Introduction to Lagrangian Dynamics

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Introduction to Lagrangian Dynamics

1.0 The generalized coordinates

A set of generalized coordinates may be defined as any set of independent

coordinates by means of which the positions of a particle in the system maybe specified.

E.g. the two Cartesian coordinates (x, y) of a particle moving in a plane can

be replaced by the two polar coordinates ( ).

Similarly, the three Cartesian coordinates (x, y, z) may be replaced by the

spherical or cylindrical coordinates ( ) and ( ) respectively.

Consider a physical system that is described by a set of generalized

coordinates. If we speak about this system without specifying for themoment just what the coordinates are, we can do this by designating each

coordinate by the letter q with a universal subscript.

A set of n generalized coordinates is then written as:

I.e. in a plane, the Cartesian coordinates (x, y) can be written as ( ).

Similarly, a particle moving in space having (x. y, z), or ( ) may

be written as: .

1.1 Configuration of a system

The configuration/state of a system of N particles may be specified by 3N

Cartesian coordinates.

I.e.

For each configuration of the system, the generalized coordinates must have

a definite set of values which will be a function of the Cartesian coordinates

and possibly also time.

I.e. ( )

( )

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Similarly, we can also have:

If a system of particles is described by a set of generalized coordinates

then the time derivative of any coordinate called the

generalized velocity can be represented as:

The Cartesian velocity component can be expressed in terms of the

generalized coordinates by differentiating equations 1 and 2

The kinetic energy of the system of N particles in terms of the Cartesian

coordinates is therefore given by:

From equation 5, the components of the linear momentum become:

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Example for classroom consideration (ask a student to try on the

board)

Consider a particle moving in a plane with coordinates (x, y).

If its Kinetic energy is given in the Cartesian and polar coordinates as:

Determine the linear momenta . In general,

1.3 Generalized force

Suppose a system of particles at positions specified by the coordinates:

are acted upon by force

Then if each particle in the system were to move to a nearby

position through a displacement the total work done on the

system can be given as:

Now expressing in terms of the generalized coordinates, we have:

Substituting the above into equation 9, we get:

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If we let

Where the generalized force and =the generalized displacement.

If the forces are derivable from a potential

Then

In terms of the generalized coordinates

Comparing equations 12 and 13,

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1.4 The Lagranges Equation

Recalling that the generalized momentum is given by:

the time rate of change of momentum is given as:

If from the Cartesian coordinates

Then from equation 15,

are functions of , from equations 3 and 4,

Substituting these into equation 16, we get:

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Differentiating equation 17 with respect to time, we get:

But from equation 18, the first part can be written as:

The second part in equation 18 can be simplified as:

Equation 18 becomes:

Equation 19 is the Lagranges equation for a non-conservative field.

If a potential energy exists so that the forces are derivable from apotential energy function (conservative field). We define the Lagrangian

function as:

. Where,

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T depends on both

V depends only on

Equation 19 becomes

Application of Lagranges Equation

1. Motion of a particle in space.

Suppose the particle has both

Note that this exists in a gravitational field, so we must use the equation:

But the Lagrangian

Now we determine,

from equation 2, and substitute them into equation 1

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This is similar for the y and z.

2. Find the equation of a simple pendulum using the Lagranges equation.

Solution

Since the system is in a conservative field.

We apply the Lagranges equation for a conservative field.

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But for a conservative force field,

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2.0 Conservation System

They are those for which the total energy E which is the sum of the Kinetic

energy and Potential energy is conserved i.e.

A classical mechanical system is conserved when there exists a function

Called the potential energy such that for any

coordinate we can write:

Where is the components of the force on particle

Example

A one dimensional particle moves in a harmonic well with force

For such a system, a potential energy exists and is given by .

Differentiating the potential energy with respect to time, gives the force for

such a harmonic oscillator. Then the total energy is

conserved.

2.1 Non-conservative systems

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Consider a block sliding on a flat surface parallel to the surface of the earth

that experiences a frictional force. If the block has some initial kinetic

energy, the speed of the block slows to zero due to the friction. This causes

some of the Kinetic energy to be dissipated into heat.

This frictional force always depends on the velocity of the object and is not

derivable from a potential energy function.

1. Student exercise

Distinguish between a conservative and a non-conservative force

field.

2.2 Central Force

Consider the diagram above.

If a mass m moving in a plane experiences a force such that

1. The force isalways directed from m towards or away from a fixed point o.

2. Its magnitude depends only on the distance r from o. i.e.

Where is the unit vector in the direction of . Where

Then this is a central force field with o as the origin.

If

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2.3 Properties of central force field

a. The path or orbit of the particle is a plane curve.

b. The angular momentum of the particle is conserved

c. The radius from the origin to the particle sweeps out equal areas in equal

times, thus the area velocity is constant (Keeplers law).

2.4 Equations of Motion

The unit vector in the direction of increasing .

Along the

Let define the position vector of a point with respect to the origin.

But

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is called the unit vector parallel to the direction of increasing

Are referred to as moving unit vectors.

2.5 Velocity of Motion

Where and are unit vectors

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2.6 Acceleration of motion

Applying the Newtons second law, the forces in the directions are:

In a central field, by definition, , therefore equation 4 becomes

Implies the quantity,

This can be written as but

This is equal to the magnitude of the moment of the

momentum or the angular momentum L.

i.e.

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In a central force field, the angular momentum is constant.

If is constant, then both rand must always lie in a

fixed plane perpendicular to the vector L.

2.7 Kinetic energy in a central field

Since the force is central force, energy is conserved.

Total energy

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Introduction to Lagrangian Dynamics

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