Chapitre 7 :Le principe fondamental de la dynamique
Transcription
Chapitre 7 :Le principe fondamental de la dynamique
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" # % $ & ) $ (=) " ' (* = $ = + " , - 0 $∀ * + ./ $ ' 1 2 , % " 1 $ * + , $ $ # 1 • 3 ' = $ 3$ ) • , " $ % (=) ) (=) ($ $ & $ ( (=) ) " ($ * $ $ & 4 , * # # 3 $ % 1 *5 $ " + % • , , ' + $ $ $ * $ $ # 1 * ' * & " * • , 6 $ $ " " 7 + $ * Ismaël Bouya http://melusine.eu.org/syracuse/immae/ $ • , & $ $ (=) ) & " 6 ) 4 4 $) "→ (=) →4 + * ( (+) 4→ (+) " ( = ) " ( + ) →4 ( + ) ) * * (=) ) 4 % ( ' 4→ ( " →4 ( = −) 4→ ' ( →4 ($ 8 $ -&1 " →4 % =− 4→ →4 & * =− 4→ * 2 9 3 6 & : % * & # + * $ # $ ! = = " + ; # * " + = " * 2 ! 1 # " 4 " • 0 • $ $ > ; " # 3 = % $ + + = * $ & # % % 1 " * # # 1 • = + % * 3% 3 1 5 % 1 #% #% $ # 1 % * 7 # #% & 1 ? $ ! # #% " $ 3 = ; $ " ' $ ' ; > + 8 $ -&- = = $ 2 ! 1 1 $ * " * 1 " = @$ =@$ # = # ' "' ' % < , # 5 ' % B ' $ " =@$ ! + − !4 = ; ; * = ! =@ + =@ + =− 4 4 4 !* 4 1 + " * 3 $ "# = "# ; "#4 4 = 4 4 C * 4 * # % * % # ; • # % 4 !4 = 1 # & " % 1 1 • C - % 6 + B - E + % ' + + = !+ $ % × 4 = 4 − 1 4 = × $ & $ 1 -1 # * % B =@ & % 3 1 * ' $ ; % 1 : 1 * # ' + ' = - D # # 1 * 1 4 * + % $ # − ! +! % A ;# ' + + + ' =− !* +∞ + + ; @ $ +∞ =− + @ ! 5 + + +! + = +∞ − ! + = @ = − ∆ +∆ % 3 $ ! + = @ = @ $ ! + = +∞ = @ % # $ ∆ % 1 * $ ∆ = @* # < 1 % & $! - 5 ' - $ # # # * # 3 + 1 #= G$ 1 #= * + = $ ' , @ − * & # $ " $ = + + − + =+ + +! + ! + + − = * 6 * ' = * !− ! ; =− − = ! $ = + +! + ! − ! − * * A • - ; % 1 # ' '% H + + 1 % # " τ F ; @ +τ π= 3 *+ @ % @ π • 6 ' ! = @ +τ ; @ +τ != @ +! + @ + τ − ! + $ @ @ =π 3 • 1 J + 1 ; 1 # = + 4→ 8− ; "→ * @ +τ = + 4→ @ @ +τ * "→ @ π 4→ 3 @ +τ # 8− $ - ; 1 "→ @ π 4→ * = π 4→ 7 4 - K 84 − " % $ 4 =π →4 @ +τ 8− = " =@ @ 8 + 84 = $ + $ 4 * 1 " $ # = 3 % !1 1 % "% & $ 1 * I • 1 # L B4M ∆" = ' 3 $ ∆" = ∆"# $ ∆"# $ ; " = " + ∆" + ∆$ $ " τ =@ ∆" = @ + % + ∆$ = @ N & 1 ' 4×+− * !@4 + ∆$ = @ + !