Phy 303: Phy 303: Classical Mechanics (2)Classical Mechanics (2)

Chapter 3Chapter 3

Lagrangian and Hamiltonian MechanicsLagrangian and Hamiltonian Mechanics

Introduction

In solving a problem in dynamics by using the Newtonian formalism, we must know all the forces acting on the studied object, because the quantity F that appears in the fundamental equation F = dP/dt is the total force acting on the object.

But in particular situations, it may be difficult or even impossible to obtain explicit expressions for all the forces acting on the object.

An alternate method of dealing with complicated problems in a dealing with complicated problems in a general mannergeneral manner is contained in Hamilton's Principle, and the equations of motion resulting from the application of this principle are called Lagrange's equations.

Hamiltonian principleHamiltonian principleIn two papers published in 1834 and 1835, Hamilton announced the

dynamical principle on which it is possible to base all of mechanics and, indeed, most of classical physics. Hamilton's Principle may be stated as follows:

Of all the possible paths along which a dynamical system may move from one point to another within a specified time interval (consistent with any constraints), the actual path followed is that which minimizes the time integral of the difference between the kinetic and potential energies.This principle is also called Principle Of Least Action.

In terms of the calculus of variations, Hamilton's Principle becomes

[1]

If we define the difference of these quantities to be

then Equation [1] becomes

[2]

[3]

The function L appearing in this expression may be identified with the function f of the variational integral (see Chapter 2),

if we make the transformations

The Euler-Lagrange equations corresponding to Equation [3] are therefore

[4]

These are the Lagrange equations of motion for the particle, and the quantity L is called the Lagrange function or Lagrangian for the particle.

Example 1:Lagrange equation of motion for the one-dimensional harmonic oscillator.

Substituting these results into Equation [4] leads to

which is identical with the equation of motion obtained using Newtonian mechanics.

Example 2: Plane pendulum

And applying Lagrange equation:

which again is identical with the Newtonian result

William Rowan Hamilton Born 4 August 1805)

Dublin

Died 2 September 1865) (aged 60)Dublin

Fields Physicist, astronomer, and mathematician

Joseph Louis Lagrange Born 25 January 1736)

Turin, Piedmont-Sardinia

Died 10 April 1813) (aged 77)Paris, FranceFrance

Fields MathematicsMathematical physics

Generalized coordinatesWe consider a general mechanical system consisting of a collection of n discrete point particles, some of which may be connected to form rigid bodies.

3n quantities must be specified to describe the positions of all the particles.

If there are m equations of constraint, then only 3n — m coordinates are independent, and the system is said to possess S = 3n — m degrees of freedom.

We give the name generalized coordinates to any set of S independent quantities that completely specifies the state of a system (their number equals the number of degrees of freedom).

The generalized coordinates are noted q1, q2, . . . , or simply as the qj. A set of independent generalized coordinates

In certain cases, we use generalized coordinates which number exceeds the number of degrees of freedom and we explicitly take into account the constraint relations through the use of the Lagrange undetermined multipliers. Such would be the case, for example, if we desired to calculate the forces of constraint.

The choice of a set of generalized coordinates to describe a system is not unique. We choose the set that gives the simplest equations of motion.

, the time derivatives of qj, are called the generalized velocities.

Important Notes on NotationImportant Notes on Notation

Cartesian coordinates are noted

Generalized coordinates are noted , 1, 2,... , number of degres of freedomjq j s s

(designates the particle)

The index i is reserved for Cartesian coordinates. xi , for i = 1,2,3 , represents either x, y, or z depending on the value of i .

The index α will be used to identify quantities associated with a given particle when using Cartesian coordinates. For example, the position vector for particle α is given by rα, and its kinetic energy Tα

Einstein’s summation convention:Whenever an index appears twice (an only twice), then a summation over this index is implied. For example,

In general, the relationships linking the Cartesian and generalized coordinates and velocities can be expressed as

We may also write the inverse transformations as

Also, there are m equations of constraint of the form

Lagrange’s equations of motion Lagrange’s equations of motion in generalized coordinatesin generalized coordinates

It follows naturally that Hamilton’s Principle can now be expressed in term of the generalized coordinates and velocities as

with Lagrange’s equations given by

if we refer to the definitions of the quantities in chapter 2 and make the identifications:

[5]

[6]

Study Examples 7.1 and 7.3 from the textbook.Study Examples 7.1 and 7.3 from the textbook.

