# 4. Hamilton's Principle and Noether's Theorem

## Introduction: Galileo and Newton

In the discussion of calculus of variations, we anticipated some basic dynamics, using the potential energy for an element of the catenary, and conservation of energy for motion along the brachistochrone. Of course, we haven't actually covered those things yet, but you're already very familiar with them from your undergraduate courses, and my aim was to give easily understood physical realizations of minimization problems, and to show how to find the minimal shapes using the calculus of variations.

At this point, we'll begin a full study of dynamics,
starting with the laws of motion. The text, Landau, begins (page 2!) by stating
that the laws come from the principle of least action, Hamilton's principle. This
is certainly one possible approach, but confronted with it for the first time,
one might well wonder where *it* came from. I prefer a gentler
introduction, more or less following the historical order: Galileo, then
Newton, then Lagrange and his colleagues, then Hamilton. The two approaches
are of course equivalent. Naturally, you've seen most of this earlier stuff before,
so here is a very brief summary.

To begin, then, with Galileo. His two major contributions to dynamics were:

1. The realization, and experimental verification, that
falling bodies have constant acceleration (provided air resistance can be
ignored) and *all falling bodies accelerate at the same rate*.

2. *Galilean relativity*. As he put it himself, if you
are in a closed room below decks in a ship moving with steady velocity, no
experiment on dropping or throwing something will look any different because of
the ship's motion: you can't detect the motion. As we would put it now, the
laws of physics are the same in all inertial frames.

*Newton's *major contributions were his *laws of
motion*, and his *law of universal gravitational attraction*.

His laws of motion:

1. *The law of inertia*: a body moving at constant
velocity will continue at that velocity unless acted on by a force. (Actually,
Galileo essentially stated this law, but just for a ball rolling on a
horizontal plane, with zero frictional drag.)

2.

3. *Action = reaction*.

In terms of Newton's laws, Galilean relativity is clear: if the ship is moving at steady velocityrelative to the shore, than an object moving at relative to the ship is moving at relative to the shore. If there is no force acting on the object, it is moving at steady velocity in both frames: both are inertial frames, defined as frames in which Newton's first law holds. And, since is constant, the acceleration is the same in both frames, so if a force is introduced the second law is the same in the two frames.

(Needless to say, all this is classical, meaning nonrelativistic, mechanics.)

Any dynamical system can be analyzed as a (possibly infinite) collection of parts, or particles, having mutual interactions, so in principle Newton's laws can provide a description of the motion developing from an initial configuration of positions and velocities.

The problem is, though, that the equations may be intractable -- we can't do the mathematics. It is evident that in fact the Cartesian coordinate positions and velocities might not be the best choice of parameters to specify the system's configuration. For example, a simple pendulum is obviously more naturally described by the angle the string makes with the vertical, as opposed to the Cartesian coordinates of the bob. After Newton, a series of French mathematicians reformulated his laws in terms of more useful coordinates -- culminating in Lagrange's equations.

The Irish mathematician Hamiltonian then established that
these improved dynamical equations could be derived using the calculus of
variations to minimize an integral of a function, the *Lagrangian*, along
a path in the system's configuration space. This integral is called the *action*,
so the rule is that the system follows the path of least action from the
initial to the final configuration.

### Derivation of Hamilton's Principle from Newton's Laws in Cartesian Co-ordinates: Calculus of Variations Done Backwards!

We've shown how, given an integrand, we can find differential equations for the path in space time between two fixed points that minimizes the corresponding path integral between those points.

Now we'll do *the reverse*: we already know the
differential equations in Cartesian coordinates describing the path taken by a
Newtonian particle in some potential. We'll show how to use that knowledge to
construct the integrand such that the action integral is a minimum along that
path. (This follows Jeffreys and Jeffreys, *Mathematical Physics*.)

We begin with the simplest nontrivial system, a particle of mass moving in one dimension from one point to another in a specified time, we'll assume it's in a time-independent potential , so

Its path can be represented as a graph against time -- for example, for a ball thrown directly upwards in a constant gravitational field this would be a parabola.

Initial and final positions are given: , , and the elapsed time is .

Notice we have *not* specified the initial velocity -- we
don't have that option. The differential equation is only second order, so its
solution is completely determined by the two (beginning and end) boundary
conditions.

We're now ready to embark on the calculus of variations in reverse.

