Giáo trình Tiếng Anh nâng cao chuyên ngành Vật lý: Phần 1

Giáo trình tập trung vào 4 môn học cơ bản của chuyên ngành Vật lý mà người học cũng được học bằng tiếng Việt nên thuận lợi trong việc tiếp thu và rất hữu ích vì các kiến thức này không những cần cho sinh viên, học viên cao học mà cho bất kỳ một người nào làm việc về Vật lý. Nội dung giáo trình được biên soạn thành 5 chương, trong đó 4 chương đầu bao gồm các nội dung về cơ học lý thuyết, điện động lực học, cơ học lượng tử, vật lý thống kê. Chương cuối cùng là phần hướng dẫn tài liệu tự học. Giáo trình được chia thành 2 phần, mời các bạn cùng tham khảo phần 1 sau đây.

NGUYỄN VĂN HÙNG

TIẾNG ANH NÂNG CAO

CHUYÊN NGÀNH VẬT LÝ

ADVANCED ENGLISH FOR PHYSICISTS

NHÀ XUẤT BẢN ĐẠI HỌC QUỐC GIA HÀ NỘI

c o n t e n t 3

CONTENT

Page

P reface 5

Giới thiệu giáo trình Tiếng A n h tĩảngcao chuyên ngành Vật l ý 7

Chapter I : T h e o re tic a l M ech an ics 9

Part 1.1: Reading, English-Vietnamese translation 9

Part 1.2: Vietnamese-English translation 40

Part 1.3: Answer the below questions in English 45

■ Part 1.4: Topics for report and discussions in English 49

Part 1.5: Supplementary materials for reading and tran slatio n 50

C h a p te r!: E le c tro d y n a m ic s 51

Part 2.1: Reading, English-Vietnamese translation 51

Part 2.2: Vietnamese-English translation 94

Part 2.3: Answer the below questions in English 100

Part 2.4: Topics for report and discussions in English 103

Part 2.5: Supplementary materials for reading and tran slatio n 104

Chapter 3: Q u a n t u m m ec h a n ic s 107

Part 3.1: Reading, English-Vietnamese translation 107

Part 3.2: Vietnamese-English translation 156

4 ADVANCED ENGLISH FOR PHYSICISTS

Part 3.3: Answer the below questions in English 161

Part 3.4: Topics for report and discussions in English 164 Part 3.5: Supplementary materials for reading and translation 166

Chapter 4: Statistical Physics 167

Part 4.1 : Reading, English-Vietnamese translation 167 Part 4.2: Vietnamese-English translation 205

Part 4.3: Answer the below questions in English 2 11 Part 4.4: Topics for report and discussions in E n g lish 214

Part 4.5: Supplementary materials for reading and translation 215 Chapter 5: Practice o f w riting scientific report or p a p er 217

Part 5.1 : Plan of writing a scientific report or paper 217

Part 5.2: An example o f a published paper 219

Appendix: New W o rd s 231

R eferen ces 251

ADVANCED ENGLISI I FOR PHYSICISTS 5

PREFEACE

This teaching material “Advanced English for Physicists” has been written based on teaching materials of the author (Prof Dr sc Nguyen Van H ung) for undergraduate (H onors Program, 3rd year) and Master’s students o f the faculty

of Physics, Hanoi University of Science in the recent years

A Purposes: This teaching material has purposes to train the following skills o f students on:

1 Reading, translating and writing English physics materials;

2 Speaking and discussion in English on fundamental subjects of physics

B Subjects for teaching and training: Main subjects o f Physics have been chosen: Theoretical Mechanics, Electrodynamics, Q u an tu m Mechanics and Statistical Physics T h e knowledge of these subjects with their new words and use in Physics is important not only for students but also for all physicists

C Training procedure is realized in each chapter by the following steps:

1 Reading and English-Vietnamese translation

2 Vietnamese-English translation

3 Answer questions in English

4 Presentation and discussions of physics topics in English

5 Selfstudy o f students using supplementary materials

6 Writing and presenting a scientific report or paper

D T h e new words concerning present teaching material has been provided to help students and teachers in using this material

