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1.2 ANALYSIS METHOD

1.2.3 DESCRIPTION- LAGRANGIAN AND EULARIAN

 There are two approahe! to "e!r#$e the %ot#o& o' a ()#" a&" #t! a!!o#ate"

propert#e!.

1. La*ra&*#a& approah2. E)+er#a& approah

, La*ra&*#a& approah

P#t)re a ()#" (ow where eah ()#" part#+e arr#e! #t! ow& propert#e! !)h

a! "e&!#t/ %o%e&t)%/ et. A! the part#+e a"0a&e! #t! propert#e! %a

ha&*e #& t#%e. The proe")re o' "e!r#$#&* the e&t#re (ow $ reor"#&* the

"eta#+e" h#!tor#e! o' eah ()#" part#+e #! the La*ra&*#a& "e!r#pt#o&.

 The ()#" pare+! are +a$e++e" $ !o%e t#%e-#&"epe&"e&t 0etor e+" x0.

O'te&/ x0 #! ho!e& to $e the e&ter o' %a!! o' the pare+! at !o%e #t#a+

t#%e t 0. It #! ho!e& #& th#! part#)+ar %a&&er to ao)&t 'or the po!!#$+e

ha&*e! o' the !hape o0er t#%e. There'ore the e&ter o' %a!! #! a *oo"

para%eter#4at#o& o' the (ow 0e+o#t u o' the pare+. I& the La*ra&*#a&

"e!r#pt#o&/ the (ow #! "e!r#$e" $ a ')&t#o& Ux0/ t *#0#&* the po!#t#o& o' 

the pare+ +a$e+e" x0 at t#%e t .

 The La*ra&*#a& "e!r#pt#o& #! !#%p+e to )&"er!ta&" o&!er0at#o& o' %a!!

a&" Newto&5! +aw! app+ "#ret+ to eah ()#" part#+e. Howe0er/ #t #!

o%p)tat#o&a++ e6pe&!#0e to 7eep tra7 o' the tra8etor#e! o' a++ the ()#"

part#+e! #& a (ow a&" there'ore the La*ra&*#a& "e!r#pt#o& #! )!e" o&+ #&

!o%e &)%er#a+ !#%)+at#o&!.

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9#*-La*ra&*#a& "e!r#pt#o&

, E)+er#a& "e!r#pt#o&

Rather tha& 'o++ow#&* eah ()#" part#+e we a& reor" the e0o+)t#o& o' the

(ow propert#e! at e0er po#&t #& !pae a! t#%e 0ar#e!. Th#! #! the E)+er#a&"e!r#pt#o&. It #! a e+" "e!r#pt#o&. A pro$e 6e" #& !pae #! a& e6a%p+e o' 

a& E)+er#a& %ea!)r#&* "e0#e.

 Th#! %ea&! that the (ow propert#e! at a !pe#e" +oat#o& "epe&" o& the+oat#o& a&" o& t#%e. The +oat#o&! are "e!r#$e" #& oor"#&ate !!te%!. I&the E)+er#a& !pe#at#o& o' a e+"/ #t #! repre!e&te" a! a ')&t#o& o' po!#t#o& x a&" t#%e t . 9or e6a%p+e/ the (ow 0e+o#t #! repre!e&te" $ a')&t#o& )6/t.

  The E)+er#a& "e!r#pt#o& #! har"er to )&"er!ta&" how "o we app+ the

o&!er0at#o& +aw!: Howe0er/ #t t)r&! o)t that #t #! %athe%at#a++ !#%p+er to

app+. 9or th#! rea!o&/ #& 9+)#" Meha! we )!e %a#&+ )!e the E)+er#a&

"e!r#pt#o&.

9#*-E)+er#a& "e!r#pt#o&

https://en.wikipedia.org/wiki/Field_(physics)https://en.wikipedia.org/wiki/Flow_velocityhttps://en.wikipedia.org/wiki/Flow_velocityhttps://en.wikipedia.org/wiki/Field_(physics)