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Variational Principles

and Lagrange’s Equations

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Definitions

Lagrangian density:

Lagrangian:

Action:

How to find the special value for action corresponding to observable ?

Joseph Louis

Lagrange/

Giuseppe Luigi

Lagrangia

(1736 – 1813)

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Variational principle

Maupertuis: Least Action Principle

Hamilton: Hamilton’s Variational Principle

Feynman: Quantum-Mechanical Path Integral Approach

Pierre-Louis Moreau

de Maupertuis

(1698 – 1759)

Sir William Rowan

Hamilton

(1805 – 1865)

Richard Phillips

Feynman

(1918 – 1988)

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Functionals

Functional: given any function f(x), produces a number S

Action is a functional:

Examples of finding special values of functionals using variational approach:

shortest distance between two points on a plane;

the brachistochrone problem;

minimum surface of revolution;

etc.

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Shortest distance between two points on a plane

An element of length on a plane is

Total length of any curve going between points 1 and 2 is

The condition that the curve is the shortest path is that the functional I takes its minimum value

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The brachistochrone problem

Find a curve joining two points, along which a particle falling from rest under the influence of gravity travels from the highest to the lowest point in the least time

Brachistochrone solution: the value of the functional t [y(x)] takes its minimum value

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Calculus of variations

Consider a functional of the following type

What function y(x) yields a stationary value (minimum, maximum, or saddle) of J ?

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Calculus of variations

Assume that function y₀(x) yields a stationary value and consider all possible functions in the form:

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Calculus of variations

In this case our functional becomes a function of α:

Stationary value condition:

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Stationary value

1

2

3

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Stationary value

1

2

3

u

dv

u

v

du

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Stationary value

1

2

3

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Stationary value

1

2

3

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Stationary value

arbitrary

Trivial … 😐

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Stationary value

arbitrary

Nontrivial !!! ☺

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Shortest distance between two points on a plane

Straight line! ☺

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The brachistochrone problem

Scary! ☠

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Recipe

1. Bring together structure and fields

2. Relate this togetherness to the entire system

3. Make them fit best when the fields have observable dependencies:

Structure

Fields

Structure

Physical Laws

Best Fit

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Back to trajectories and Lagrangians

How to find the special values for action corresponding to observable trajectories ?

We look for a stationary action using variational principle

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Recipe

1. Bring together structure and fields

2. Relate this togetherness to the entire system

3. Make them fit best when the fields have observable dependencies:

Structure

Fields

Structure

Physical Laws

Best Fit

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Back to trajectories and Lagrangians

For open systems, we cannot apply variational principle in a consistent way, since integration in not well defined for them

We look for a stationary action using variational principle for closed systems:

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Stationary value

Nontrivial !!! ☺

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Simplest non-trivial case

Let’s start with the simplest non-trivial result of the variational calculus and see if it can yield observable trajectories

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Stationary value

Nontrivial !!! ☺

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Euler- Lagrange equations

These equations are called the Euler- Lagrange equations

Joseph Louis

Lagrange

(1736 – 1813)

Leonhard Euler

(1707 – 1783)

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Recipe

1. Bring together structure and fields

2. Relate this togetherness to the entire system

3. Make them fit best when the fields have observable dependencies:

Structure

Fields

Structure

Physical Laws

Best Fit

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How to construct Lagrangians?

Let us recall some kindergarten stuff

On our – classical-mechanical – level, we know several types of fundamental interactions:

Gravitational
Electromagnetic
That’s it

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Gravitation

For a particle in a gravitational field, the trajectory is described via 2nd Newton’s Law:

This system can be approximated as closed

The structure (symmetry) of the system is described by the gravitational potential

Sir Isaac Newton

(1643 – 1727)

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Electromagnetic field

For a charged particle in an electromagnetic field, the trajectory is described via 2nd Newton’s Law:

This system can be approximated as closed

The structure (symmetry) of the system is described by the scalar and vector potentials

Really???

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Electromagnetic field

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Electromagnetic field

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Electromagnetic field

Lorentz force! ☺

Hendrik Lorentz

(1853-1928)

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Kindergarten

Thereby:

In component form

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How to construct Lagrangians?

Kindergarten stuff:

The “kindergarten equations” look very similar to the Euler-Lagrange equations! We may be on the right track! ☺

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Gravitation

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Gravitation

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Electromagnetism

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Bottom line

We successfully demonstrated applicability of our recipe

This approach works not just in classical mechanics only, but in all other fields of physics

Structure

Physical Laws

Best Fit

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Some philosophy

de Maupertuis on the principle of least action (“Essai de cosmologie”, 1750): “In all the changes that take place in the universe, the sum of the products of each body multiplied by the distance it moves and by the speed with which it moves is the least that is possible.”

How does an object know in advance

what trajectory corresponds to a

stationary action???