Eg 7.3:Eg 7.3: see textbook see textbook

Other example: The double pendulum.

Consider the case of two particles of mass m1 and m2 each attached at the end of a mass less rod of length l1 and l2 , respectively. The second rod is also attached to the first particle (see Figure ). Derive the equations of motion for the two particles.

Solution:

It is desirable to use cylindrical coordinates for this problem. We have two degrees of freedom, and we will choose θ1 and θ2 as the generalized coordinates. Starting with Cartesian coordinates, we write an expression for the kinetic and potential energies for the system (take U=0 at y=0).

And from Lagrange’s equations we get

or

Lagrange’s equations with Lagrange’s equations with undetermined multipliersundetermined multipliersIf the constraint relations for a problem are given in differential form, we

can incorporate them directly into Lagrange's equations by using the Lagrange undetermined multipliers (see Chapter 2)

That is, for constraints expressible as

the Lagrange equations (see chap 2) are

[7]

[8]

The undetermined multipliers are closely related to the forces of constraint.

The generalized forces of constraint Qj are given by

)(tk

[9]

Study Example 7.9 from the textbook.Study Example 7.9 from the textbook.

(see textbook)

Equivalence of Lagrange’s and Equivalence of Lagrange’s and Newton’s equationsNewton’s equations

We now explicitly demonstrate the equivalence of Lagrange’s and Newton’s equations.

Let us choose the generalized coordinates to be the rectangular coordinates. Lagrange's equations (for a single particle) then become

We also have: for a conservative system:

and

so Lagrange’s Equations yield the Newtonian equations, as required:

[10]

A theorem concerning the kinetic A theorem concerning the kinetic energyenergyIn a Cartesian coordinates system the kinetic energy of a system of particles is expressed as

where a summation over i is implied.Using equations relating the two systems of coordinates:

An important case occurs when a system is scleronomic, i.e., there is no explicit dependency on time in the coordinate transformation, we then have

and the kinetic energy can be written in the form

where a summation on i is still impliedWe see that the kinetic energy is also a quadratic function of the (generalized) velocities. If we next differentiate equation [11] with respect to and then multiply it by (and summing), we get

[11]

lq

lq[12]

Conservation theoremsConservation theoremsConsider a general Lagrangian L; the total time derivative of L is

But from Lagrange’s equations,

(there is summation on j)

where we have introduced a new function H called the Hamiltonian of the system

[13]

[14]

In cases where the Lagrangian is not explicitly dependent on time we find that

cste.0 Hdt

dH

If we are in presence of a scleronomic system, then

[15]

Equation [15] can be written as

(from [12])

The Hamiltonian of the system is equaled to the total energy only if the following conditions are met:1. The system is scleronomic; ie the equations of the transformation connecting the Cartesian and generalized coordinates must be independent of time (the kinetic energy is then a quadratic function of the generalized velocities).2. The potential energy must be velocity independent.

The Hamiltonian of the system is conserved (but not necessarily equal to the total energy) when the Lagrangian is not explicitly dependent on time.

Canonical equations of motion – Canonical equations of motion – Hamiltonian mechanics.Hamiltonian mechanics.if the potential energy of a system is velocity independent, then the linear momentum components in rectangular coordinates are given by

By analogy, in the case in which the Lagrangian is expressed in generalized coordinates, we define the generalized momenta as

[16]

The Lagrange equations of motion are then expressed by

[17]

Using the definition [16] of the generalized momenta, Equation [14] for the Hamiltonian may be written as

From [16], the generalized velocities can be expressed as

Thus we may make a change of variables from the set to the set and express the Hamiltonian as

[18]