lectures on classical mechanics-1
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PBC Lecture Notes Series in Mechanics:
by Dr. Abhijit Kar Gupta, e-mail: kg.abhi@gmail.com
1
Classical Mechanics: Lecture-1 (degrees of freedom, constraints)
To describe the motion of a single particle or a system of particles we have to know the
coordinates. If N particles are moving freely in three dimensions, the motion is described
by N3 space coordinates. But all the coordinates are not independent when there are
constraints in the system. This means there are restrictions imposed on the coordinates;
change in one coordinate affects other coordinates. The ‘restrictions’ can be expressed in
terms of equations or appropriate conditions.
Examples:
Let us think of a simple pendulum oscillating in a two dimensional plane ( yx, ):
Next, we consider a particle moving on the surface of a sphere. The three
Cartesian coordinates satisfy: 2222 azyx . Thus in this case we have 2
independent coordinates. If we know x and y we can know 222 yxaz .
But if we say, the particle can be anywhere inside the sphere, we have the
following inequality condition: 2222 azyx .
In this case the constraint is not in the form of an equation as before.
Degrees of freedom:
Number of independent coordinates that are required to describe the motion of a system
is called degrees of freedom.
In a system of N -particles, if there are k -equations of constraints, we have
kNn 3 number of independent coordinates. n degrees of freedom.
Configuration space:
The motion of a system of N -particles having n -degrees of freedom can be imagined by
the motion of a point in the space of n -coordinates ( n -dimensional hypothetical space).
This space is called configuration space.
Classification of constraints: Constraints can be of broadly two types.
Holonomic
Non-Holonomic
X
Y
l
),( yx
The motion of the pendulum bob is
such that we have 222 lyx . This
equation is a relation between two
coordinates and thus is a constraint.
Therefore, the motion is described by a
single independent coordinate.
PBC Lecture Notes Series in Mechanics:
by Dr. Abhijit Kar Gupta, e-mail: kg.abhi@gmail.com
2
Holonomic:
A Holonomic constraint is one that can be expressed in the form of an equation relating
the coordinates: 0)..,,,( 111 tzyxf
Example : simple pendulum, 222 lyx ; particle on a sphere, 2222 azyx .
A general example of holonomic constraint is a rigid body where the distance
between any two points is fixed: 22)( ijji crr .
Non-holonomic: The constraint which can not be expressed in the form of an equation relating the
coordinates is called non-holonomic.
Example : The motion of a particle inside anywhere a sphere, 2222 azyx .
The constraints are also classified into the following way:
Scleronomic: where the constraints are independent of time.
Rheonomic: where the constraints have explicit dependence on time.
More examples on the types of constraints:
#1. An object sliding down an inclined plane:
#2. An object sliding down an inclined plane where the inclination angle of the plane
varies with time: constraint, tx
ytan , here t
Type of constraint: Holonomic and Rheonomic.
X
Constraint: tanx
y
Type of constraint:
Holonomic and Scleronomic
Y
PBC Lecture Notes Series in Mechanics:
by Dr. Abhijit Kar Gupta, e-mail: kg.abhi@gmail.com
3
#3. A disc rolling down an inclined plane (not slipping):
Note: Sometimes the constraints are written in terms of velocities and momentum etc. But they
can be integrated to find relations among coordinates.
#4. A disc rolling on the XY-plane:
#5. A sphere rolling over a larger sphere:
v
X
Y
Z
X
s
Y Constraint (No slipping condition):
dt
da
dt
ds
Integrating, as const., where
s is the distance measured on the
inclined plane.
Type of constraint:
Holonomic and Scleronomic
Condition of rolling:
av
Also we can write,
cosvx
; sinvy
Thus we have,
0.cos dadx ……(1) and
0.sin dady ……..(2).
The above two differential
equations for constraints are not
exact differential. Thus there can
not be any algebraic equation
relating the coordinates. Therefore,
the type of constrain is
Non-holonomic.
The constraint changes when the small
sphere is detached from the larger sphere
while rolling down. Thus we can not write
an equation relating the coordinates for the
entire process. The system is Non-
holonomic.
Since there is no explicit time dependence,
the system is scleronomic.
PBC Lecture Notes Series in Mechanics:
by Dr. Abhijit Kar Gupta, e-mail: kg.abhi@gmail.com
4
NOTE:
Mind that for HOLONOMIC constraint, you have to have an equation relating the
coordinates (more precisely, an algebraic equation). In case you have a differential
equation and you know this can be integrated to convert into an algebraic one, you will
still consider this to be holonomic. One particularly interesting example is a cylinder (or a
disc or a sphere) rolling down an inclined plane. When you consider the relation between
the linear speed with the radius and angle that is created due to rotation, you have a
constraint of the kind as just said. Here it is assumed that the object is rolling down
straight over the inclined plane.
In case of a disc (or sphere) rolling on a horizontal plane, the object is rolling but its path
is not restricted to a straight line always, it is wandering around over the plane making a
curved path. So there is one additional coordinate (an angle) which is needed to describe
the motion but that is not known. So if you write the differential equations here, one
along x-axis and another along y-axis, that additional angle appears there. As the angles
are unknown, the differential equations can not be integrated to make an algebraic
equation in this case like before. So this is NON-HOLONOMIC.
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