Let us spend a little time today digressing
to a topic in dynamical systems more properly taught in courses on physics namely Hamiltonian
systems and I would like to bring out to you how special Hamiltonian systems are and we
studied Hamiltonian systems from the point of view not so much of mechanics as of our
general study of dynamical systems and nonlinear dynamics before we talk about Hamiltonian
systems there is another class of dynamical systems which I would like to dispose off
or at least mention very briefly. In passing and perhaps we might come back
to such a system little later in the course and this is a class of dynamical systems called
gradient systems. We call our particular kind of dynamical system
we focused on namely those which satisfy autonomous couple differential equations of the form
f of x where F is a prescribed vector field it is a function which has really got n components
in n dimensional phase space F1 F2 F3 up to F sub N and these are functions of the coordinates
of the variables x1 x2 up to x sub n therefore to specify the dynamical system you have to
specify a vector field at every point in some region of phase space a gradient system is
one where. But this vector field has a very special form
if this vector field is of the form the gradient of some scalar function of the variables x
I call this a gradient system what's special about it of course is the fact that instead
of specifying a vector field which means all its components n functions f1through to xn
the F sub n we need to specify just a single scalar function ? and everything else all
the components of F are determined in terms of this single scalar function ?, so clearly
it is a very special kind of dynamical system. Not all dynamical systems can be written in
this form but those that can clearly enjoy a little superior status among dynamical systems
because a single scalar function suffices to tell you the flow at all points, now of
course the component of this if I take the ith component this implies that xi. ?? / ?x,I
and ? is a function of the variables x that is what is meant by a gradient system if you
give me the function ? and I set ? of x equal to a constant it would in general be a surface
in this n dimensional phase space, so if schematically this is what the surface looks like at a particular
point x and this corresponds to the surface ? equal to a constant c say then what is the
direction of the gradient of ? at this point the gradient is defined.
As the normal derivative the gradient of any scalar field is the normal derivative it is
the direction in which this function ? has the maximum rate of change that direction
along which it changes most rapidly therefore it is normal to the direction of the level
surfaces of ?, so if I equal to constant is a surface the level surface ? equal to constant
then this is the direction of the gradient of ? at that point which means
the flow lines or the phase trajectories of a gradient system are always normal to the
level surfaces ? equal to constant at each point.
That is a useful factor now e.0 come across too many gradient systems they are somewhat
rare in a certain sense but when they when they do occur then it is very useful to give
a geometrical interpretation for the flow lines in this in this sense namely the lines
? equal to the flow lines are normal to the level surfaces of the scalar function phi
single scalar function ? this is another very important class of dynamical systems called
Hamiltonian systems for which once again a single scalar function determines the vector
field F on the right hand side and now let me define a Hamiltonian system. These are systems which are always even dimensional
in a very specific manner they described by a set of dynamical variables which occur in
pairs therefore the total number of variables is always an even number the first n variables
are generally denoted by the Q's q1 q2 upto qn and these are generally called usually
called generalized coordinates and along with them go a pair a set of variables
p1 p2 up to pn which are called generalized momentum therefore the full set of dynamical
variables x. Is a set of two and variables q1 to qn p1
to pn to n dynamical variables the phase space of a Hamiltonian system is always even dimensional
it is conventional to call Q's the generalized coordinates degrees of freedom, so this system
has n degrees of freedom and it has to end dynamical variables something very special
connects each Q to its corresponding P called its conjugate momentum P and that is a structure
called the possum bracket and I will define this as we go along very shortly for the moment.
We will use this terminology which says that the Q's and PS are conjugate to each other
q3 is conjugate to p3 q 4 is conjugate to p4 and, so on the more strict term is canonically
conjugate in a standard manner and we will explain as we go along what we mean by this
the equations of motion are of particular interest they call Hamilton's equations of
motion and the most common occurrence of Hamiltonian systems is a set of particles moving for example
under the influence of forces. Which they exert on each other conservative
forces which they exert on each other or a set of particles moving under the influence
of some conservative external force or both mechanical systems very often turn out to
be Hamiltonian systems in the absence of friction in the absence of dissipation, so Hamiltonian
systems will in the sense we understand it will be conservative dynamical systems with
the special cases of conservative dynamical systems and the equations of motion which
correspond to our dynamical equations go by the name of Hamilton's equations.
