M.TECH
ADVANCED ENGINEERING MATHAMETICS
5MARK:
1. Classical mechanics
Classical mechanics is based on two fundamental principles: the {\bf Galileo relativity} principle
\index{Galileo relativity} and the fundamental principle of dynamics. Let us state Galileo
relativity principle:
Principle: Galileo relativity principle. Classical mechanics laws (in particular Newton's law of
motion) have the same form in every frame in uniform translation with respect to each other.
Such frames are called Galilean frames or inertial frames.
In classical mechanics the time interval separating two events is independent of the movement of
the reference frame. Distance between two points of a rigid body is independent of the
movement of the reference frame.
Remark: Classical mechanics laws are invariant by transformations belonging to Galileo
transformation group. A Galileo transformation of coordinates can be written:
Following Gallilean relativity, the light speed should depend on the Galilean reference frame
considered. In 1881, the experiment of Michelson and Morley attempting to measure this
dependance fails.
2.
Show that a rocket of mass
that ejects at speed (with respect to itself) a part of its mass
by time unit
moves in the sense opposed to . Give the movement law for a rocket with
speed zero at time
, located in a earth gravitational field considered as constant.
For a cylindrical coordinates system, metrics
Calculate the Christoffel symbols
of the space is:
defined by:
3. Doppler effect. Consider a light source
moving at constant speed with respect to reference
frame . Using wave four-vector
give the relation between frequencies measured by
an experimentator moving with and another experimentater attached to . What about sound
waves
Exercice:
Consider a unit mass in a three dimensional reference frame whose metrics is:
Show that the kinetic energy of the system is:
Show that the fundamental equation of dynamics is written here (forces are assumed to derive
from a potential ) :
20mark:
1.Relativistic mechanics (Special relativity)
Relativistic mechanics in the special case introduced by Einstein, as he was 26 years old, is based
on the following postulate:
Postulate: All the laws of Universe ({\it i. e. }laws of mechanics and electromagnetism) are the
same in all Galilean reference frames.
Because Einstein believes in the Maxwell equations (and because the Michelson Morley
experiment fails) has to be a constant. So Einstein postulates:
Postulate: The light speed in vacuum
speed is an upper bound.
is the same in every Galilean reference frame. This
We will see how the physical laws have to be modified to obey to those postulates later
on\footnote{The fundamental laws of dyanmics is deeply modified (see section secdynasperel
(see section secdynasperel), but as guessed by Einstein Maxwell laws obey to the special
relativity postulates (see section seceqmaxcov.}. The existence of a universal speed, the light
speed, modifies deeply space--time structure. \index{space--time} It yields to precise the
metrics\index{metrics} (see appendix chaptens for an introduction to the notion of metrics)
adopted in special relativity. Let us consider two Galilean reference frames characterized by
coordinates:
Then:
and
. Assume that at
both coordinate system coincide.
that is to say:
and
Quantity
is thus:
is thus an invariant. The most natural metrics that should equip space--time
It is postulated that this metrics should be invariant by Galilean change of coordinates.
Postulate: Metrics
reference frame.
is invariant by change of Galilean
Let us now look for the representation of a transformation of space--time that keeps unchanged
this metrics. We look for transformations such that:\index{Lorentz transformation}
is invariant. From, the metrics, a "position vector" have to be defined. It is called four-vector
position, and two formalisms are possible to define it:\index{four--vector}.

Either coordinates of four-vector position are taken equal to
space is equipped by pseudo scalar product defined by matrix:
and
Then:
where

represents the transposed of four-vector position
.
Or coordinates of four-vector position are taken equal to
is equipped by pseudo scalar product defined by matrix:
and space
Then:
where
represents the transposed four-vector position
.
Once the formalism is chosen, the representation of transformations ({\it i. e.,} the matrices), that
leaves the pseudo-norm invariant can be investigated (see ([#References|references])). Here we
will just exhibit such matrices. In first formalism, condition that pseudo-product scalar is
invariant implies that:
thus
cond
Following matrix suits:
where
(
is the speed of the reference frame) and
. The inverse of
2.Eigen Time:
If mobile travels at velocity in reference frame
travelling the mobile by:
and are referenced in
by:
So, one gets the relation verified by ::
so
Velocity four-vector
Velocity four-vector is defined by:
, then events A and B that are referenced in
:
where
is the classical speed.
Other four-vectors
Here are some other four-vectors (expressed using first formalism):

