Text 8. Hamiltonian Bifurcations

The energy-momentum method has also been used in the context of Hamiltonian bifurcation problems. We shall give some simple examples of this in §1.8. One such context is that of free boundary problems building on the work of Lewis, Marsden, Montgomery, and Ratiu [1986], which gives a Hamiltonian structure for dynamic free boundary problems (surface waves, liquid drops, etc.), generalizing Hamil­tonian structures found by Zakharov. Along with the Arnold method itself, this is used for a study of the bifurcations of such problems in Lewis, Marsden, and Ratiu [1987], Lewis [1989, 1992], Kruse, Marsden, and Scheurle [1993], and other references cited therein.

Converse to the Energy—Momentum Method. Because of the block structure mentioned, it has also been possible to prove, in a sense, a con­verse of the energy-momentum method. That is, if the second variation is indefinite, then the system is unstable. One cannot, of course, hope to do this literally as stated, since there are many systems (e.g., gyroscopic system mentioned earlier – an explicit example is given in Exercise 1.7-4) that are formally unstable, and yet their linearizations have eigenvalues lying on the imaginary axis. Most of these are presumably unstable due to "Arnold diffusion," but of course this is a very delicate situation to prove analytically. Instead, the technique is to show that with the addition of dissipation, the system is destabilized. This idea of dissipation-induced instability goes back to Thomson and Tait in the last century. In the con­text of the energy-momentum method, Bloch, Krishnaprasad, Marsden, and Ratiu [1994, 1996] show that with the addition of appropriate dissipa­tion, the indefiniteness of the second variation is sufficient to induce linear instability in the problem.

 

Слова к тексту

 

bifurcation	[,baıfə´keı∫n]	раздвоение, разветвление; бифуркация
boundary	[´baυndərı]	граница
surface	[´sз:fıs]	поверхность
wave	[weıv]	волна
liquid	[´lıkwıd]	жидкий; жидкость
drop	[drכp]	капля
generalize	[´dζenrəlaız]	обобщать
reference	[´refrəns]	ссылка
cite	[saıt]	ссылаться; цитировать
therein	[,ðeər´ın]	там, в том, в нем, в них
converse	[´kכnvз:s]	обратное утверждение; обратная теорема
mention	[´men∫n]	упоминать; ссылаться на
literally	[´lıtrəlı]	точно, буквально
state	[steıt]	излагать, формулировать
gyroscopic	[,dζaırə´skכpık]	гироскопический
explicit	[ık´splısıt]	ясный, подробный; точный определенный
yet	[jet]	все же
imaginary	[ı´mædζınrı]	воображаемый; мнимый
presumably	[prı´zju:məblı]	предположительно
delicate	[´delıkət]	деликатный, затруднительный
technique	[tek´ni:k]	техника; метод; способ
addition	[ə´dı∫n]	прибавление
dissipation	[,dısı´peı∫n]	рассеивание
destabilize	[di:´steıbəlaız]	дестабилизировать
induce	[ın´dju:s]	вызывать, стимулировать
appropriate	[ə´prəυprıət]	соответствующий
indefiniteness	[ın´defənətnəs]	неопределенность
sufficient	[sə´fı∫nt]	достаточный
linear	[´lınıə]	линейный

 

Letters and Sounds

 

Task 1. Answer the following questions

1. Обратите внимание на словосочетания, в которых существительное пишется с заглавной буквы: Page 5, Exercise 10. Какую синтаксическую роль в них выполняет количественное числительное?

 

2. Как читается сочетание al в словах always, already? Можете ли вы привести примеры других подобных слов? Как вы думаете, почему в словах algebra, algorithm, alphabet буква а читается как обычно (в чем отличие от первого случая)? Как читается a в словах alike, allow, along? Почему?

Как читаются сочетания all, alt, alse; alk; alm, alf, alve? Приведите примеры слов с этими буквосочетаниями.