Trivially, multiplying both sides of the equation of motion by an arbitrary infinitesimal function the equality still holds:

and in fact if *this *equation is true for *arbitrary
*, the original equation
of motion holds throughout, because we can always choose a nonzero only in
the neighborhood of a particular time , from which the
original equation must be true at that .

By analogy with Fermat's principle in the preceding section, we can picture this as a slight variation in the path from the Newtonian trajectory, , and take the variation zero at the fixed ends, .

In Fermat's case, the integrated time elapsed along the path was minimized -- there was zero change to first order on going to a neighboring path. Developing the analogy, we're looking for some dynamical quantity that has zero change to first order on going to a neighboring path having the same endpoints in space and time. We've fixed the time, what's left to integrate along the path?

For such a simple system, we don't have many options! As we've discussed above, the equation of motion is equivalent to (putting in an overall minus sign that will prove convenient)

Integrating the first term by parts (recalling at the endpoints):

,

using the standard notationfor kinetic energy.

The second term integrates trivially:

establishing that on making an infinitesimal variation from
the physical path (the one that satisfies Newton's laws) there is zero first
order change in the integral of *kinetic energy minus potential energy*.

The standard notation is

The integralis
called the *action integral*, (also known as Hamilton's Principal
Function) and the integrand is called the *Lagrangian*.

**This equation is
Hamilton's Principle.**

The derivation can be extended straightforwardly to a
particle in three dimensions, in fact to interacting
particles in three dimensions. We shall assume that the forces on particles
can be derived from potentials, including possibly time-dependent potentials,
but we exclude frictional energy dissipation in this course. (It can be
handled -- see for example Vujanovic and Jones, *Variational Methods in Nonconservative
Phenomena*, Academic press, 1989.)

### But Why?

Fermat's principle was easy to believe once it was clear that light was a wave. Imagining that the wave really propagates along all paths, and for light the phase change along a particular path is simply the time taken to travel that path measured in units of the light wave oscillation time. That means that if neighboring paths have the same length to first order the light waves along them will add coherently, otherwise they will interfere and essentially cancel. So the path of least time is heavily favored, and when we look on a scale much greater than the wavelength of the light, we don't even see the diffraction effects caused by imperfect cancellation, the light rays might as well be streams of particles, mysteriously choosing the path of least time.

So what has this to do with Hamilton's principle? Everything. A standard method in quantum mechanics these days is the so-called sum over paths, for example to find the probability amplitude for an electron to go from one point to another in a given time under a given potential, you can sum over all possible paths it might take, multiplying each path by a phase factor: and that phase factor is none other than Hamilton's action integral divided by Planck's constant, So the true wave nature of all systems in quantum mechanics ensures that in the classical limit the well-defined path of a dynamical system will be that of least action. (This is covered in my notes on Quantum Mechanics.)

*Historical footnote*: Lagrange developed these
methods in a classic book that Hamilton called a "scientific poem". Lagrange
thought mechanics properly belonged to pure mathematics, it was a kind of
geometry in four dimensions (space and time). Hamilton was the first to use
the principle of least action to derive Lagrange's equations in the present
form. He built up the least action formalism directly from Fermat's principle,
considered in a medium where the velocity of light varies with position and
with direction of the ray. He saw mechanics as represented by geometrical
optics in an appropriate space of higher dimensions. But it didn't apparently
occur to him that this might be because it was really a wave theory! (See
Arnold, *Mathematical Methods of Classical Mechanics*, for details.)

### Lagrange's Equations from Hamilton's Principle Using Calculus of Variations

We started with Newton's equations of motion, expressed in Cartesian coordinates of particle positions. For many systems, these equations are mathematically intractable. Running the calculus of variations argument in reverse, we established Hamilton's principle: the system moves along the path through configuration space for which the action integral, with integrand the Lagrangian is a minimum.

We're now free to *begin* from Hamilton's principle,
expressing the Lagrangian in variables that more naturally describe the system,
taking advantage of any symmetries (such as using angle variables for rotationally
invariant systems). Also, some forces do not need to be included in the
description of the system: a simple pendulum is fully specified by its position
and velocity, we do not need to know the tension in the string, although that *would*
appear in a Newtonian analysis. The greater efficiency (and elegance) of the
Lagrangian method, for most problems, will become evident on working through
actual examples.

We'll define a set of *generalized coordinates*** ** by requiring
that they give a complete description of the configuration of the system (where
everything is in space). The state of the system is specified by this set plus
the corresponding velocities . For example,
the -coordinate of a
particular particleis
given by some function of the *'*s, and the
corresponding velocity component .