four-vector position :

four-vector wave:

four-vector nabla:
General relativity
There exists two ways to tackle laws of Nature discovery problem:
1. First method can be called {
"phenomenological"}. A good example of phenomenological theory is quantum mechanics
theory. This method consists in starting from known facts (from experiments) to infer laws.
Observable notion is then a fundamental notion.
1. There exist another method less "anthropocentric" whose advantages had been underlined at
century 17 by philosophers like Descartes.
It is the method called {\it a priori}. It has been used by Einstein to propose his relativity theory.
It consists in starting from principles that are believed to be true and to look for laws that obey to
those principles.
3.Fundamental principle of classical mechanics
Let us state the fundamental principle of classical dynamics for a material point, or particle[1]. A
material point is classically described by its mass
, its position , and its velocity . It
undergoes external actions modelled by forces
. The momentum
of the particle is
denoted by .\index{momentum}
Principle: The fundamental principle of dynamics (or Newton's equation of motion)
\index{Newton's equation of motion} states that the time derivative of momentum is equal to the
sum of all external forces\index{force}:
Least action principle
Principle: Least action principle: \index{least action principle} The function
the trajectory of a particle with potential energy
is defined by
, where
describing
yields a constant action, where the action
is the Lagrangian of the particle:
.
This principle can be taken as the basis of material point classical mechanics. But it can also be
seen as a consequence of the fundamental dynamics principle presented previously
[ma:equad:Arnold83].
Let us multiply by
and integrate over time:
Using Green's theorem (integration by parts):
is a bilinear form.
Defining the Lagrangian
by:
the previous equation can be written
meaning that the action
is constant.
4. Description by energies
Laws of motion does not tell anything about how to model forces. The force modelization is
often physicist's job. Here are two examples of forces expressions:
1. weight
. is a vector describing gravitational field around the material point
of mass
considered.
2. electromagnetic force
, where is particle's charge, is the
electric field, the magnetic field and the particle's velocity.
This two last forces expressions directly come from physical postulates. However, for other
interactions like elastic forces, friction, freedom given to physicist is much greater.
An efficient method to modelize such complex interactions is to use the energy (or power)
concept. At chapter chapelectromag, the duality between forces and energy is presented in the
case of electromagnetic interaction. At chapters chapapproxconti and chapenermilcon, the
concept of energy is developed for the description of continuous media.
Let us recall here some definitions associated to the description of interactions by forces.
Elementary work of a force for an elementary displacement is:
Instantaneous power emitted by a force
to a material point of velocity is:
Potential energy gained by the particle during time
that it needs to move of
is:
Note that potential energy can be defined only if force field have conservative circulation[2].
This is the case for weight, for electric force but not for friction. A system that undergoes only
conservative forces is hamiltonian. The equations that govern its dynamics are the Hamilton
equations: {IMP/label|eqhampa1}}
where function
is called hamiltonian of the system. For a particle with a potential
energy
, the hamiltonian is:
where is particle's momentum and its position. By extension, every system whose dynamics
can be described by equations eqhampa1 and eqhampa2 is called hamiltonian \index{hamiltonian
system} even if
is not of the form given by equation eqformhami.
Dynamics in special relativity
It has been seen that Lorentz transformations acts on time. Classical dynamics laws have to be
modified to take into into account this fact and maintain their invariance under Lorentz
transformations as required by relativity postulates. Price to pay is a modification of momentum
and energy notions. Let us impose a linear dependence between the impulsion four-vector and
velocity four-vector;
where
is the rest mass of the particle,
, with
is the classical speed of the particle
. Let us call ``relativistic momentum quantity:
,
and "relativistic energy" quantity:
Four-vector
can thus be written:
Thus, Einstein associates an energy to a mass since at rest:
This is the matter--energy equivalence . \index{matter--energy equivalence} Fundamental
dynamics principle is thus written in the special relativity formalism:
where
is force four-vector.