E T h e author welcomes any comments to improve this teaching material

February 2007

The author

ADVANCED ENGLISH FOR PHYSICISTS 7

• Giúp các giáo vièn tham gia giảng dậy có giáo trình đế việc giảng dậy

và học tập dược thuận lợi hơn

• Cung cãp cho sinh vicn, học viên cao học và nghiên cứu sinh chuyên ngành Vật lý một tài liệu học tập, trong đó đói với học viên cao học giáo viên sẽ láy m ột sổ phán chính, rút gọn cho thích hợp với thời lượng cũng như mục đích của việc học

• Thông qua việc học có thé nâng cao nhận thức của người học vé những ván dé cơ bàn của chuyên ngành Vật lý

• Đói với toàn bộ giáo trình có thé đạt mục đích ròn luyện cho người học các kỹ năng quan trọng như: đọc, dịch xuôi, ngược, nói, viét các ván đé Vật lý bằng tiéng Anh

II NỘI DUNG CỬA GIÁO TRÌNH

• Giáo trình góm 5 chương với phán Phụ lục dành cho các từ mới chuyên mòn Vật lý liên quan đén các tài liệu được dạy và cuói cùng là

8 PREFRACE

• Việc ròn luyện các kỹ năng của người học được thế hiện trong việc giàng dạy của từng chương, bao góm:

- T ập dọc và dịch các tài liệu Vật lý từ Anh sang Việt

- T ập dịch các tài liệu Vật lý từ Việt sang Anh

- T ập đặt cầu và trả lời các vấn đé Vật lý bâng tiếng Anh

- Tập trình bày và thảo luận các nội dung Vật lý bằng tiếng Anil

- Chương 5 dành cho việc rèn luyện người học viét và trình bày một bản báo cáo hay một bài báo vé Vật lý bảng tiéng Anh

- Lưu ý là với cao học vì thời lượng hạn chế trong 30 tiết học (45phút/tiét) thì giáo viên chi có the ròn luyện hai kỷ năng thứ nhát

và thứ hai với nội dung cũng rút gọn

• Chương trình tập trung vào 4 mòn học cơ bản của chuycn ngành Vật

lý mà người học cũng được học bâng tiếng Việt nên thuận lợi trong việc tiép thu và rát hữu ích vì các kicn thức này không những cẩn cho sinh viên, học viên cao học mà cho bát kỳ một người nào làm việc vé Vật lý Nó được thế hiện trong 4 chương đáu của giáo trình:

- Chương I: C ơ học lý thuyét (Theoretical Mechanics)

- Chương II: Điộn dộng lực học (Electrodynamics)

- Chương III: C ơ học lượng tử (Q uantum Mechanics)

- Chương IV: Vật lý thống kê (Statistical Physics)

- T rong cuối mỗi chương đéu có hướng dẫn các tài liệu tự học hay dược dùng khi chương trình có thèm thời lượng

Giáo trình được viét dựa trên các tài liệu của tác giả đã giảng dạy cho các sinh viên H ệ dào tạo Cử nhân khoa học tài năng, học viên cao học thuộc chuyên ngành Vật lý trong những năm gán đây ncn đã rút được m ột só kinh nghiệm thực tế T uy nhiên, đế dậy theo giáo trình này người dạy cấn có một sỗ kién thức Vật lý nhát định

THEORETICAL MECHANICS

C H A P T E R 1

PA R T 1.1: Reading, English- Vietnamese translation

Sec 1 G eneralized coordinates

Frame o f reference:

In order to describe the motion o f mechanical system, it is necessary to specify its position in space as a function of time Obviously, it is meaningful to speak o f the relative position of any point For instance, the position of a flying aircraft is given relative to some coordinate system fixed with respect to the earth; the motion o f a charged particle in an accelerator is given relative to the accelerator, etc T h e system, relative to which the motion is described, is called a frame o f reference

10 CHAPTER 1 THEORETICAL MECHANICS

Specification o f time

As will be shown later, specification of time in general case is also connected with definding the frame of reference in which it is given T he intuitive conception o f universal, unique time, to which we are accustomed in every day life, is, to a cetain extent, an approximation that is only true when the relative speeds of all material particle are small in comparison with the velocity of light T he mechanics o f such slow movements is term ed Newtonian, since Isaac Newton was the first to formulate its laws