And they read as follows qi . is ? /? Pi of a certain scalar function H of all the Q's
and PS in general and let me abbreviate that by just writing it as q,p that stands for
all two and variables in general and this is true for each eye from 1 to N qi. and the
rate of change of Pi is again given by the partial derivative of the Hamiltonian but
with respect to the conjugate generalized coordinate qi with a minus sign and this set
of equations is valid for each eye it is not a gradient system because the gradient system
would imply. A gradient system would imply that each time
derivative of each dynamical variable is the partial derivative of some scalar function
with respect to the same variable, but that is not happening here the time derivatives
of the Q's are partial derivatives with respect to the conjugate P's and vice-versa but with
a minus sign and this minus sign is crucial it is very, very important this is the general
structure of equations when you look at particles moving in a potential for instance yeah there
is a question. What exactly the question is where do these
things come from this is the definition of a Hamiltonian system but as soon as I finish
defining this I will go back to ordinary mechanics and show you that ordinary mechanical problems
in the absence of frictional forces these are the defining equations these are Hamilton's
equations they define a Hamiltonian system the original motivation was of course the
observation that Newton's equations of motion in the case of conservative forces are essentially
Hamiltonian equations and this is a generalization of the idea.
Which already one already has from Newton's equations of motion, so let me do this in
a formal way and then we go back in a minute and see that ordinary mechanics is indeed
a special case of Hamilton's equations of motion with a specific prescription for the
Hamiltonian function but right now I am doing this in a very abstract way I am simply saying
that if a dynamical system is even dimensional and has this kind of structure with certain
additional properties which I am going to write down regarding the way the Q's and PS
are related to each other then it is called a Hamiltonian system okay.
So let us step back and ask is this a gradient system no not as it stands but you still have
this wonderful property that the entire vector field on the right hand side is actually defined
in terms of determined by a single scalar function the Hamiltonian function that is
going to cause a great deal of creator a lot of simplification it is going to lead to a
lot of simplification and very interesting properties, now let us go back and ask what
happens if I take a single particle which is non relativistic.
Ordinary Newtonian mechanics moving in some conservative potential or field of force what
would you say are the equations. Of motion a single particle moving in some potential
I could write down Newton's equations of motion and the way to write it down in our language
is to say that the mass times the velocity which is dr/dt is the momentum p and the rate
of change of momentum dp over dt is equal to the force on the particle if this force
is time independent no explicit time dependence and depends only on the instantaneous coordinate
of the particle then it's some F of R and if this force is conservative which means
it is the gradient of some scalar potential which has the connotation of a potential energy.
Of the particle then this is equal to minus the gradient of a potential V which is a function
of the coordinates of the particle this is what you would write down as normal equations
of motion I have just written Newton's equations down but as a set of coupled equations the
r/ dt is p/ m I could bring this to the other side and dp /dt is some function of R it certainly
is a dynamical system in the sense we understand dynamical systems what is the dimensionality
of the phase space here it is six dimensional because three coordinates and three momentum
variables. So six coordinates and it is exactly the structure
that we had x. is some f of x some vector field on the right hand side, so we have written
it as two coupled vector equations three dimensional vector equations in this fashion the question
is it of that form is it of this form at all can we write it in the form of Hamilton's
equations by hindsight we know we can because we know that in ordinary mechanics with conservative
forces the Hamiltonian of the particle as the connotation of the total energy of the
particle. Which is the sum of it is kinetic energy and
its but initial energy, so it right H as a function of r and p and these would be q1
q2 and q3 in this case and this the components of p would be p1 p2 and p3 these are Cartesian
components then this is equal to p2 over twice the mass of the particle plus the potential
V of r and if I call these components q1 q2 andq3
and similarly this has components p1p2 and p3 it is immediately clear that these equations
here really imply that qi. is indeed equal to ? H /? pi.
