Classification of Equations Characteristics

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1 Clssiiion o Eqions Cheisis Consie n elemen o li moing in wo imensionl spe enoe s poin P elow. The ph o P is inie he line. The posiion ile is s so h n inemenl isne long is s. Le he goening eqions e epesene he ollowing elions, Hee,, e inepenen iles whih ol epesen eloi omponens. The ems,,,, e oeiiens whih ol e nions o,. ee poin in he omin, n ll hei eiies e inie wih niqe les. The ojeie will e o eemine i hee e lines in he omin long whih, he eiie h, lhogh inie, e ineeminn heisi lines. In spesoni low, hese lines wol e h lines n shok wes. long sh lines eiies o, e ineeminn. oss sh lines hese eiies e isoninos. These n e on elizing h, e nions o, so h hei ieenils n e wien s, ll o eqions n e wien s n eiie n e se o eemine he pesene o heisi lines, s n emple / is se. Cme s le,,, The eiie is ineeminn when / /, o les when he eeminn o is zeo. This mens h i he pope ieion o P is hosen,,, he le o will e ienill zeo. The eeminn o is, s

2 This eqion n e wien s, This eqion heeoe poies he slopes o he heisi lines emning om P, i he eis. This epession is qi in /. Theeoe, he slopes e, 4 ± The ne o hese heisi lines ies s he le o he esiminn hnges, D 4. Eqion Clssiiion i D > - wo isin, el heisi lines eis, he goening eqions e lle, Hpeoli i D - one line eiss, he goening eqions e lle, Poli i D < - he heisi lines e omple n o no eis in eli, he goening eqions e lle, Ellipi Eigenle eho o ssems o eqions genel poee n e omle o ssems o eqions se on he llion o eigenles o mi. Consie he ssem elow, iing he solion eo s, he eqions eome o [ ] [ ] Soling, [ ] [ ] seing, [ ] [ ] [ ] N so h [ ] N Digonlizing he N mi, we n eple i wih he Eigeneo mi,

3 3 [ N] [ λ ] so [ λ ] Repling he N mi wih he eigeneo mi eeiel eoples ll o he eqions. e n hen sole o he iniil heisi lines, λ λ The om o he eigenles eemines he he o he eqions. I he eigenles e el, he ssem is hpeoli. I he e omple, he ssem is ellipi. I is lso possile o he mie se o eigenles in mie hpeoli-ellipi ssem. One eqion lssiiion Some polems e esie single goening eqion sh s Lple s eqion. Consieing genel eqion o he om, φ φ φ φ eφ φ g whee,,,,e,,g n e nions o,,φ,φ,φ. The lssiiion is one in simil mnne o he heisi llion one peiosl, i -4 > - wo isin, el heisi lines eis, he goening eqions e lle, Hpeoli i -4 - one line eiss, he goening eqions e lle, Poli i -4 < - he heisi lines e omple n o no eis in eli, he goening eqions e lle, Ellipi Ellipi Eqions Ellipi eqions e oen lle eqiliim polems sine he esl iel om onseion gmens. n emple o n ellipi eqion is Lple s eqion, whih n e se o sess nlsis in soli, se se he onion in soli, n poenil low polems. s n emple, onsie he one imensionl se he onion in mel o wih empees hel ie oh ens. insle q T solion T T o L o L

4 4 The goening eqion is, T The solion is sigh line whose he is eine he les he en poins onl. Noe h soes e no pesen. I one h een e, he solion ee poin on he wol he eee. This is pope o ellipi ssems, isnes he inlene oe he enie omin. In nmeil nlsis, soles o ellipi eqions ms llow inomion o popge in ll ieions one. Poli Eqions Poli eqions n e lssiie s mhing polems sine hei solions onin el heisi line. Solions e on m moing ow in n ppopie imension eemine he heisi. In he peios emple o one imensionl hee o, i he poile h een llowe o hnge in ime hogh nsien em, he ssem wol he een poli, T T α solion The neess oniions o niqe solion wol he een he en oniions s wih he o L ellipi eqions, o he poli solion n iniil solion is neee. In his emple, n iniil solion wol e neess. Poli ssems e ones wih signiin mons o issipion. In his emple he mhing ile is ime. The solion is mhe om he iniil solion o he esie en poin. zone o epenene is eslishe in his mnne sine in he emple oe, sen isne n onl ee he empee poile le poins in ime. Hpeoli Eqions Hpeoli eqions mi we-like solions n ehii omin o epenene simil o he poli eqions. Howee, his omin o epenene is in spe he hn ime. s n emple, spesoni ioil is esie hpeoli ssem o eqions. Disne nno el psem sine he onoming eloi is gee hn he spee o son. nohe emple is he seon oe we eqion, φ φ whee he onsn is he we spee. gin, he en oniions long wih n iniil poile e eqie o niqe solion. s n emple, onsie he moemen o n iniil isplemen on sing. I he iniil solion ime is eine φ,, φ,g hen he solion hs he om o wo wes moing in opposie ieions, φ, [ ] g s s

5 5 o L o L The impliion is h some poin, he le o φ epens on inomion insie he inel - o. Iniil oniions osie his omin o no ee he lol solion,. Theeoe zone o epenene is ee on,, Vles, e eee les in he omin o epenene onl, eine he heisi lines. The le, ees les in he zone o inlene, gin one he heisi lines. gin, ime is he mhing ile., zone o inlene omin o epenne - Smm Eqion Tpe Emple Coniions Domin Solion Ellipi T n onl Close omin lws smooh Poli T n Open omin in lws α iniil smooh Hpeoli φ φ on Open omin in e iniil n isoninos

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