July 08, 2018 Sunday
Bedtime Story
Nineteen Years Before Riemann's Lecture got Published
Gauss died the very next year in February
of 1855 of a heart attack after Riemann presented this lecture and probably
this could be the reason why Riemann’s work went largely ignored by the
mathematicians of his time.
Either there was no one left who had heard
Riemann inaugural lecture in Göttingen that day of 1854 or those who did know
about Riemann’s ideas failed to grasp its significance and hence to popularize
it.
It was almost two decades later in 1873 the
Riemann’s lecture was translated into English by the English mathematician
Clifford and published in the nature journal.
Clifford also reported on this paper to the
Cambridge Philosophical Society on the curved space concepts of Riemann and it
was probably he who then first speculated the gravity could be a manifestation
of the underlying geometry of the space.
Now having a very basic, non-mathematical
intuitive grasp of manifold we can proceed further with Lagrangian mechanics.
So now you will perhaps understand better
how Lagrangian mechanics operates using generalized coordinates: it relies upon
parameters that satisfy mathematical constraints such that the set of actual
configurations of the system is a manifold in the space of generalized
coordinates.
So if a single particle is constrained to
move around over a sphere, its configuration space will be a manifold (a
subset) of points on the sphere.
The welcome outcome of the use of such
restraints is doing away with the constraint forces that would be needed in the
mathematical equations.
Moreover, it helps even in reducing the
number of equations to be dealt with since the constraints are not being
calculated for each time interval.
So now let us get more mathematical and see
how the Newtonian and Lagrangian mechanical equations differ for the same
system consisting of N point particles with masses m1, m2…mN
with each particle having a position vector denoted by r1, r2
and rN.
The Cartesian coordinates will be r1
= (x1, y1, z1), r2 = (x2,
y2, z2) and so on.
In a three-dimensional space each vector of
a specific position to be accurately defined of it position will require three
coordinates, which means that there will totally of 3N coordinates to define
vector position of each point particle of the system.
Thus these 3N coordinates will uniquely
define the configuration of the system.
This 3N collection of coordinates will
characterize each point of each particle of the system but a general point in
space can be written as r = (x, y, z).
Now bringing into play the velocity of each
particle that would be represented as v1, v2,…vN,
which can be stated in terms of time derivative of its position as
v1
= dr1/dt, v2 = dr2/dt,…,vN = drN/dt
Now we can bring into play Newton Laws.
We shall continue working with the Lagrangian
mechanics in the nights to come.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night Mon Ami and my fellow cousin ape.
Advertisements
Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
No comments:
Post a Comment