Because if I want to find out what d q2 /dt is we know from this set of equations here
that d q2 / dt is p2 over m and that is precisely ? H /? p2 and it also tells us that pi. is
- ? H / ? qi since the potential energy depends only on the coordinates when you differentiate
with respect to the coordinates in Hamilton's equations you end up with the corresponding
component of the force so this is an example of a Hamiltonian system and the way I have
defined it there in general it generalizes the same idea to n degrees of freedom and
the simplest way of thinking. Of a Hamiltonian system is to look at the
special case of a set of particles moving in a conservative field of force and that
is an example the standard example of a Hamiltonian system there are many other Hamiltonian systems
as well, but that is the simplest example so Newton's laws of motion for conservative
systems are definitely Hamiltonian the prescription for writing the Hamiltonian down in conventional
mechanics is fairly straightforward you start with the Lagrangian of the particle and then
you make a certain transformation. And go over to the Hamiltonian and generally
in elementary problems it turns out to be the sum of t + V it is the sum of T and V
but T is a kinetic energy and V is the potential in that is the way we have written it down
here but it is much more general than that this CAF folding is not necessary a Hamiltonian
system is defined by the set of equations Hamilton's equations that I wrote down the
general equations together with a certain connection a certain structure called the
Poisson bracket relationship between the Q's and PS and let me define the Poisson bracket. Given the dynamical variables q 1through qn
and p1 through pn if I look at any function of these dynamical variables let us call them
a of q and p and any other function of all the dynamical variables B of Q and P then
the Poisson bracket of a with B is denoted by curly brackets A, B and by definition it
is equal to a summation over all the degrees of freedom 1 to N ?? a / ?? qi ?? B/ ?? pi
minus ??A / ??Pi this is the definition of the Poisson bracket of a with B we need to
understand a little more about the structure involved here.
But it is these partial derivatives in this particular combination if the Q's by assumption
the Q's are independent of each other and the peas are independent of each other then. It follows if these are independent dynamical
variables it follows that ??qi / ??qj is 0 unless I equal to J in which case it is equal
to1 therefore this is equal to the ?? ij similarly ??pi / ??pj is ?? ij because
the ps are independent of each other unless I equal to J in which case it is identically
equal to 1 and the Q's and ps are independent dynamical variables, so ?? qi / ?? pj is always
equal to 0, so you have two sets of dynamical variables half of them are regarded as generalized
coordinates. And the other half as generalized momentum
they are independent dynamical variables you need all of them to describe the dynamical
system, so this is not surprising and the fact that this relationship or that holds
code simply says that different Q's are independent variables you need that many independent variables
to describe the system once you have that you could ask what is the Poisson bracket
of particular Q with a particular p, let us figure that out. So what is the Poisson bracket of qk with
pl for instance what is this equal to all you have to do is to plug it in here and do
a summation over I and compute what the answer is using the statements, we had earlier about
the P's and Q's so what do you get well this is going to become a delta ki and this is
going to become a ??LiI so it is zero unless K is equal to I and this term becomes zero
unless L is equal to I, so the answer is ?? kl because this term would anyway be zero if
you differentiate the Q. With respect to a P or a P with respect to
Q the answer is zero and therefore it is equal to the Canonical ?? what is the Poisson bracket
of QK with QL both these quantities are, now Q's in the numerator and at least one of these
will vanish therefore this is zero and the same is true for pk this set of relationships
between the generalized coordinates and the generalized moment a of a dynamical system
of a Hamiltonian system are called canonical Poisson brackets, so these are the canonical
Standard Poisson bracket relations you might ask why am i defining a Poisson bracket.
