 3

1

.N

j j k

j kk

er x

1

,M

CH

j j

j

f

P

CHf

Subset M of

dimension n

Thus and as a consequence

and j Subset M of dimension n

Page 16: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

16

Lagrange equations with ideal holonomic

constraints

• The ideal constraints forces do not appear in Lagrange equations

* CHf a f a f a * ,Q

*( ) , 1, , .d T T

Q ndt q q

Page 17: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

17

Lagrange equations with ideal non-holonomic constraints

* ,CH CNf f f f

CN

r r

r

f a B

1

( , ) ( , ) 0, 1, , ;n

r rB q t q B q t r n

,CN CN

r r r

r r

f f b

1

, ;n

r rb B a a a a

TPM.

*

1

( ) , 1, , ,

( , ) ( , ) 0, 1, , ;

r r

r

n

r r

d T TQ B n

dt q q

B q t q B q t r n

• In addition, let suppose we have ideal non-holonomic constrains of type:

Page 18: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

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Lagrange function:

• In addition, let suppose that the given forces can be derived from the

generalized potential as

* ,U d U

fr dt v

( , , )L q q t

( , , )U r v t

• Let . Then, 1( , , , )nr r q q t

* * ,a

U d UQ f a

q dt q

• We define the Lagrange function as .

Then, the Lagrange equations can be finally written as:

( , , ) ( , , ) ( , , )L q q t T q q t U q q t

3 3

1 1

, ,N N

k k

k kj j

U U U Ue e

r x v x

1

( ) , 1, , ,

( , ) ( , ) 0, 1, , ;

r r

r

n

r r

d L LB n

dt q q

B q t q B q t r n

Page 19: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

Definición de sistema lagrangiano:

U sistema de partículas sin ligaduras (o con ligaduras holónomas

ideales) bajo la acción de unas fuerzas que derivan de un potencial

generalizado se dice que es un sistema lagrangiano.

Un sistema físico cuya evolución en el tiempo ( ) se

determina a partir de las ecuaciones

para una cierta función , se dice que es un sistema

lagrangiano.

0, 1,2,3,j j

d L Lj

dt q q

( ), 1,2,3,jq t j

( , , )L q q t

Page 20: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

20

Introducción a las leyes de conservación o

constantes del movimiento.

Definición:

Las ecuaciones de Euler-Lagrange constituyen un sistema de

ecuaciones diferenciales ordinarias (EDO) de segundo orden en las

coordenadas generalizadas . Cualquier que

permanece constante durante el movimiento del sistema se llama

constante del movimiento o integral primera :

1 2, ,q q q ( , , )q q t

donde son las trayectorias o soluciones de las ecuaciones del

movimiento.

( )q t

Si es una integral primera deberá verificar : ( , , )q q t

( )

0,j j

j j q q t

dq q

dt t q q

Page 21: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

21

Casos “triviales” de leyes de conservación Energía

1)

Supongamos un sistema con la lagrangiana, , independiente del

tiempo, y sometido a dos, una o ninguna ligadura no holónoma ideal sin

término independiente, es decir . Bajo estas

condiciones la función

( , ) ,i

i i

LE q q q L

q

( , )L q q

es una constante del movimiento o integral primera.

( , ) 0ri i

i

B q t q

Demostración:

El sistema de ecuaciones que determina el movimiento es

,r ri

ri i

d L LB

dt q q

( , ) 0,ri i

i

B q t q y derivando E se obtiene

i i i i

i i i ii i i i

dE L d L L L Lq q q q

dt q dt q q q t

0,i r ri i

i r ii i

d L Lq B q

dt q q

Page 22: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

22

Caso particular de 1)

Demostración:

( )jk j

jk k

L Tc q q

q q

• Si , con (función solo de las q) y una

función cuadrática homogénea de las velocidades generalizadas, es

decir, , entonces .

L T U ( )U U q ( , )T q q

12

,

( )ij i j

i j

T c q q q ( , )E q q T U

( , ) ( ) 2 ,i ij i j

i i ji

LE q q q T U c q q q T U T T U T U

q

Page 23: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

23

Demostración:

a)

i i

d d

dt q q dt

b) Es consecuencia directa de a) y 1)

i i

d d d

dt q dt q dt

j j

j ji j i j

dq q

dt q t q q t q

2 2

j j

j ji j i i j

dq q

dt q q q t q q

2 2

j

ji i i j

dq

dt q q t q q

2 2 2 2

0, ( ).j j i

j ji j i i i j

q q q tq t q q q t q q

2)

Supongamos un sistema lagrangiano con lagrangiana , tal que

admite la descomposición

a) Las soluciones de las ecuaciones de Lagrange con lagrangiana

son idénticas a las soluciones con lagrangiana .

b) Si no depende de t explícitamente, , el sistema tiene la

ley de conservación: .

( , , )L q q t

( , , ) ( , , ) ( , ).d

L q q t L q q t q tdt

LL

( , ) i

i i

LE q q q L

q

L ( , )L q q

Page 24: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

Definición: En un sistema lagrangiano se denomina coordenada

cíclica a aquella coordenada generalizada que no aparece

explícitamente en la expresión de la lagrangiana.

Definición de momento canónico conjugado a una

coordenada .

Supongamos un sistema lagrangiano con lagrangiana . El momento

canónico, , conjugado a la coordenada es la función de ,

definida por:

, ,q q t

( , , ) .j

j

Lp q q t

q

jqjpL

jq

Ley de conservación: El momento canónico conjugado a una

coordenada cíclica es una constante del movimiento.

Demostración: sea la coordenada cíclica, De la

ecuación de Lagrange correspondiente a esa coordenada se tiene:

kq 0.k

L

q

0, ( , , ) .k

k k

d L Lp q q t const

dt q q

Page 25: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

Ejemplo 1 (ejercicio nº2 de apuntes).