@ 4 1 = 1 = ∆"# = ∆"#4 ; # ∆"# $ # • ; % - + + =@ + 4 =@$ " 8 + 84 = @ 84 4 8 = 3 4 = 8 = 84 3 # ! 1 ! 6 ' • & " + σ+ $∀ + = σ+ + + σ+ ; • 5 = = ∧ !+ !+ + + !+ " = + * ∧ !+ + * 4 6 ' & " " ∆ =@ 5 =@$ ; σ∆ 5 ∆ σ∆ = ∆ ∆ = ∆ + + !+ = ' 5 + - 5 ∆ ∧ !+ ∆ ⋅ ⋅ " + ≡ 1 σ+ = @$ = @$ σ+ = σ∆ = = @$ # " $ ∀∆ ∆ 2 ! - ∧ !+ =@ " + !+ σ∆ + + !+ = " " * + " " % * ;# ' σ∆ = ∆ σ ∆ = % ∆θ ' $ % ∆θ = ∆ L % M ; θ θ • P • 5 ' • , • 6 6 • 6,; % $ $ ) ( $ ) (* 1 + = + E O &∆ 6 ' % ∆θ = ; + *3 # # θ *; = $ % =− θ+ 1 θ ≈θ • + θ = θ@ θ = @$ %∆ θ4 =− + = + θ θ θ* θ 1 # = θ =@ %∆ $ θ + 3 4 +ω + ϕ $ > ω = % ∆θ + θ* θ * θ* − = θ* 1 $ # 5θ =@ $ =@ + $ % @ α 3 3 • % α 7 1 > ϕ +α α * $ >ϕ * *3 ; " ! =@ ' ≤ & * @ • θ α = + * , • , • ) ($ ) ( + • - 6 ' = α +' @=− α+& ; &= α* =− θ* H 1 α* - 6 % ∆θ = ' + =@ 4 F 5 & + ' = +' × =@ 4 - & × $ ;# > ' = − − =' $ ' =− α$ = F 4 4 F = F α α 4 # α< > α$ 4 α* • 3 + 1 * < # / θ $ $ Q 3 • • • • % $ # " % $ = θ + " % $ % =− 4 + " 5 ' B B , % $ ) ($ ) 4( $ ) ( $ ) ( + 6 ' σ∆ = ∆ A 5 = θ= ; * " + ' # * #" R * & ∆= ∆ σ ∆ = % ∆θ + 3 %∆ * + 4 × − 4 4× =− + 4 + 4− 4 %∆ + + 4 4 $ = = +%∆ + + 4 %∆ + 4 − + 4 4 θ 4 4 + 4 4 % %∆$ % + * % * @ - 4 % ϕ θ α - + θ = ϕ +α , 5 ' % B * " ) ($ ) ( $ ) ( + - 6 ' σ∆ = ∆ #" & =@ ; σ∆ = =@ 3 $ σ ∆ = % ∆α + θ× +%∆ + ; 4 α+ $ ∆α = − - ϕ = @$ 4 + ϕ %∆ 4 + ∆ϕ = 4π % 4π % + ∆4 ; % ' $ + # α =− $ $ ! % ' ∆ %' &) ( ; #" + & −) ( # − − − − + : * %∆ 4 ! -7 − − 1 # + HH − ⊥ + # # # $ % − − ⊥ + #" E S $ " T* #" * $ " * 4 , ∆ θ 6,; + = + $ − = − ++− θ 4 + θ #1 θ − + % θ θ4 $ 7 θ '% 3 7 #" + #1 * * = * 7 < 6 σ+ σ+ = + " = % +Ω * 5 " *! 2 + $ #" σ+ " = Ω ∧σ + # $ * U 3 λ* ' ' $ " % $ "% $ 4 '+ + −' + 3 ) ( = )' ( + # # $ '+ + 4 "% $ 3 = !