Formules de Lagrange

Jean-Paul Molina d'aprs TD ECP 1

Soit un point P dont la position est connue dans un repre R par { qi } paramtres indpendants. i [ 1..n ] On a

i i i iR R

OP d OP d OP d OP(P/R). V(P/R) . V(P/R). V(P/R).

q dt q dt q dt q

= =

uuur uuur uuur uuuruuuuuuuuur uuuuuuuuuur uuuuuuuuuur uuuuuuuuuur

Or , on peut crire 'ii

OPV(P/R) q

q

=

uuuruuuuuuuuuur avec ' ii

dqq

dt= , en utilisant la convention

d'Einstein.

Alors, on en dduit immdiatement que 'i i

V(P/R) OPq q

=

uuuuuuuuuur uuur

De mme ' 'j ji j i i j iR

d OP OP OP V(P/R)q q

dt q q q q q q

= = =

uuur uuur uuur uuuuuuuuuur

Ainsi

'i i iR

OP d V(P/R) V(P/R)(P/R). V(P/R). V(P/R).

q dt q q

=

uuur uuuuuuuuuur uuuuuuuuuuruuuuuuuuur uuuuuuuuuur uuuuuuuuuur

Ce qui donne finalement

2 2

'i i i

V(P/R) V(P/R)OP 1 d(P/R).

q 2 dt q q

=

uuuuuuuuuur uuuuuuuuuuruuuruuuuuuuuur

ou

2 2

' 'i i i

V(P/R) V(P/R)V(P/R) 1 d(P/R).

q 2 dt q q

=

uuuuuuuuuur uuuuuuuuuuruuuuuuuuuuruuuuuuuuur

Formules de Lagrange

Jean-Paul Molina d'aprs TD ECP 2

Exemple : Soit un point M se dplaant sur une sphre S de centre O.

On associe cette sphre le repre 0 0 0 0R (i ,j ,k )uuruuruur

M est connu par sa longitude et sa latitude , donc 2 paramtres.

Soit un repre R(M,i,j,k)rurur

tel que OM Ri=uuuur r

et jr tangent au mridien.

2

0V(M/R ) R( 'cos ') = + uuuuuuuuuuuur

v j 0 0V(M/R ) V(M/R )

Rcos R' '

= =

uuuuuuuuuuuur uuuuuuuuuuuurr r

[ ] v00 0V(M/R ) d

( M / R ). Rcos . ( M / R ) R 'cos' dt

= =

uuuuuuuuuuuuruuuuuuuuuuur r uuuuuuuuuuur

[ ] j00 0V(M/R ) d

( M / R ). R . ( M / R ) R ' R 'cos sin' dt

= = +

uuuuuuuuuuuuruuuuuuuuuuur r uuuuuuuuuuur

On en dduit

v

j0

0

. (M / R ) R( "cos 2 ' 'sin )

. (M / R ) R( " 'sin cos )

=

= +

r uuuuuuuuuuurr uuuuuuuuuuur

2 paramtres donc 2 rsultats seulement. On va donc calculer la 3me composante de faon classique, sachant que :

ii i i0 0 0 0

R 0 R0

d d d. (M/R ) . V(M/R ) .V(M/R ) V(M/R ).

dt dt dt

= =

rruuuuuuuuuuur r uuuuuuuuuuuur r uuuuuuuuuuuur uuuuuuuuuuuur

ik j

R 0

d'cos '

dt

= +

r r r i 0. (M/R ) R( ' 'cos ) = +

r uuuuuuuuuuur

0

R 0

0

diV(M/R ) R

dt

R( 'k 'k) i R( 'cos v 'j)

=

= + = +

ruuuuuuuuuuuur

uur ur r r r