Why is this what is the use of this and we will see very shortly what the practical use
of this is but meanwhile this relationship here this way of defining Poisson brackets
has got mathematical structure to it very deep mathematical structure notice it is a
bilinear operation in some sense you give me two functions A and B of the dynamical
variables of a Hamiltonian system and I do something which involves both of them it is
a bilinear structure in general the right hand side is some other function of all the
dynamical variables. And it is also got certain interesting properties
of the following kind A with B + C if you took two functions the sum of these two functions
as the second member of the Poisson bracket it is quite clear this is equal to A with
B plus a with C it is immediately obvious that this is, so if I multiply B by some constant
which is independent of all the dynamical variables then of course the constant comes
out and a with KB is equal to K times a with P but K is some constant
and this quantity is also anti-symmetric. So you have a natural anti symmetric bilinear
structure here yes absolutely there is nothing great about it at this stage the fact is that
these things are enormous implications once you identify a Hamiltonian system then you
guaranteed certain relationships automatically, so what we have done is the other way about
I started with mechanical examples and I told you Cartesian coordinates are independent
of each other and I wrote down certain relationships between them but the fact is that structure
generalizes much more general context than just the usual ones you are used to.
So simply that it is a very useful and a very general frame work for understanding a huge
class of dynamical systems of conservative dynamical systems right now we are talking
about some mathematical properties of Poisson brackets and we are writing, now relationships
which are fairly trivial they follow fairly straight forwardly from the independence of
the coordinates and the moment a between each other plus the canonical Poisson bracket relationships,
but these have deeper implications as we will see. What is the Poisson bracket of a with a product
of two functions so if I took A width BC and asked what this is equal to you would have
to use BC here and BC here and use the chain rule of differentiation repeatedly and then
the simple exercise to show that it is A with b times c plus b times let us see this is
a chain rule for differentiating functions that is all that's been used and then it leads
to an identity of this kind what would this be what could A B with C D be equal to what
will you get on the right hand side. You get four terms so first treat a B as a
unit and use this formula for decomposing it into Poisson brackets of AB with C and
AB with D and then use the formula again for decomposing this into two more terms, so you
get a set of four relationships so this is four terms on the right hand side
there is one more property of Poisson brackets which is crucial importance and that is the
following if you took three functions a b c of the dynamical variables and computed
a with the Poisson bracket of B with C and then took the Poisson bracket of a with this
plus the Poisson bracket of B. With the Poisson bracket of C with A plus
the process on bracket of see with the Poisson bracket of a with B, what would you guess
is the answer it is zero it turns out to be identically zero on using this property, so
there is this anti symmetry when you take three of these guys the anti symmetry of the
Poisson bracket leads to this property it is easy to establish do these properties you
have got a set of objects a b c etc. Which are functions of the phase space variables
in a Hamiltonian system and among those functions i have defined a certain bilinear structure
called the Poisson bracket. I take two the two of them at a time and i
create something a new function that function is an T has this anti symmetry property and
it also has this property which is called the echo be identity and it has the usual properties such as a
with the Poisson bracket of A with a scalar times B constant times B is the constant times
that of a with P the Poisson bracket of a with B it has the anti symmetry property and
it has the property a with B + C is a with B +A with C does this remind you of something
does this structure remind you of something these four properties -they remind you of
anything else any other example where you have exactly the same sort of properties yes.
If you took ordinary vectors in three-dimensional space in Euclidean space and it took the cross
product of two vectors what happens then do they have identical properties. If I took vectors A B C etc. And I computed
the cross product of two vectors is this not equal to one minus B cross a and is it not
true that a cross some scalar times B is K times a cross Band certainly this is true
this is true - and is this true, if I replace the Poisson bracket operation by the vector
product or the cross product of two vectors is it not true that A x B x C + cyclic permutations
turns out to be 0 identically on the right hand side, so exactly the same set of properties
mathematically a bilinear operation which is anti-symmetric satisfies these properties.
And satisfies the ecobee identity anything else you think of any other structure which
does exactly the same any other class of objects which have similar properties suppose ABC
etc. square matrices n by n matrices of some given order and instead of the Poisson bracket
instead of A, B as the Poisson bracket I replaced it by the commentator of the two matrices
this stands for a B - B a then the commentator of two matrices is anti-symmetric the commentator
of A with B is minus the commentator of B. With A and this property is valid for commentators
and, so is this any set of elements for which such a bilinear product or bilinear operation
exists satisfying the anti symmetry property these properties as well as the ecobee identity
is called alley algebra, so it says functions of the dynamical variables of a dynamic Hamiltonian
system formally algebra under the Poisson bracket operation a very useful concept we are not going to
deal, so much with the algebraic properties here in this course.
