Appedix: The Laplace Trasform The Laplace trasform is a powerful method that ca be used to solve differetial equatio, ad other mathematical problems. Its stregth lies i the fact that it allows the trasformatio of a differetial equatio to a algebraic equatio. The oe-sided Laplace trasform is defied as follows L x( t) X s x( t)e st dt, (I.) where the variable s is defied as cotaiig both a real ad imagiary part, i.e., s σ + iω with σ such that e st remais fiite as t. Referrig to equatio (I.) we say that X ( s) is the Laplace trasform of x( t). We also assumed that the time variable t starts at, but this could be chaged to ay other value (e.g., t ). For example, we calculate the Laplace trasform of a few simple fuctios [ ] Ae st dt L A A s e st A s, s > t L e at e at e st dt e s+a, s > a. (I.) A particularly importat trasform is that of a impulse of time duratio τ defied as x( t) τ, < t < τ, t > τ (I.3) with τ X s τ τ st e e st dt sτ If we ow take the limit of equatio (I.4) whe τ, we get lim X s τ sτ sτ ( e ). (I.4) sτ ( [ sτ ] ). (I.5) The limit of a fuctio such as, defied i equatio (I.3), whe the duratio of the impulse is take to be ifiitely small while keepig the area of the impulse costat is I
called a Dirac or delta fuctio. It is usually simply writte as δ t property that δ t for t ad δ ( t) for t, but, ad has the δ ( t t )dt. (I.6) Equally importat is the trasform of a step or Heaviside fuctio, represeted by H ( t), t >, t <. (I.7) Sice our defiitio of the Laplace trasform trucates ay fuctios that are o-zero for t <, the Laplace trasform of the step fuctio was evaluated i equatio (I.) ad foud to be L H ( t) s. (I.8) The list of trasforms appearig i Table I. ca be similarly verified. Table I. Laplace trasform pairs L Aδ ( t) A, s > L A H ( t) A s, s > L e at H ( t), L t H ( t) s > a!, s >,,, 3,... + s! +, s > a,,,3,... L t e at H ( t) L si( ωt) H ( t) ω s + ω, s > L cos( ωt) H ( t) s s + ω, s > ω L e at si( ωt) H ( t) ( ) + ω, s > a L e at cos( ωt) H ( t) ( ) + ω, s > a. II
The Laplace trasform also possesses other importat properties, some of which are (assumig that every time fuctio is zero for t < i what follows) I. Liearity. If A ad B are costats L A II. Trasform of derivatives. + By( t) AX s + BY ( s) (I.9) L d dt d e st dt x( t)e st dt + s sx ( s) x x( t)e st dt (I.) where we itegrated by parts, ad x the trasform of higher derivatives ca be show to give is the iitial coditio x( t). Similarly, L d dt L d dt s X s s X s III. Trasform of primitive of fuctios. L t sx dx( t) dt t d k x( t) s k k dt k x( τ )dτ t { x( τ )dτ } e st dt t x ( τ s { )dτ } e st + s X ( s) + t x ( τ s s )dτ { } t t x( t)e st dt (I.) (I.) where we agai itegrated by parts. The trasform of higher primitives is give by L... x( τ )( dτ ) X s + s k {... s x( τ )( dτ ) } k k+ t (I.3) IV. Time shiftig. Sice x( t) for t <, we ca write III
L x( t τ ) x( t τ )e st dt x( t τ )e st dt x( λ)e s( λ +τ ) dλ e sτ x( λ)e sλ dλ e sτ X s τ (I.4) where we made the substitutio λ t τ. V. Multiplicatio by a expoetial. L e at x( t) e at x( t)e st dt X x( t)e ( s+a)t dt (I.5) The residue theorem Oce a fuctio or a equatio has bee trasformed i the Laplace domai, the modified for oe purpose or aother, it will evetually eed to be trasformed back to the time domai. Although a iverse Laplace trasform ca be mathematically defied, it is always more coveiet ad easier to use the so-called residue theorem to go from the Laplace to the time domai. This theorem is stated as follows. Give a fuctio X ( s), m for which the deomiator ca be writte as a product of factors of the type j (where a j is called a pole of order m), we ca write L X ( s) ( m )! lim d m s a j ( ds m j ) m X ( s)e st, t > j (I.6), ad the quatity i betwee where is the umber of poles i the deomiator of X s the curly braces is called the residue of X ( s)e st at the pole a j of order m. Let s cosider a few examples IV