 

3. Как обычно читается буквосочетание еа? Приведите примеры. Как еще оно может читаться перед d, th? Приведите примеры на разные случаи чтения этих сочетаний. Как читаются буквосочетания ealth, easure? Приведите примеры.

 

4. Как обычно читается буква о в открытом и закрытом слоге? В чем особенность чтения слов post, host, ghost, (al)most; roll, scroll? Вспомните, как читается сочетание old.

 

5. Какие вы знаете односложные глаголы на –ie? Как пишется IV форма этих глаголов? Как пишется IV форма глагола to dye (красить)? Почему?

 

6. Знаете ли вы, что означают и как читаются следующие сокращения: etc., e. g., i. e., viz.?

 

Task 2. Say and write what sound is pronounced in the following words:

 

1) h al ves, 2) al most, 3) l ea ther, 4) c o st, 5) a lone, 6) st ea dy, 7) f al se, 8) inst all, 9) c al m, 10) tr ea sure, 11) p o st, 12) ch al k, 13) l o st, 14) s al t, 15) h ea lth, 16) b o ld, 17) d yei ng, 18) he r ea d, 19) he r ea ds, 20) he l ea ds.

 

Task 3. Read aloud the following words from Text 8

 

Bifurcation, dissipation, references, cited, these, converse, other, second, gyroscopic, technique, imaginary, explicit, situation, of course, delicate, mentioned, analytically, generalizing, induces, variation, technique, sufficient, exercise, energy, addition, eigenvalues, gyroscopic, method, liquid, some, because, example, axis, structure, bifurcation, unstable, prove, work, examples, last.

 

Grammar Activity

 

Task 1. Answer the following questions

Как обычно переводится местоимение these? Назовите его форму единственного числа. Как переводится these в следующем контексте: There are many systems that… Most of these…?

 

Task 2. Translate the phrases from Text 8

 

The energy-momentum method; examples of this; Hamiltonian bifurcations problems; found by Zakharov; free boundary problems; some simple examples; the context of Hamiltonian bifurcation problems; found by Zakharov; a study of the bifurcations of such problems; a delicate situation to prove analytically; a converse of the energy-momentum method; the Arnold method itself; the addition of dissipation; the second variation; this idea of dissipation-induced instability; the context of the energy-momentum method; surface waves; the addition of appropriate dissipation; references cited therein; the indefiniteness of the second variation; in the last century; most of these; problems building on the work of Lewis; generalizing Hamiltonian structures; the block structure mentioned; systems that are formally unstable; to do this as stated; gyroscopic system mentioned earlier; Arnold diffusion; eigenvalues lying on the imaginary axis; dissipation-induced instability; the work of Lewis; other references; to do this literally as stated; examples of this; liquid drops; their linearizations

The method has been used… We shall give… It has been possible… This idea goes back to Thomson… The technique is to show that… This is used… It is given… The system is destabilized… If the second variation is indefinite, then the system is unstable… One cannot hope to do this… They show that… There are many systems… This is a very delicate situation… It has also been used… It is sufficient to induce… Most of these are presumably unstable… One such context is that of free boundary problems…

 

Word Formation

 

Translate the words from Text 8

 

Structure, dynamic, variation, therein, delicate, linearizations, analytically, situation, dissipation, instability, unstable, indefiniteness, Hamiltonian, appropriate, sufficient, linear, destabilized, literally, addition, indefinite, imaginary, formally, gyroscopic, presumably, bifurcation, converse, diffusion, destabilized, indefiniteness, the energy-momentum method.

 

Words and Phrases

 

Task 1. Answer the following questions

 

1. Какие значения имеет слово yet? Приведите примеры.

 

2. Чем различаются because и because of? Приведите примеры их употребления. Какие это части речи?

 

Task 2. Translate with a dictionary

 

Are you ready? – Not yet. This problem is more difficult yet. Should I return yet? He was shocked and yet glad to see her. It is the best computer yet invented. He did his homework in one copy-book, wrote poems in another one and drew in yet another. She was crying because she had got a bad mark. She was crying because of a bad mark. She was crying because of the five problems, she had solved only one.