But it is a useful thing to know that there is something deeper a deeper mathematical
structure to Hamiltonian systems under the Poisson bracket operation, so that is responsible
a lot of the properties actually are mathematical properties which have physical implications
which follow from this li algebraic structure of the functions of the dynamical variables
under the Poisson bracket operation there is one immediate use of the Poisson bracket
and that is the follower in conservative mechanical systems.
Such as those determined as where the Hamiltonian is simply determined as a kinetic energy plus
the potential energy we know that the energy is a constant of the motion I now show you
that the Hamiltonian itself for autonomous systems is a constant of the motion identically
that is easy to see because. If I start with H as a function of all the
Q's and PS and I compute D over DT of this quantity this is equal to by definition since
it is a function of the Q's and PS this is equal to a summation over the degrees of freedom
?? H over ?? Q I multiplied by Qi dot plus ?? H over ?? P I multiplied by P I dot since
it is a function of the independent variables Q 1 to Q 1 and P 1to P n its time derivative
total derivative is just this set of partial derivatives multiplied by these total derivatives,
but now we use Hamilton's equations of motion. Therefore on a solution set on a solution
of the equations of motion this becomes equal to the ? ?? H over ?? Qi and what was the
equation of motion for Qi . ?? H / ?? pi + ?? H / ?? Pi and what was the equation of
motion for Pi. – the all important -and of course this is identically equal to zero
which implies that the Hamiltonian of a Hamiltonian system an autonomous Hamiltonian system is
a C is a constant of the motion you will appreciate the importance of that relative minus sign
in producing this result what is the equation of motion of any function
of the dynamical variables that too can be written in a very simple way and that will
bring out the use of the Poisson bracket so let us suppose a is a function of the Q's
and PS and I ask what is the rate of change of this function of the Q's and PS. And I use exactly the same technique as I
used earlier what would this become this would become a summation from I equal to 1 to N
?? a over ?? qi times qi dot but qi .is ?? H over ?? Pi put that in I + ?? A over ?? Pi.
dot which is- ?? H / ?? QI, so they can get minus but what is this equal to the Poisson
bracket of a with H that is a remarkable fact it says the rate of change of any function
of the dynamical variables is just the Poisson bracket of that function with the Hamiltonian
of the system that tells you the fundamental role played by the Hamiltonian of all the
functions of the dynamical variables. Of such a system the Hamiltonian is predominant
plays a fundamental role because it controls the rate of change with time of every other
function when is this a constant of the motion yes when a Poisson commutes with H when the
Poisson bracket of a function with the Hamiltonian is zero it vanishes identically then the function
is a constant of the motion and the converse is true as well as long as we are dealing
with autonomous Hamiltonian systems of course you could generalize Hamiltonian systems to
non autonomous systems and then this is no longer true because.