Answer: quantum-mechanical path

integral approach

Pierre-Louis Moreau

de Maupertuis

(1698 – 1759)

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Some philosophy

Feynman: “Is it true that the particle doesn't just "take the right path" but that it looks at all the other possible trajectories? ... The miracle of it all is, of course, that it does just that. ... It isn't that a particle takes the path of least action but that it smells all the paths in the neighborhood and chooses the one that has the least action ...”

Richard Phillips

Feynman

(1918 – 1988)

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Some philosophy

Dyson: “In 1949, Dick Feynman told me about his "sum over histories" version of quantum mechanics. "The electron does anything it likes," he said. "It just goes in any direction at any speed, forward or backward in time, however it likes, and then you add up the amplitudes and it gives you the wave-function." I said to him, "You're crazy." But he wasn't.”

Freeman John Dyson

(born 1923)

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Some philosophy

Philosophical meaning of the Lagrangian formalism: structure of a system determines its observable behavior

So, that's it?

Why do we need all this?

In addition to the deep philosophical meaning, Lagrangian formalism offers great many advantages compared to the Newtonian approach

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Lagrangian approach: extra goodies

It is scalar (Newtonian – vectorial)

Allows introduction of configuration space and efficient description of systems with constrains

Becomes relatively simpler as the mechanical system becomes more complex

Applicable outside Newtonian mechanics

Relates conservation laws with symmetries

Scale invariance applications

Gauge invariance applications

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Simple example

Projectile motion

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Another example

Another Lagrangian

What is going on?!

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Gauge invariance

For the Lagrangians of the type

And functions of the type

Let’s introduce a transformation (gauge transformation):

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Gauge invariance

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Gauge invariance

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Gauge invariance

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Back to the question: How to construct Lagrangians?

Ambiguity: different Lagrangians result in the same equations of motion

How to select a Lagrangian appropriately?

It is a matter of taste and art

It is a question of symmetries of the physical system one wishes to describe

Conventionally, and for expediency, for most applications in classical mechanics:

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Cylindrically symmetric potential

Motion in a potential that depends only on the distance to the z axis

It is convenient to work in cylindrical coordinates

Then

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Cylindrically symmetric potential

How to rewrite the equations of motion in cylindrical coordinates?

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Generalized coordinates

Instead of re-deriving the Euler-Lagrange equations explicitly for each problem (e.g. cylindrical coordinates), we introduce a concept of generalized coordinates

Let us consider a set of coordinates

Assume that the Euler-Lagrange equations hold for these variables

Consider a new set of (generalized) coordinates

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Generalized coordinates

We can, in theory, invert these equations:

Let us do some calculations:

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Generalized coordinates

The Euler-Lagrange equations are the same in generalized coordinates!!!

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Generalized coordinates

If the Euler-Lagrange equations are true for one set of coordinates, then they are also true for the other set

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Cylindrically symmetric potential

Radial force causes a change in radial momentum and a centripetal acceleration

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Cylindrically symmetric potential

Angular momentum relative to the z axis is a constant

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Cylindrically symmetric potential

Axial component of velocity does not change

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Symmetries and conservation laws

The most beautiful and useful illustration of the “structure vs observed behavior” philosophy is the link between symmetries and conservation laws

Conjugate momentum for coordinate :

If Lagrangian does not depend on a certain coordinate, this coordinate is called cyclic (ignorable)

For cyclic coordinates, conjugate momenta are conserved

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Symmetries and conservation laws

For cyclic coordinates, conjugate momenta are conserved

p = const

p ≠ const

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Cylindrically symmetric potential

Cyclic coordinates:

Rotational symmetry Translational symmetry

Conjugate momenta:

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Electromagnetism

Conjugate momenta:

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Noether’s theorem

Relationship between Lagrangian symmetries and conserved quantities was formalized only in 1915 by Emmy Noether:

“For each symmetry of the Lagrangian, there is a conserved quantity”

Let the Lagrangian be invariant under

the change of coordinates:

α is a small parameter. This invariance

has to hold to the first order in α

Emmy Noether/

Amalie Nöther

(1882 – 1935)

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Noether’s theorem

Invariance of the Lagrangian:

Using the Euler-Lagrange equations

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Example

Motion in an x-y plane of a mass on a spring (zero equilibrium length):

The Lagrangian is invariant (to the first order in α) under the following change of coordinates:

Then, from Noether’s theorem it follows that

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Example

In polar coordinates:

The conserved quantity:

Angular momentum in the x-y plane is conserved

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Example

For the same problem, we can start with a Lagrangian expressed in polar coordinates:

The Lagrangian is invariant (to any order in α) under the following change of coordinates:

The conserved quantity from Noether’s theorem:

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Back to trajectories and Lagrangians

How to find the special values for action corresponding to observable trajectories ?

We look for a stationary action using variational principle

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Stationary value

1

2

3

u

dv

u

v

du

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What is H?

