CLASSICAL MECHANICS | Constraints| Generalised co-ordinates| Degrees of freedom

Constraints

In practice, the motion of a particle or system of particles generally restricted in some ways e.g.

 

1. i) The motion of rigid body is always such that the distance between two particles remain unchanged.

 

         ii) The motion of simple pendulum/point mass is such that the point mass and point of suspension always remain constant.

 

          iii) The motion of gas molecules within a container is restore by the walls of the container.

Definition of constraints: The limitations on  geometrical restrictions on the motion of the particle or system of particles generally known as Constraints

 

Classifications of constraints :

Constraints may be clarified into four groups .

Holonomic constraints: If the conditions of constraints can be expressed as equations connecting the coordinates of the particles

$(r_1,r_2––––)$

and time (t) having the form

                                                 

$f(r_1,r_{2,}––––r_n,t) = 0$

 

 Then the constraints are called holonomic constraints.

 

Example:

1. Motion of a rigid body –                                                                                                                                       Here the distance between two particles always constant: i.e.

                                       

${(r_{i }– r_k)}^2 =cons\tan t$

…………………………for all values of i and k

2. Motion of a point mass of a simple pendulum –

Here the distance between point mass and point of suspension is always constant i.e.

                                       

${\mathrm{x}}^2+ {\mathrm{y}}^2={\mathrm{z}}^2 = \mathrm{constant}$

 

……………… i.e. In X-Z plane the motion is restricted

 

3. Motion of a body on a smooth inclined or a horizontal plane

 The constraints equation 

                                                                  y – xtanθ = 0 

4. A cylinder, rolling without slipping down a rough inclined plane–

                         The constraint equation 

                                                     ds = adθ

 

                   Integrating      s = a.θ + constant

                                             (s – aθ)= constant

 

Non-holonomic constraints

If the conditions of constraints can be expressed as equations connecting ire coordinates and time   t (may or may not) having the form, 

                   

$f( r_1, r_2 ,––––––––,t) \ne 0$

 

Then the constraints are called non-holonomic constraints. ( When the constraints are not holonomic form, then it is called non-holonomic constraints.

 

Examples: Gas molecules within a spherical container Constraining equation – 

                 

${r_i}^2\le a^2$

 

 

Motion of a particle placed on the surface of a solid sphere

 

Constraining equation,       r ≤ a ·

The other constraints are:

Scleronomic constraints

Constraints are independent of time are called scleronomic constraints .

 

Example: 1,2,3,4,5,6

 

Rheonomic constraints

Constraints dependent of time exphitry are called rheonomic constraints.

Example :

Pendulum in a moving lift – the equation of constraint explicitly involve the time.

A beam is sliding rotating wire. Constraining equation,     

 

$y – x\tan \omega t = 0$

 

 Generalised co-ordinates

The elimination of the dependent c0-ordinates can be expressed by introducing a set of (3N – k) independent variables

${\mathrm{q}}_1, {\mathrm{q}}_{2 }–––––– {\mathrm{q}}_{3\mathrm{N}–\mathrm{k}}$

in terms of old coordinates ,

${\mathrm{r}}_1, {\mathrm{r}}_2–––– {\mathrm{r}}_{\mathrm{N}}$

are expressed by the equations of the form,

${\mathrm{r}}_{\mathrm{i}}=\mathrm{f}({\mathrm{q}}_1, {\mathrm{q}}_2 –––––{\mathrm{q}}_{3\mathrm{N}–\mathrm{k}},\mathrm{t})$

containing the constraints in them explicitly. These q’s are called generalised-coordinates.

 

In order to choose a suitable sets of generalised coordinates for a given system, we should be guided by the guided by the following three principles

1. Their values determine the configuration of the system.
2. They may be varied arbitarily and independently of each other without violating the constraints on the system.
3.  There is no uniqueness in the choice of generalised co-ordinates . Our choice should be reasonable mathematical simplification of the problem . Definition : The set of independent coordinates sufficient in number to specify the system configuration is called generalised coordinates.Advantages:1.Unlike the cartesian coordinates, the generalised coordinates need not belong to any particular particle,in a group of three.2.They can be selected according to our conveince and symmetry of the system
    All sorts of quantities may be used to serve as generalised coordinates. They may be positional, angular or any kind of variables.
4.  Number of independent variables being minimised, constraints are automatically obeyed .

 

 Examples:                                                                                                               

1  Simple pendulum

                                     

$–––––\left(\theta \right)$

 

 2 fly wheel

$–––––––––( \theta )$

 

  3 A particle moving on surface of a sphere

                                             

$––––––( \theta ,\varphi )$

 

  4 A particle moving on the inside surface of a cone

                                                   

$––––––––(r,\theta )$

 

 Degrees of freedom

The minimum number of independent variables necessary to describe the configuration of dynamical system is called degrees of freedom (d.o.f) ·

Example : On Three dimensional space, a free particle has 3 degrees of freedom, because its position at any instant can be specified by its three positional co-ordinates may be either

                                                            (x,y,z) of a cartesian system 

                                               or     

$(r,\theta ,\varphi )$

a spherical polar system

                                                or     

$(\rho ,\varphi ,z )$

of a bylindrical polar system

Remember that:

A dynamical system of N particles subjected to k constraints will have f degrees of freedom i.e.

                                                          f = 3N – k

 

Degrees of freedom in different cases –

i) A free particle N=1 , k=0 , so f =3

ii) N free particles N=N,  k=0,  so  f = 3N 

iii) Fixed pendulum of a simple pendulum – N=1,  k=3 , so  f=0 

iv) Bob of conical pendulum – N=1, k=1, so  f= 2

v) A dumbell –     N=2, k=1,  so  f= 5     (no. of particles = 2)

vi) Three point masses connected by 3 rigid massless rod-   N= 3, k=3  so f=6                                                                 

vii) A rigid body fixed at a point –         f=6–3 = 3

viii) A very thin straight rod –  f= 3

and ix) A rigid linear tri- atomic molecule – N = 2, k =3  so    f =3

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CLASSICAL MECHANICS

CLASSICAL MECHANICS