If you have a non autonomous and let me take a digression for a second and talk about non
autonomous systems although would imply that there is also explicit time
dependence on this side what would happen then what would happen then what would the
right-hand side be modified to plus the partial derivative of H with respect to time the explicit
time dependence gets differentiated and of course this portion vanishes identically and
you are left therefore with is equal to ?? H / ?? T
And if H is non autonomous that right-hand side is not identically 0 and therefore the
Hamiltonian is not a constant of the motion not surprising because it is actually explicitly
a function of time and it is not a constant of the motion in this sense what happens if
I consider a function a of the dynamical variables which also has an explicit time dependence
independent of whether the system is autonomous or not, so independent of this
I consider a function which has got explicit time dependence. What should I add on the right-hand side I
should add plus ?? a over ?? T therefore it is this plus ?? a over ?? T we need to modify
this statement then we already saw that you could have constants of the motion which had
explicit time dependence because the time dependence from the explicit T dependence
could be cancelled by that of dynamical variables the other dynamical variables in this combination,
so we now come to the conclusion that a of Q P T is a constant of the motion provided
A, H. Is equal to - ?? a so the right hand side
vanishes then the total derivative of this function with respect to time vanishes and
you have a constant of the motion so this is the test this then is the test of when
something is a constant of the motion we are going to look at some other remarkable properties
of Hamiltonian flows or Hamiltonian dynamical systems and the most important one of them
for our purposes is the fact that it is a conservative system in the sense of what we
meant by a conservative system namely. That volume elements do not change in phase
space under the flow let us verify that recall that when the dynamical system was given by. x is f of x then this was conservative, if
del . F vanished identically see but this was identically equal to zero then we called
it a conservative dynamical system well we now have a Hamiltonian system for which our
x is really Q 1 to Q n P 1 to P n it is a ton dimensional dynamical system what is the
gradient operator in this in this terminology in terms of the Q's and Ps the gradient operator
is one which has components which are the partial derivatives with respect to all the
dynamical variables so it simply stands for ?? / ?? q1 ?? / ?? q2.
Now another qn p1 and recall our equations of motion the equations of motion were Qi
.was ?? H / ?? Pi but Pi .was minus ?? H / ?? Q therefore if I simply compute this divergence
here it gives us a very simple result therefore ??. F in this case was equal to what it is
equal to ?? / ?? q1 F 1 but F 1 is ?? H / ?? P1 and soon so this is equal to summation I equal
to 1 to N ?? /r ?? Q 1 qi ?? H / ?? Pi that takes care of the first n contributions to
del dot F and the remaining n are ?? / ?? P1 times minus Delta H over ?? q 1 and so on. So you also have minus ?? e over ?? Pi ?? H
or ?? Q ion using Hamilton's equations of motion but what is this equal to it is identically
0 since you can take partial derivatives here in either order this function is integral
then you can take from partial derivatives in either order therefore this vanishes identically
which says that Hamiltonian systems have flows such that the volume elements in these systems
are preserved in time the flow is like that of an incompressible fluid or the volume is
preserved in Hamiltonian flows in phase space and this goes by the name of here or Preserve
is a better word. Again a fact of great significance it is not
just volume elements that are preserved many other things are made retained as constants
also under Hamiltonian flows so it is very restrictive these conditions the conditions
which are imposed by this set of equations plus the Poisson bracket structure between
Q's and PS are actually very restrictive and they have very deep implications there is
a very rich mathematical structure to Hamiltonian flows some of which we will uncover subsequently
okay I start by saying. It is not it is not a result of this structure
no not at all what is not a result of this structure it is not I am not saying it is
an additional constraint I am saying that you define a Hamiltonian system as a set of
two and dynamical variables in which the variables are paired off pair wise that each q with
its conjugate P such that the Poisson bracket of each Q with the p with its conjugate p
is one and he Poisson bracket of each Q with all the other variables is zero and similarly
for the piece together with that you specify a certain function of all the dynamical variables
called the Hamiltonian. And the equations of motion have this structure
that is my definition of a Hamiltonian system and from that everything else follows including
the fact that the Hamiltonian is a constant of the motion for autonomous systems the fact
that any dynamical variable or function of any dynamical variable its rate of change
with time is given by the Poisson bracket of this variable with the Hamiltonian for
explicitly time independent variable functions and you will theorem which says volume elements
are preserved various other things are preserved as well we would not go into that for instance.
If this is a volume elements schematically you have some volume element of this kind
in phase space in two n dimensional phase space you could project this volume element
onto each kewpie plane, so I take each cue and it is conjugate P and that forms a two-dimensional
plane and I project the shadow of this on to the two onto this QP plane and I take the
sum of all these shadows of these areas so I project on to q1 p 1 q 2 p 2 q 3 p 3 up
to q and p N and I take the sum of all the areas and it turns out as this volume element
flows not only does it maintain it is volume the size.