N ew ton’s laws permit a determitation o f the position of a mechanical system at an arbitrary instant o f time, if the positions and velocities o f all points

of the system are known at some initial instant, and also if the forces acting in the system are known

Degrees o f freedom o f a mechanical system

The num ber o f independent parameters defining the position of a mechanical system in space is termed the number o f its degrees o f freedom

The position o f particle in space relative to other bodies is difined with the aid o f three independent parameters, for example, its Cartesian coordinates The position o f a system consisting o f N particles is determined, in general, by 3N independent parameters

However, if the distribution o f points is fixded in any way, then the number

o f degrees o f freedom may be less than 3N For example, if two point are

ADVANCED ENGLISH FO R PHYSICISTS 11

constrained by some form o f rigid nondeformable coupling, then, upon the six Cartesian coordinates o f these points, Xi.yi, Z1.x2.y2, Z2, is imposed condition

(x2- x i )2+(y2- y i )2+(z2- z , ) 2= R?2, ( l l )

where R12 is the given distance between the points It follows that the Cartesian coordinates are no longer in d e p e n d e n t parameters: a relationship exists between them Only five o f the six values X j, ,z2 are now independent Inother words, a system o f two particles, separared by fixed distance, has five degrees of freedom If we consider three particles which are rigidly fixed in a triangle, then the coordinates o f the third particle must satify the two equations:

(x3-xi)2+(y3-yi)2+(z3-zi)2= R 2,, (1.2)

(x3-x2)2+(y3-y2)2+(z3-z2) 2= R32 (1.3)

T hus the nine coordinates of the vertices of the rigid triangle are defined by the three equations ( l l ) , (1.2) and (1.3), and hence only six of the nine quantities are independent T h e triangle has six degrees o f freedom

12 CHAPTER 1 THEORETICAL MECHANICS

T h e position o f a rigid body in space is defined by three points which do not lie on the same straight line T hese three points, as we have just seen, have six degrees o f freedom It follows that any rigid body has six degrees of freedom It should be noted that only such m otions o f the rigid body are considered as, for example, the rotation o f a top, where no noticeable deformation occurs that can affect its motion

Generalized coordinates

It is not always convenient to describe the position o f a system in Cartesian coordinates As we have already seen, when rigid constraints exist, Cartesian coordinates m ust satisfy supplem entary equations In addition, the choice of coordinate system is arbitrary and should be determined primarily on the basis

of expediency For instance, if the forces depend only on the distances between particles, it is reasonable to introduce these distances into dynamical equations explicitly and not by means o f Cartesian coordinates

ADVANCED ENGLISH FOR PHYSICISTS 13

In other words, a mechanical system can be described by coordinates whose num ber is equal to the num ber o f degrees o f freedom o f the system These coordinates may sometimes coincide with the Cartesian coordinates of some of the particles For example, in a system o f two rigidly connected points, these coordinates can be chosen in the following way: the position o f one o f the points is given in Cartesian coordinates, after which the other point will always

be situated on a sphere whose centre is the first point T h e position o f the second point on the sphere may be given by its longitude and latitude T ogether with the three Cartesian coordinates o f the first point, the latitude and longitude of the second point completely define the position o f such a system in space

14 CHAPTER 1 THEORETICAL MECHANICS

For three rigidly bound points, it is necessary, in accordance with the method just described, to specify the position of one side o f the triangle and the angle of rotation o f the third vertex about that side

T he independent parameters which define the position o f a mechanical system in space are called its generalized coordinates We will represent them by the symbol qa, where the subscript a signifies the num ber o f the degree o f freedom

As in the case of Cartesian coordinates, the choice o f generalized coordinates is to a considerable extent arbitrary It must be chosen so that the dynamical laws o f motion o f the system can be formulated as conveniently as possible

Sec 2 Lagrange's Equation

In this section, equations o f motion will be obtained in terms o f arbitrary generalized coordinates In such form they are especially convenient in theoretical physics