L lim d s a! ds e at, t > L ( ) d lim s a! ds e st ( ) te at, t > e st (I.7) ad fially L ( ) + ω lim s a+iω d! ds + lim s a iω L ( s + ( a iω )) d! ds iω ( ) s + ( a + iω ) + ω est ( s + ( a + iω )) iωe ( a+iω )t iωe ( a iω )t + iω iω e at cos( ωt), t >. ( ) ( ) + ω est (I.8) These results ca be verified agaist the examples preseted i Table I.. Applicatio to the damped oscillator problem Let s ow solve a few cases ivolvig the equatio of motio of a damped oscillator with differet types of drivig iput. The equatio to solve is I. f ( t) Aδ ( t). x ( t) + β x ( t) + ω x( t) f ( t) (I.9) + β x ( t) + ω x( t) L L f t (I.) Usig the liearity property of the Laplace trasform ad Table I., we get ( s X ( s) sx x ) + β sx ( s) x + ω X s A, (I.) V
or X ( s) s ( + βs + ω ) A + x ( s + β) + x. (I.) I everythig that will follow, we will assume that x x. We ow solve equatio (I.) X ( s) A s + βs + ω A s + β β ( ( ω )) s + β + β ω ( ) (I.3) We ow use the residue theorem stated i equatio (I.6) β β ( ω ) t A e β + β ( ω ) t β ω e β ω e βt A β ω e β ω t e β ω t, t > (I.4) A close examiatio of equatio (I.4) shows that the respose of the damped oscillator to a Dirac fuctio is othig more tha the complemetary solutio of the equatio of motio. I the case of the uderdamped oscillator ( β < ω ), we fid that x( t) A e βt si( ω t), t > (I.5) ω with ω ω β. II. f ( t) A H ( t) I this case, we have (assumig that β < ω, ad ω ω β ) X ( s) s ( + βs + ω ) A s (I.6) VI
Figure I. Respose to a Dirac fuctio drivig iput. x( t) A s s + βs + ω A s s + ( β iω ) ( s + ( β + iω )) e ( β iω )t A ω + iω iω β A ω ω t + e β +iω ( iω + β) iω e βt β + ω cos( ω t φ), t > (I.7) with β φ ta. ω (I.8) Figure I. Respose to a step fuctio as drivig iput. VII
The Laplace trasform ca be systematically applied to more complicated types of problems ad drivig fuctios (periodic or ot). It is also importat to realize that the solutio to a give problem provided by the applicatio of the Laplace trasform icludes both the complemetary ad the particular solutios. The Two-sided Laplace Trasform It is geerally the case i physics that a fuctio is ot limited to t but ca exist for times both positive ad egative. We ca the geeralize the oe-sided Laplace trasform give i equatio (I.) with its two-sided versio X s Usig our previous otatio for the oe-sided trasform L s we ca write for the two-sided trasform e st dt. (I.9) e st dt (I.3) X s e st dt + x( t)e st dt L s + L x t s. t> t< (I.3) I equatio (I.3) we made the depedecy of the Laplace trasform o the parameter s explicit by addig ( s) to the left-had side. The last term o the right-had side of the secod of equatios (I.3) ca be ascertaied with (usig λ t ) L x t s x t e st dt x( λ)e sλ dλ x( λ)e sλ dλ. (I.3) For example, if we calculate the two-sided Laplace trasform of the followig fuctio x( t) e t, t < e 3t, t (I.33) we fid (usig Table I.) VIII
X ( s) s! + s<! s + 3 s> 3 5, for 3 < s <. s + s 5 (I.34) We must also determie the proper relatio to calculate iverse Laplace trasforms for two-sided fuctios. Usig equatio (I.3) ad the residue theorem for oe-sided fuctios (i.e., equatio (I.6)) we ca write L X ( s) x t ( m )! lim d m s a j ( ds m j ) m X ( s)e st, t >. j (I.35) We ca ow look at egative time values by chagig t t, ad x( t) ( m )! lim d m s a j ( ds m j ) m X ( s)e st, t <. (I.36) j We fially also chage s s to fid x( t) j j ( m )! lim s a j d s ( m )! lim d m s a j m X s m j d m ( ) m ds m ( ) m s a j e st m X s e st ( m )! lim d m s a s a j ( ds m j ) m X ( s)e st, t <. j (I.37) That is, the iverse Laplace trasform for a fuctio defied oly for times t < is similar to that for a fuctio defied for t >, except for the overall egative sig. For example, if we calculate the iverse Laplace trasform of equatio (I.34) we fid 5( s ) + 5( s + 3) s + s 5 et H t s + s 5 e 3t H t e t H ( t) e 3t H ( t), (I.38) which is the same as equatio (I.33). Fially, we ote that because the two-sided Laplace trasform cosists of a itegral performed over the domai < t < the depedecies o derivatives ad itegrals IX
evaluated at t (see equatios (I.) ad (I.3)) do ot appear i the results of calculatios. Notably we have dt d... x ( τ { )( dτ ) }e st dt e st dt s X ( s) X ( s). s (I.39) X