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Notices

13. Lecture on Mechanics: Specification of State

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About Monday Classes and Problem Sessions

From now onwards we will have the following arrangement for the classes. On Wednesdays  and Fridays there will be lectures and on Mondays both the lectures will be reserved for problem solving sessions. Afternoon class we will have tutorials like we had on the last Monday. This means that the problems will be given to you in advance and you have to solve the  problems in the class and submit before leaving. In addition one person will be called to solve one question the problem on the board, then next problem will be solved by another person an so on. Others will have the choice of doing it yourself on the paper or of following the solution being worked out on the board. The person who comes to the board will get all the help from me and from Vijay. This will ensure that every one finishes the problems in time. I will call you randomly and will make sure that every one comes to the board at least once and solves at least one problem eventually sometime during the semester. So that is the arrangement for the afternoon session. What you do in the afternoon session is to be submitted. All tutorials will be evaluated and returned. They will make up one unit of assessment of 20 marks.


Morning sessions will be something similar again. We will be solving problems except that you work in groups. Again the problems will be given in advance. What is discussed in the mornings need not be submitted. Last Monday we could discuss only one problem in one hour that is not very efficient. I want this discussion session to be kept under control. So what I suggest is the following. For the morning sessions you work on the problems in groups, discuss and thrash them out in advance. One member of a group will be asked to present solutions of one or two problems, then next a member from some other group chosen randomly will be asked to solve next problem and so on. This is primarily to encourage group activity. You should have some amount of activity discussing Physics among yourself. In India we lack team work. In many organizations people are expected to  work as a team. You can see results where ever there is team work. You are expected to solve the problems in advance and come prepared to present solutions. This will also help you to prepare in facing an audience of fifty to sixty and in making a presentation of your ideas to them. The solutions will have to be presented and explained to the class as if you are the teachers and you are explaining it to the class. This part will be evaluated in some way which will be known to you on Monday. The part that should make it interesting for you is that the whole class will be participating in evaluation in the morning sessions. Thus all of you will have to submit a evaluation report for every Monday morning session. This will be considered as group activity and your evaluation will be included for grading. All group activities put together will constitute one unit of evaluation and will carry 20 marks.  The problems in the afternoon sessions will be primarily from H.C. Verma's book.  The problems in the morning sessions will be on the lectures which I will be giving on Wednesdays and Fridays. The problems in the morning sessions will be similar to what you have solved in the tutorial on rotations and you may not find them in any other book. These problems will be designed to give you supplementary material and provide you help in understanding the lectures. 

The other two lectures, on Wednesdays and Fridays will constitute a complete, coherent course by itself, delivered in a logical order. It will be at a slightly higher level. An attempt will be made to consolidate and upgrade what you have already done in 12th standard. Also I will make an effort to present new topics which are not really difficult and are accessible to you
with your level of Mathematics and Physics. Many of these problems do not  form part of the text books at 12+ level. Traditionally, the topics, I hope to cover, are included in higher level courses, typically in M.Sc. or B.Sc. honours level courses in Mechanics. What ever can be brought to your level will be included. You do not get worried in the sense that the course will become Hi-Fi or an advanced course. I am only trying to optimise the returns you get for what you already know. 


I am attempting to take you to higher level without being too demanding on preparation needed. The results on rotations that have been derived for you is an example, it makes use of only vector algebra and some calculus and does not use any matrices. Normally, I have presented this part of lectures using matrices to M.Sc. students. Only yesterday I discovered that the central result on transformation of coordinates under a rotation can be made much simpler and transparent and now it will be part of an assignment.  


I will also make an effort to put a draft of lecture notes in advance on the Moodle Course Site. Later the draft will be modified and will be brought as  close to the class room lecture as is possible. I will try to synchronise the lecture delivery in the class and the internet and reduce the phase difference as much as possible. Right now I have already put some lecture notes. Some of them are older versions. Some others are up to date. What I put up on the internet will be almost same, within about 95 to 98\% of what I do in the class. So that you get everything. There may be some improvements and and some  2 to 3 % additional remarks may be added. The only problem is time lag and I will make every effort to minimize it.

 

Let us then begin with the main topic of discussion In the  Newtonian formalism, the Newton's Law play a central role. We have seenthat in order to use the laws to predict the behaviour of mechanical systems, we need to introduce frames of reference. It was also noted that the use of vectors offers important advantages.

Structure of Physical Theories
Here we give an overview of the general  structure of formalism of Newtonian Mechanics. There are following four major components to be described in the language of the Newtonian Mechanics.

  •   State of a physical system
  •   Physical quantities of the dynamical variables
  •   The laws of motion or the laws of evolution
  •   The  forces or the interactions


States of Physical system
The idea of {\it state} of  a physical system has already been introduced in the previous lecture. One of you had observed that the displacement is a state function and does not depend on the path. 

What is meant by state of a system?  If you are already familiar with the concept I will build upon your understanding.

Dialogue with students

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The state concept is an important concept that will keep coming again and again. It is the information about the system that is needed to specify it at a  particular time. The specification of the state of a point particle in \underline{Newtonian mechanics} at any time, $t_0$,  means specifying the values  of its position vector, $\vec{x}$ and momentum vector $\vec{p}$ at that time. One criterion is that knowing the state at any time enables us to  compute every other observable quantity such as energy momentum angular momentum of the system.