$ =λ −' + + $ #> $ =@$ ' + ' =λ '+ =@ + =@ % 1 * = 5 ' + $ $ $ ' = @$ ' = @$ ' = λ * 1 - > 1 * α ϕ $ ' =' ' =' α α ' =' α ϕ= ϕ= ϕ= #" - 1 $ Q % $ $ & ' =' ϕ =@* α= = '@ * α ' = '@ α ' = '@ α = '@ ; 3 ' = '@ * 4 4 4 =λ =λ + = * < $ # '@ =λ ; = - 1 + 4 λ * +ψ $ # > = '@ 3 +& 3 λ '@ λ '@ +ψ + ( '@ $ψ $ ( * ; 1 #1 =− $ 4 3 $ @= ; $ )= 4 ψ = @* * λ '@ λ )* 4'@ +( 4 = $ + = + '@ '= α # E U $ ' +J -&- $ θ & # θ = ϕ+ $ $ $ $ * ) *( E 4 * & * " 1 * 3 ) (E $θ * # " % % ' & E & $ ) *()' (−4 * U 1 • 5 ' • , • ' % ) ($ ) (* $ * A • θ • 6 %θ = − & θ × 4 * + 4 5 *4 + 4 4 • = θ =− θ θ4 = − θ+ 4 4 =@ = × +− *θ 4 = ' * + &* θ* + θ θ =− θ* +3 θ < @ θ* − θ− $' = θ *θ 4 = θ+ θ × 4 4 θ+ &= *θ = 4 θ = ' + 3 3 1 % ; 1 θ 1 + '1 + 4 θ θ− 4 • 3 4 + 4 4 <I * 4 $ -&- " * ' ,' G ; 1 π π HH 3 0 • ) 1 ' : ( = ) (+) (+ ) ( G G 1 $) " ( =) (* @ τ$ '* F • - 6, !+ = ; = !+ + ! + $ τ − ! + $@ = 5 ! + $τ = ; 4 π & θ= × • 6, 6 4 I θ= ; =π @ =@ 6 4 τ θ +τ − θ +@ = *; I π$ 8 θ +τ = -&- I π = !+ θ= • ) ' * 5# 1 ( = ) ( +) (+) ($ ' ≤ & = ) + ( = )' ( + ) ( ' * 7 * 4 ,' % # " 1 π π • • Q; ) - 6, - 6 !+ = ! + $τ = π + π + *; % ∆θ = & #" %∆ = + ( = ) (+) (+) ( 4 4 + 4 A 4 $ < < π $ ! + $τ = θ +τ = × * 4 4 < π = 4π * 4 π = π +π $ θ +τ = π * $ = < 4 π × * θ +τ = <π * * I λ Q N% • 3 N% 1 " G * 3 • & # % '+ + % % −' + + =@$ '+ =@ + $& # • 3 ; 3 ' = −λ =λ = • ; 5 N% 1 = '+ =@ * = '× *; θ= $ >θ # θ =λ * +' θ =λ * + 4 θ= =@ = @$ ' =@$ ' = 1 # G * ' =λ * $ ; ; 8 8 8+ % $ λ $ 4 4 4 = λ * + ) +# * & = @$ * ; 7 # = @$ λ$ & % = λ 4 N% $ " $ E * + N% *+ − $ Q G % % θ = % -5 - @ & τ* $ • ;# ' + = + ' $ + ; ! +τ = τ τ + @ =π +@+ 3 + @ τ τ @ = π + π' + π & @ % = $ % +τ = π + π ' @ =π& # ; 1 π' = @ * ' ≤ *& $ ! +τ = ;# > π • 3 % ∆θ = ' + *+ − + $ θ +τ = 4 * ' & % ∆θ +τ = ; F 4 −+ 4 * τ @ ' ++ −+ τ @ = + −+ π π 1 • - 6 + 6 ' = + $ =' $ & = ' % ∆θ = ' $ & θ= F' 4 O • J ! = ! +, ∈ + , ∧Ω = + + θ = !+ ! =+ + θ = '! ≤ @ $ !4 ! ! ≤ @$ π 3 ≤@ π +τ + θ +τ = @ $ 1 - + F 4 − + π += = @$ F * 1 =− θ$ 3 =' $ θ= F =− ' 4 ' = @$ = @$ θ = @$ 8 ; * F' 4 -&- # 1 > ! +τ > @ • +< - F ' <@ ' =− & =− =− $ = +τ − F' F F θ= =− θ = θ +τ − $ 4 4 4 Q # 3 = + ; % % * + θ+ + Fπ 1 1 T 4 > ! +τ = @ • 3 ; ' 4− + 4 = ! +τ × =@$ >@ = ' $ V W$ ' % %