N ewton’s second law

M otion in mechanics consists in changes in mutual configuration of bodies in time In other words, it is described in terms o f the mutual distances, or

lengths, and intervals of time As was shown in the preceding section, all motion

is relative; it can be specified only in relation to some definite frame of reference

In accordance with the level of knowledge o f his time, Newton regarded the concept o f length and time interval as absolute, which is to say that these quantities are the same in all frames of reference As will be shown later,

N ew ton’s assumption was an approximation It holds w?hen the relative speeds

of all the particles are small compared with the velocity o f light; here Newtonian mechanics is based on a vast quantity of experimental facts

16 CHAPTER 1 THEORETICAL MECHANICS

In formulating the laws o f motion a very convenient concept is th e material particle, that is, a body w'hose position is completely defined by three: Cartesian coordinates Strictly speaking, this idealization is n o t applicable to a n y body Nevertheless, it is in every way reasonable when the m otion ot a body is sufficiently well defined by the displacement in space of any of its particles (fo r example, the centre of gravity o f the body) and is independent of rotations o r deformations the body

If we start with the concept of a particle as the fundamental entity of mechanics, then the law of motion (Newton’s second law) is formulated thus:

T he quantity m involved in equation (2.1) characterizes the particle and iscalled its mass.

Force and mass

Equality (2.1) is the definition of force However, it should not to be regarded as a simple identity or designation, because (2.1) establishes the form

of the interaction between bodies in mechanics and thereby actually describes a certain law o f nature T he interaction is expressed in the form o f a differential equation that includes only the second derivatives of coordinates with respect to the time (and not derivatives, say, of the fourth order)

In addition, certain limiting assumptions are usually made in relation to the force In Newtonian mechanics it is assumed that forces depend only on the mutual arrangement of the bodies at the instant to which the equality refers and

do not depend on the configuration o f the bodies at the previous times As we shall see later, this supposition about the character of interaction forces is valid only when the speeds of the bodies are small compared with the velocity of light

18 CHAPTER I THEO RETICAL M EC H A N ICS

The quantity m in equation (2.1) is a characteristic o f the body, its ma:ss Mass may be determined by comparing the accelerations which the force imparts to different bodies; the greater the acceleration, the less the mass In order to measure mass, some body must be regarded as a standard T he choice

of a standard body is completely independent o f the choice o f standards of length and time This is what makes the dimension (or unit of measurement) of mass a special dimension, not related to the dimensions o f length and time

T he properties o f mass are established experimentally Firstly, it can be shown that the mass of two equal quantities of the same substance is equal to twice the mass o f each quantity For example, one can take two identical scale weights and note that a stretched spring gives them equal accelerations If we join two such weights and subject them to the action o f the same spring which has been stretched by the same the amount as for each weight separately,, the acceleration will be found to be one half what it was It follows that the overall mass o f the weights is twice as great, since the force depends only on the tension

of the spring and could not have changed Thus, mass is an additive quantity, that

is, one in which the whole is equal to the sum o f the quantities o f each part taken separately Experiment shows that the principle o f additivity of mass is also applied to bodies consisting of different substances

ADVANCED ENGLIS1 I FOR PHYSICISTS 19

In addition, in Newtonian mechanics, the mass o f a body is a constant quantity which does not change with motion

It must not be forgotten that the additivity and constancy of masses are properties that follow only from experimental facts which relate to very specific forms of motion For example, a very important law, that o f the conservation of mass in chemical transformation involving rearrangement o f the molecules and atoms of a body, was established by M V Lomonosov experimentally

20 CHAPTER 1 THEORETICAL M EC H A N IC S

Like all laws deduced from experiment, the principle o f additivity o f nnass has a definite degree o f precision For such strong interactions as take place in the atomic nucleus, the breakdown of the additivity o f mass is apparent

We may note that if instead of subjecting a body to the force o f a stretched spring it were subjected to the action of gravity, then the acceleration of a b*ody

o f double mass would be equal to the acceleration of each body separately F rom this we conclude that the force of gravity is itself proportional to the mass of a body Hence, in a vacuum, in the absence of air resistance, all bodies fall with the same acceleration