Let me take an example that is familiar to you. The example of an ideal gas. For this case the state is completely specified by giving  any two of the three quantities pressure, volume and temperature. If you know temperature and volume you can compute pressure or internal energy or any other thermodynamic function. That forms one requirement on specification of the state.

The other criterion is that state specification should be such that knowing
the physical laws, given the laws of dynamics, it should be possible in
principle to find the state of the system at a later time.

These two criteria which specify the state of a system. So I summaries the two
requirements:
  •  The state is plete specification of the system at a time, complete in the sense that any other physically measurable quantity of the system can be computed.
  • Using the equations governing the time evolution  and the knowledge of the system at a particular initial instance of time allows us find the state at a later time .
Let me now give consider examples. Consider the simplest example of a point particle. 

 

Dialogue with students

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For a simple pendulum the state is completely specified by the angle that the string makes with vertical and the angular velocity at that position. It can also be specified by means of the horizontal displacement and the velocity.

As other examples of specifying the state, for a system of $n$- particles, the state will be {\it completely specified} by giving position and momentum vectors  of all the particles.  
Now we will take a little more complicated example. May be you can think of some examples.
  •  A moving cart
  • A/S:=  A satellite
  • Q/Me:=In all these examples we treat the bodies as point particles  unless we want more details.}
  • A/S := Rotating rigid bodies.
  • Q/Me:= This is a good example. Examples of rigid bodies are flywheel, spinning top. We will discuss it later in the course.
  • Q/Me:=Charged particle, that is also a point particle. This is   slightly different kind of system. I will keep it. From the  point of view of mechanics, it is not different from a point  particles.
  • R/Me:=Give me more complicated examples.
OK Let us proceed. For a rigid body, we will see that besides the position and momentum of the centre of mass of the body, one needs to consider three parameters that specify the orientation of the body and its angular velocity to specify the  state of the system.

A more complex example is that of a vibrating string held fixed at the end points.
 
 
 SOem text herebetween two accordions.
 

 

Dialugues with students

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Thus we need to know the displacement and the velocity of each point of the string. How many pieces of information are needed for a vibrating string? One, two, three, four ? How many ? We need infinite number of values,  two for each point of the string. We need position and velocity of each point and the number of points of the string is infinity.

The state of different systems will be specified in different ways. For the same system it may happen that the state can be specified in several different ways, remember the ideal gas. 

Different theories may be needed  to explain different phenomena involving the same system. In such a case, the answer depends on the system but also what theory is needed to describe the system. For example, we are studying mechanics now. Today, for an electron going round the nucleus the state of the electron is given by its position and velocity.  But in the fourth semester, when you learn quantum mechanics, you will describe the state of the electron by a "wave function". 

Physical Observables
These are functions of coordinates and velocities and are measurable quantities such as energy momentum and angular momentum. \subsection*{Laws of Motion and Time evolution} The Newton's laws of motion govern the behaviour of mechanical systems. These laws tell us when the system will be in equilibrium and when it will be moving or changing with time. The specification of the state refers to the system at one time only. Most often we are interested in the changes that take place with time, how position or velocity is changing with time. The laws of  a physical theory, such as Newtonian Mechanics, tells all about this change, also referred to as {\it time evolution}. {\it Predicting  time  evolution of a system means finding out the state of the system at a later time $t$, when the state of the system is known at an initial time $t_0$. The three Laws gives us all the information covering the 'rules of the game' to obtain the time evolution of a system.}

Interactions
For a physical system the knowledge of the laws of motion and the state of the system at initial time alone is not sufficient to predict  future behaviour of the system. What else is needed. It I give you the position and velocity of the earth at an initial time $t_0$ and all the mathematical machinery that you want,  can you predict the position of the Earth after time $t_0$ using the Laws of motion? Is there any other information that is needed? The answer is that we need to know about the {\it force} that the Sun exerts on the Earth to compute the Earth's orbit. The {\it interactions} is  a general term for the concept, such as force, which controls the behaviour of a system and will be different for different systems, and from one situation to another. A charged particle falling in Earth's gravitational field experiences a different kind of interactions than the same particle moving in an electric or a magnetic field.



It must be emphasised that all the components, described  above, will be present in all the major formalisms of physical theories}. For example, we need a description of state, laws and EOM, and knowledge of interactions to complete description of systems in all major theories such as the ones listed below.

  • Classical Mechanics 
         States : Generalised coordinates
         Laws, EOM : Action Principle Lagrangian EOM, Hamiltonian EOM
         Interactions : Potential Energy, Lagrangian, Hamiltonian or Action
  • Statistical Mechanics, Thermodynamics
         States: Variables such as pressure, temperature and Volume for a gas\\
         Laws, EOM : Laws of Thermodynamics, ...
         Interactions : Hamiltonian
  •  Quantum mechanical system,
         States: Wave function ; 
         EOM: Schrodinger equation;
         Interactions: Hamiltonian, Lagrangian 
  • Charges in presence of electromagnetic fields 

 

 Here I have  a few questions for you to think :

  1.  What will be needed to describe the state of torsional pendulum at a given time? state of vibrating string at a given time?
  2.  Do you think equations of motion for a vibrating string can be derived from    Newton's Laws?
  3. You have an experiment on torsional pendulum in Physics-1 lab and an experiment on Kater's Pendulum. Do you think that the EOM of motion for these also can be obtained from the Newton's Laws?