The magnitude of this volume is not changed it might get distorted but it maintains this
volume, but in addition the sum of all those shadows those areas is also preserved it is
also constant that is a very remarkable fact so it is not very intuitively very clear why
this should be, so but it follows all these properties follow from the structure of these
equations it is not a gradient system as you can see but I could write, it as something
which looks like a gradient system and there is a mathematical structure here called the
simplistic structure which I am not getting into right now.
But I was a little later on mention a few properties of this extra structure as we go
along and as a prelude to that let me do the following let me define a matrix Ja. 2n / 2n matrix with the following kinds of
elements so let me define this J as a 2nby 2n matrix with n by n blocks here and I have
the null matrix here this stands for n by n null matrix I have the unit matrix here
and this stands for the N by n unit matrix minus the unit matrix here and the null matrix
once again here, so it is partitioned into four blocks in by n blocks and it looks like
this and I call this matrix J it is a numerical matrix it has just zeros and ones or minus
ones as the elements this is a diagonal block here one here.
And this is a diagnosed with minus one zero these are null matrices then it is not hard
to see that the equations of motion in this case in the Hamiltonian case which are x i
qi dot is ?? H / ?? Pi and P I got is - ?? H / ?? q I this set of equations can be written
almost in gradient form in the form x dot is equal to J times the gradient of H where
by the gradient I mean A2 and component differential operator with components ?? / ?? q1 up to
?? / ?? q 1 and then ?? / ?? P1 up to PN and once I put this matrix J here then it is nothing
but the gradient. This twists in such a fashion that the Q's
the rate of change of the Q's depends on the derivative with respect to the PS and vice
versa with a minus sign that is the reason for this minus sign here, so if I wrote this
x as a column vector to n-dimensional column vector q1 up to PN and similarly for the gradient
of Hand then multiplied it by J you would end up with just this so this J does the required
trick of converting the part that the rate of change of the Q's will depend on derivatives
with respect to the peas and the peas with respect.
To the Q's but with a minus sign and this structure has a name it is called the simplistic
this prop this matrix J has remarkable properties what is the square of J well it is immediately
clear from this that the transpose of je is minus J because these guys would go there
and those fellows would come here and what do you think is the square of J I leave you
to figure this out incidentally when n is 1 then a large number of the properties become
quite transparent because then you have a 2 by 2 matrix which is essentially for n equal
to 1implies J = 0 1 - 1 0 and of course you could square this very trivially I leave you
to figure out what J squared is then we look at some.
Interesting properties of Hamiltonian systems in this language the advantage is that it
looks very much like a gradient system so you can write these things down much more
compactly in this form one final comment we started with mechanical systems and said you
have the coordinates and you have the moment a and they are quite physically distinct from
each other, but now in a in an abstract setting when we talk about Hamiltonian systems and
systems defined by just a set of 2nvariables of this kind pair it off two at a time then
the distinction between what you call moment. A and what you call coordinates can be actually
lost completely you could make changes of variables of these dynamical variables in
such a way that what looks like moment a initially could turn out the coordinates belong to the
set of coordinates and vice-versa provided of course you put in factors for the right
dimensionalities of these variables I bring this up because the conventional notion of
momentum as a linear momentum is very restrictive because after all you could also have an angular
momentum you could have generalized coordinates which are angles and the conjugate.
Momentum would be angular momentum so the momentum variables could have dimensions of
angular momentum not necessarily linear momentum and the generalized coordinates could be angles
which are dimensionless they do not have to be lengths so the advantage of looking at
it this way is that you can go beyond the normal mechanical examples and look at the
actual structure of the dynamical per se which is very intricate very interesting and has
very crucial properties. So we have seen today to summarize that Hamiltonian
systems forma special class of conservative dynamical systems autonomous Hamiltonian systems
Louisville's theorem asserts that the volume elements are preserved in phase space which
conforms to a definition of what we said was a conservative system the rate of change of
any dynamical function of the dynamical variables is given by the Poisson bracket of these variables
with the Hamiltonian and the Hamiltonian itself is a constant of the motion and the Poisson
bracket idea enables us to identify other constants of the motion we will take it from
this point.