Inertial frames o f reference

In equation (2.1) we have to do with the acceleration o f a particle T h e re is

no sense in talking about acceleration without stating to which frame }f reference it is referred For this reason there arises a difficulty in stating the caiu;e

of the acceleration This cause may be either interaction between bodies o r it may be due do some distinctive properties o f the reference frame itself For example, the jolt which a passenger experiences when a carriage suddenly stiops

is evidence that the carriage is in nonuniform motion relative to the earth

ADVANCED ENGLISH FOR PHYSICISTS 21

Let us consider a set of bodies not affected by any other bodies, that is, one tha is sufficiently far away from them W e can suppose that frame o f reference exiits such that all accelerations of the set o f bodies considered arise only as a result o f the interaction between the bodies This can be verified if the forces satiify N ew ton’s third law, i.e., if they are equal and opposite in sign for any pair

of {-articles (it is assumed that the forces occur instantaneously, and this is true onl/ w'hen the speeds o f the particles are small compared with the speed of transmission of the interaction)

A frame o f reference for which the acceleration o f a certain set o f particles defends only on the interaction between these particles is called inertial frame (o' inertial coordinate system) A free particle, not subject to the action o f any otker body, moves, relative to such a reference frame, uniformly in a straight

line or, in everyday language, by it own m omentum If in a given fra m e.1 of reference N ew ton’s T hird law is not satisfied we can conclude that this is m o t

an inertial system

22 CHAPTER 1 THEORETICAL MECHANKICS

Thus, a stone thrown directly downward from a tall tower is deflectted towards the east from the direction of the force o f gravity T his direction can be independently established with the aid of a suspended weight It follows that tthe stone has a com ponent o f acceleration which is not caused by the force of tthe earth’s attraction From this we conclude that the frame o f reference fixed in tthe earth is noninertial T he noninertiality is, in this case, due to the diurnal ro ta tio n

of the earth

On thcforcc o f friction

In every day life we constantly observe the action o f force that arise from direct contact between bodies T he sliding and rolling o f rigid bodies give rise to force o f friction T he action o f the force cause a transition o f the macroscopic

ADVANCED ENGLISH FOR PHYSICISTS 23

motion of the constituent atom s and molecules This is perceived as generation o f heat Actually, w hen a body slides an extraordinarily complex process of interaction occurs between the atoms in the surface layer A description o f this interaction in the simple terms o f frictional force is a very convenient idealization for the mechanics of macroscopic motion, but, naturally, does not give us a full picture of the process T he concept o f frictional force arise as a result o f a certain averaging of all the elementary interactions which occur betw een bodies in contact

In this part, which is concerned only with elementary laws, we shall not consider averaged interactions where motion is transferred to the internal, microscopic, degrees o f freedom o f atoms and molecules Here, we shall study only those interactions which can be completely expressed with the aid of elementary laws o f mechanics and which do not requyre an appeal to any statistical concepts connected with internal, thermal motion

24 CHAPTER 1 THEORETICAL M E C H A N I CS

Ideal rigid constraints

Bodies in contact also give rise to force of interaction which cam be reduced to the kinematic properties of rigid constraints If rigid contains actt in a system they force the particles to move on definite surfaces Thus, in Sec 1 we considered the motion o f a single particle on a sphere, at the centre o f which» was another particle

This kind o f interaction between particles does not cause a transition of the motion to the internal, microscopic degrees of freedom o f bodies In o th er words, motion which is limited by rigid constraints is completely described b y its own macroscopic generalized coordinate qa

ADVANCED ENGLISH FOR PHYSICISTS 25

If the limitations imposed by the constraints distort the motion, they thereby cause accelerations (curvilinear motion is always accelerated motion since velocity is a vector quantity) This acceleration can be formally attributed

to forces which are called reaction forces of rigid constraints

Reaction forces change only the direction of velocity o f a particle but not its magnitude If they were to alter the magnitude of the velocity, this would produce a change also in the kinetic energy of the particle According to the law

of conservation of energy, heat would then be generated B u t this was exclude from consideration from the very start