Dos partículas de masas M1 y M2 están unidas a través de un hilo ideal que

pasa por un agujero taladrado en el plano horizontal (sin rozamiento) de la figura.

Determinar: Variedad de configuración, fuerzas generalizadas, ecuaciones de

Lagrange, etc.

x

y

z

2

1

g

Page 26: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

x

y

z

2

1

g

hT

Partícula 1: Partícula 2: ( , )x y z

Espacio de config. Cartesiano:

1 2 3, , , ,x x x x y z

1 2 3 1 1 2 1 2, , , , ; ;m m m M M M m M M

Espacio vect. de config: 1 2 31,0,0 , 0,1,0 , 0,0,1 ,e e e

1 1 2

1 2 1 2 1 21 1 2 2 3 3; ; ;M M M

M M M M M Me e e e e e

1 2 1 2 1 2

1 1 2

1 2 3

1 2 3; ; ;M M M M M M

M M Me e e e e e

2M g

1 2 3;r xe ye ze

hT

3 2 312 1 2

2 2 2 2( ) ;CH

h h h

x yf M ge f T e T e T M g e

x y x y

Ecuación de ligadura: 2 2 2 2

1 0,x y z x y z

2 2 2 311

2 2 2 2,

x ye e x y e

x y x y

2 311

2 2 2 2;CH h

h

h

x y Tf T e e e

Tx y x y

Ligadura ideal:

1 1 1;CHf ;h hT T

Page 27: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

Coordenadas generalizadas: (no usamos la ligadura en la parametrización de la

variedad de configuración)

( , , )q x y z

x

y

z

2

1

g

2 2 2 21 1 11 22 2 2( )T mv M x y M z

1 1 2 22 2 2 2

3 3 2

; ;

;

h h

h

x yQ f e T Q f e T

x y x y

Q f e T M g

1

2

3

,

,

,

d T TQ

dt x x

d T TQ

dt y y

d T TQ

dt z z

2M g

hT

hT

2 2

1 0,x y z

1 2 3;dr

v xe ye zedt

12 2

12 2

2 2

,

,

,

h

h

h

xM x T

x y

yM y T

x y

M z T M g

1 1 2 2 3 3, , ,r

a e a e a ex

Page 28: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

Coordenadas generalizadas: (usamos la ligadura en la parametrización de la

variedad de configuración!!!!)

( , )q

x

y

z

2

1

g

2 312 1 2

1 2 3 2

cos sin ( ) ( sin cos ) 0,

(cos sin ) ,

h h hQ f a T e T e T M g e e e

Q f a f e e e M g

,

,

d T TQ

dt

d T TQ

dt

2M g

hT

hT

,dr

v a adt

1 2 3( , ) (cos sin ) ( ) ;r e e e

cos , sin , ,x y z

1 2 3 1 2cos sin , ( sin cos ),r r

a e e e a e e

2 2 2 2 2 2 2 21 1 1 1 11 2 1 2 22 2 2 2 2( ) ( ) ,T mv M M M M M

2

2

2

1 2 2 2

( )0,

( ) ,

dM

dt

M M M M g

Page 29: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

Obtened la lagrangiana usando

y 2 leyes de conservación.

( , )q

x

y

z

2

1

g

2M g

hT

hT

0, ( , ) .L L L

E q q L T U constt

2 2 21 11 2 2 22 2

( ) ( )L T U M M M M g

2 2 ( ),U M g z M g

2

20, .L L

p M const

Page 30: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

30

Determinación de fuerzas de ligadura holónomas ideales

* ,CH CNf f f f

1

,CN

r r

r

f b

• Supongamos que el sistema (N partículas: espacio de configuración 3N) posee M

ligaduras holónomas y ligaduras no holónomas (ambas ideales):

( , ) 0,

1,2, , .

j r t

j M

3

1

( , ) ( , ) 0,

1, , 3 .

N

r rA r t x A r t

r n N M

1

,M

CH

j j

j

f

Page 31: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

1

( ) ( ) , 1, , 3 1 ,

( , ) ( , ) 0, 1, , ;

( , ) 0,

M M r r

r

n

r r

M

d T TQ a B n N M

dt q q

B q t q B q t r n

q t

31

Determinación de fuerzas de ligadura holónomas ideales

1 2, , , ; 3 1 .nq q q q n N M

• Las ecuaciones de Lagrange se pueden plantear con coordenadas generalizadas

sobre una variedad de configuración de dimension con ,

dependiendo del nº de ligaduras holónomas que tomemos en la parametrización. n 3 3N M n N

• Las fuerzas de ligadura holónomas (los ) no apareceran si se escoge la

variedad de configuración de dimensión mínima posible: . 3n n N M

• Las fuerzas de ligadura no holónomas (los ) siempre apareceran cualquiera

que sea la dimensión de la variedad de configuración. r

j

• Supongamos que queremos determinar la fuerza de ligadura hónoma ejercida por

la ligadura, por ejemplo, : j M CH

M M Mf

( , ) 0,

1,2, , 1.

j r t

j M

( , ), , 1,2, ,r q t a n

Page 32: Lecture 1: Lagrange equations for N particles Geometrical ... · 1 Lecture 1: Lagrange equations for N particles Geometrical derivation of Lagrangian Mechanics (James Casey. Am. J

32

Potencial de fuezas giroscópicas

0Q q

• Def: Se denominan fuerzas giroscópicas a aquellas cuya potencia es nula:

1 2, ,q q q

( )d d

Q q q qq dt q q dt

; ; ;q q qq q q q

• Ejemplo: El potencial generalizado es giroscópico. ( , ) ( )U q q q q

Q q

0.q q