Let us expand the Lagrangian in powers of :

From calculus, for a homogeneous function f of degree n (Euler’s theorem) :

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What is H?

If the Lagrangian has a form:

Then

For electromagnetism:

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Conservation of energy

In the field formalism, the conservation of H is a part of Noether’s theorem

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The brachistochrone problem

Similarly to the “H-trick”:

Scary! ☠

!!!

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The brachistochrone problem

Change of variables:

Parametric solution

(cycloid)

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Scale invariance

For Lagrangians of the following form:

And homogeneous L₀of degree k

Introducing scale and time transformations

Then

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Scale invariance

Therefore, after transformations

If

Then

The Euler-Lagrange equations after transformations

The same!

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Scale invariance

So, the Euler-Lagrange equations after transformations are the same if

Free fall

Let us recall

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Scale invariance

So, the Euler-Lagrange equations after transformations are the same if

Mass on a spring

Let us recall

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Scale invariance

So, the Euler-Lagrange equations after transformations are the same if

Kepler’s problem

Let us recall 3rd Kepler’s law

Johannes Kepler

(1571-1630)

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How about open systems?

For some systems we can neglect their interaction with the outside world and formulate their behavior in terms of Lagrangian formalism

For some systems we can not do it

Approach: to describe the system without “leaks” and “feeds” and then add them to the description of the system

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How about open systems?

For open systems, we first describe the system without “leaks” and “feeds”

After that we add “leaks” and “feeds” to the description of the system

Q: Non-conservative generalized forces

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Generalized forces

Forces

1: Conservative (Potential)

2: Non-conservative

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Generalized forces

In principle, there is no need to introduce generalized forces for a closed system fully described by a Lagrangian

Feynman: “…The principle of least action

only works for conservative systems —

where all forces can be gotten from a potential function. … On a microscopic level — on the deepest level of physics — there are no non-conservative forces. Non-conservative forces, like friction, appear only because we neglect microscopic complications — there are just too many particles to analyze.”

So, introduction of non-conservative forces is a result of the open-system approach

Richard Phillips

Feynman

(1918 – 1988)

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Degrees of freedom

The number of degrees of freedom is the number of independent coordinates that must be specified in order to define uniquely the state of the system

For a system of N free particle there are 3N degrees of freedom (3N coordinates)

N

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Constraints

We can impose k constraints on the system

The number of degrees of freedom is reduced to 3N – k = s

It is convenient to think of the remaining s independent coordinates as the coordinates of a single point in an s-dimensional space: configuration space

N

k

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Types of constraints

Holonomic (integrable) constraints can be expressed in the form:

Nonholonomic constraints cannot be expressed in this form

Rheonomous constraints – contain time dependence explicitly

Scleronomous constraints – do not contain time dependence explicitly

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Analysis of systems with holonomic constraints

Elimination of variables using constraints equations

Use of independent generalized coordinates

Lagrange’s multiplier method

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Double 2D pendulum

An example of a holonomic scleronomous constraint

The trajectories of the system are very complex

Lagrangian approach produces equations of motion

We need 2 independent generalized coordinates

(N = 2, k = 2 + 2, s = 3 N – k = 2)

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Double 2D pendulum

Relative to the pivot, the Cartesian coordinates

Taking the time derivative, and then squaring

Lagrangian in Cartesian coordinates:

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Double 2D pendulum

Lagrangian in new coordinates:

The equations of motion:

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Double 2D pendulum

Special case

The equations of motion:

More fun at:

http://www.mathstat.dal.ca/~selinger/lagrange/doublependulum.html

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Lagrange’s multiplier method

Used when constraint reactions are the object of interest

Instead of considering 3N - k variables and equations, this method deals with 3N + k variables

As a results, we obtain 3N trajectories and k constraint reactions

Lagrange’s multiplier method can be applied to some nonholonomic constraints

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Lagrange’s multiplier method

Let us explicitly incorporate constraints into the structure of our system

For observable trajectories

So

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Lagrange’s multiplier method

- constraint reactions

Now we have 3N + k equations for and

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Application to a nonholonomic case

A particle on a smooth hemisphere

One nonholonomic constraint:

While the particle remains on the sphere, the constraint is holonomic

And the reaction from the surface is not zero

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Application to a nonholonomic case

Constraint equation in cylindrical coordinates:

New Lagrangian in cylindrical coordinates:

Equations of motion

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Application to a nonholonomic case

Constraint equation in cylindrical coordinates:

New Lagrangian in cylindrical coordinates:

Equations of motion

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Application to a nonholonomic case

Constraint equation in cylindrical coordinates:

New Lagrangian in cylindrical coordinates:

Equations of motion

Trivial

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Application to a nonholonomic case

Constraint reaction:

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Application to a nonholonomic case

Constraint reaction:

Reaction disappears when

The particle becomes airborne