'T o summarize, the reaction force of ideally rigid constrains do not change the kinetic energy of a system In other words, they do not perform any work on

it, since work performed on a system is equyvalent to changing its kinetic energy (ifheat is not g en erated )

In order that a force should not perform work, it must be perpendicular to the displacement For this reason the reaction forces o f constraints are perpendicular to the direction o f particle velocity at each given instant of time

26 CHAPTER 1 TH EO RETICA L M ECHAM ICS

However, in problems o f mechanics the reaction forces are not initially given, as are the functions o f particle position T hey are determ ined by integrating equations (2.1), with account taken o f constraint condition Therefore, it is best to formulate the equations o f mechanics so as to exclude constraint reaction entirely It turns out that if we go over to generalized coordinates, the num ber of which is equal to the n um ber o f degrees o f freedom

of the system, then the constraint reaction disappear from the equations In this section we shall make such a transition and wall obtain the equations of mechanics in terms of the generalized coordinates o f the system

ADVANCED ENGLISH FOR PHYSICISTS 27

The transformationJrotn rectangular to generalized coordinate

VVc take a system with total of 3N = n Cartesian coordinates of which v are independent W e shall always denote Cartesian coordinate by the same letter x1; understanding by this symbol all the coordinates x, y, z ; this mean that i varies from 1 to 3N, that is, from 1 to n The generalized coordinates we denote by

q ( l < a < v ) Since the generalized coordinates completely specify the position of the system, x, are their unique function:

Xi = X i ( q i , q 2 , q 3 , q a , q r ) ( 2 2 )

From this it is easy to obtain an expression for the Cartesian components

of velocity Differentiating the function of many variables x ,( qa) with respect

28 CHAPTER 1 THEORETICAL M E C H A N IC S

not convenient to use this convention for the Latin characters which denote th e

dx ■Cartesian coordinates) T hen the velocity - j- 5- can be rewritten thus:

dXj _ dxj dqa

dt 5qa dtHere the summation sign is omitted

ADVANCED ENGLISH FOR PFIYSIC1STS 29

30 CHAPTER 1 THEORETICAL MECHANJI CS

Potential o f force

W e now consider components of force In many cases, th e th re e components o f the vector o f a force acting on a particle can be expresse d in terms o f one scalar function U according to the formula:

o \ i

Such a function can always be chosen for the force o f N ew to n ian attraction, and for electrostatic and elastic forces T he function U is calledl the potential o f the force

It is clear that by far not every system o f forces can, in the general caste, be represented by a set o f partial derivatives (2.7), since if

ADVANCED ENGLISH EOR PHYSICISTS 31

Expression (2.7) defines the potential function U to the accuracy of an arbitraty constant term U is also called the potential energy of the system Forcxanple, the gravitational force F = - mg, where g~980 c m /se c : is the acce eration o f a freely falling body and z is the height to which it has been raised It can be calculated from any level, which in the given case corresponds to a dete'mination of U to the accuracy of a constant term A more precise expression for tie force o f gravity than F = - mg (with allowance made for its dependence on heigit also admits o f a potential, which we shall derive a little later

if the displacements are compatible with the constraints Indeed, (2.8) expresses precisely the work performed by the reaction forces for certain possible displacement o f the system; but this work has been shown to be equal to zero.32 CHAPTER 1 TH EO RETICA L M E C H A N IC S

ADVANCED ENGLISH FOR PHYSICISTS 33

ot a general displacement

dx *After multiplication by — L and summation, the left-hand side of equation

dqy(2.S) can be written in a more compact form, without resorting to explicit Cartesian coordinates It is precisely the purpose of this section to give such an improved notation T o do this we express the kinetic energy in terms of • generalized coordinates:

34 CHAPTER 1 THEO RETICA L M ECHANIICS

in the given position in which the coordinate has a given value q„, it is possible to

ADVANCED ENGLISH FOR PHYSICISTS 35

imp:rt to the system an arbitrary velocity q„ permitted by the constraints

cHTNaturally, qa and q ^ a are also independent It follows that in calculating -all tie remaining velocities qp*a and all the coordinates, including qa, should beregarded as constant

Let us calculate the derivative - In the double summation (2.13), the

by 5 and in the second by a T hey can be combined, replacing (3 by a in the firs' summation; naturally the value o f the sum does not change due to renaming

o ftie summation sign T h e n we obtain

+ qa L m i ^ W r qp

Here we have had to write down the derivatives o f each o f the three factors

of all the term in the summ ation (2.14) separately

36 CHAPTER 1 THEORETICAL MECHAN7ICS

dt 3qy dqy g dqa dqy ^ ^

However, the expression on the right-hand side o f (2.17) can also be

dx ■

obtained from (2.9) if we multiply its left-hand side by — - and sum over i For

dqy d\J

this reason, (2.17) in accordance with (2.10), is equal t o - Thus we find

dqv

( 2 1 g )

dt dqv dqy dqy

In mechanics it is usual to consider interaction force that are independent

o f pirticle velocities In this case U does not involve q a , so that (2.18) may berew itten in the following form:

d dL dL _ ,

- = 0 , U a < v

38 CHAPTER 1 THEORETICAL M ECfiA IN ICS

These equations are called Lagrange’s equations Naturally, in (2.21)) L is

considered to be expressed in terms of qa and q a , the Cartesian co o rd in atesbeing excluded It turns out that this type o f equation holds also in cashes w h e n the forces depend on the velocities

The rules f o r form ing Lagrange's equations

Since the derivation o f equation (2.2 1) from N ew ton’s second law is; not readily evident we shall give the order o f operations which, for this given system, lead to the Lagrange’s equations

1) T h e Cartesian coordinates are expressed in terms o f gen<eralli;ed coordinates:

Xj = Xi(qi, ,qa, qr)

2) T h e Cartesian velocity components are expressed in term s of generalized velocities:

d x:

ADVANCED ENGLISH FOR PHYSICISTS 39

3) T h e coordinates are substituted-in the expression for potential energy

so that it is defined in relation to generalized coordinates:

5) T he partial derivatives - a n d -are found

5<la Sqa

6) Lagrange’s equations (2.21) are formed according to the number of degrees o f freedom

40 CHAPTER 1 THEORETICAL M EC H A N IC S

PART 1.2: Vietnamese-English translation

1 Hệ quy chiếu là một hệ mà đói với nó chuyến động được mô tả T h í dụ,

vị trí cùa m ột xe lửa được cho đói với hệ tọa độ gán với nhà ga

2 Bậc tự do của m ột hệ cơ học là số các tọa độ độc lập xác định vị trí của

hệ đó trong không gian

3 Vị trí của một vật rân trong không gian luôn được xác định bởi ba điểm không nằm trên m ột đường thẳng Ba điếm này có sáu bậc tự do T ừ đây rút ra rằng m ột vật rân bát kỳ có sáu bậc tự do

4 Các tham sổ độc lập xác định vị trí của của hệ cơ học trong không gian được gọi là tọa độ suy rộng Chúng ta biếu diỗn chúng bởi ký hiệu qa,

trolly đó chỉ số cc ký hiệu số bậc tự do của hệ.

5 Chat hạt là vật the mà vị trí của nó hoàn toàn được xác định bởi ba tọa

độ Dcscartcs Đó là một khái niệm rát thuận tiện đói với mò tả các định luật chuyển dộng

6 Tương tác dược biếu diên dưới dạng phương trình vi phân chỉ chứa dạo hàm bậc hai của tọa liộ theo thời gian chứ không chứa các đạo hàm bậc cao hơn

7 N^ười ta láy hai vật thế dõng kich thước, trọng lượng và một lò xo bị căng sẽ cho chúng các gia tóc như nhau Nếu nỗi chúng với nhau và dưa chủng vào chuyển dộng bởi cùng một lò xo bị càng với cùng một lực như đỗi với từng vật riêng biệt, thì gia tóc nhận dược sẽ là một nửa gia tóc mà nó có T ừ dầy rút ra ràng khói lượng tống của các vật thế là lớn gấp hai lấn do lực chi phụ thuộc vào dộ căng của cùa dây lò xo và không

bị thay dồi