The Lagrangian will depend on all these variables in general, and also possibly on time explicitly, for example if there is a time-dependent external potential. (But usually that isn't the case.)

Hamilton's principle gives

that is,

Integrating by parts,

Requiring the path deviation to be zero at the endpoints gives Lagrange's equations:

### Non-uniqueness of the Lagrangian

The Lagrangian is not uniquely defined: *two Lagrangians
differing by the total derivative with respect to time of some function will
give the same identical equations on minimizing the action*,

and since are all fixed, the integral over is trivially independent of path variations, and varying the path to minimize gives the same result as minimizing. This turns out to be important later -- it gives us a useful new tool to change the variables in the Lagrangian.

### First Integral: Energy Conservation and the Hamiltonian

Since Lagrange's equations are precisely a calculus of variations
result, it follows from our earlier discussion that *if the Lagrangian has no
explicit time dependence* then:

(This is just the first integral discussed earlier, now withvariables.)

This constant of motion is called the *energy* of the
system, and denoted by.
We say the energy is *conserved*, even in the presence of external
potentials -- provided those potentials are time-independent.

(We'll just mention that the function on the left-hand side, is the Hamiltonian. We don't discuss it further at this point because, as we'll find out, it is more naturally treated in other variables.)

We'll now look at a couple of simple examples of the Lagrangian approach.

### Example 1: One Degree of Freedom: Atwood's Machine

In 1784, the Rev. George Atwood, tutor at Trinity College, Cambridge, came up with a great demo for finding. It's still with us.

The traditional Newtonian solution of this problem is to write for the two masses, then eliminate the tension. (To keep things simple, we'll neglect the rotational inertia of the top pulley.)

The Lagrangian approach is, of course, to write down the Lagrangian, and derive the equation of motion.

Measuring gravitational potential energy from the top wheel axle, the potential energy is

and the Lagrangian

Lagrange's equation:

gives the equation of motion in just one step.

It's usually pretty easy to figure out the kinetic energy and potential energy of a system, and thereby write down the Lagrangian. This is definitely less work than the Newtonian approach, which involves constraint forces, such as the tension in the string. This force doesn't even appear in the Lagrangian approach! Other constraint forces, such as the normal force for a bead on a wire, or the normal force for a particle moving on a surface, or the tension in the string of a pendulum -- none of these forces appear in the Lagrangian. Notice, though, that these forces never do any work.

On the other hand, if you actually are interested in the tension in the string (will it break?) you use the Newtonian method, or maybe work backwards from the Lagrangian solution.

### Example 2: Lagrangian Formulation of the Central Force Problem

A simple example of Lagrangian mechanics is provided by the central force problem, a mass acted on by a force .

To contrast the Newtonian and Lagrangian approaches, we'll first look at the problem using just . To take advantage of the rotational symmetry we'll use coordinates, and find the expression for acceleration by the standard trick of differentiating the complex number twice, to get

The second equation integrates immediately to give

a constant, the angular momentum. This can then be used to eliminatein the first equation, giving a differential equation for.

The Lagrangian approach, on the other hand, is first to write

and put it into the equations

Note now that since doesn't depend on , the second equation gives immediately:

and in fact the angular momentum, we'll call it.

The first integral (see above) gives another constant:

This is just

the energy.

Angular momentum conservation, then gives

giving a first-order differential equation for the radial motion as a function of time. We'll deal with this in more detail later. Note that it is equivalent to a particle moving in one dimension in the original potential plus an effective potential from the angular momentum term:

This can be understood by realizing that for a fixed
angular momentum, the closer the particle approaches the center the greater its
speed in the tangential direction must be, so, to conserve total energy, its
speed in the radial direction has to go down, unless it is in a *very*
strongly attractive potential (the usual gravitational or electrostatic
potential isn't strong enough) so the radial motion is equivalent to that with
the existing potential plus the term, often
termed the "centrifugal barrier".

*Exercise*: how strong must the potential be to
overcome the centrifugal barrier? (This can happen in a black hole!)

### Generalized Momenta and Forces

For the above orbital Lagrangian, the momentum in the -direction, and the angular momentum associated with the variable

The **generalized momenta** for a mechanical system are
defined by

(*Warning*: these generalized momenta are an essential
part of the formalism, but do not always directly correspond to the *physical*
momentum of a particle, an example being a charged particle in a magnetic
field, see my quantum notes.)

Less frequently used are the generalized *forces*, defined to make the
Lagrange equations look Newtonian,

### Conservation Laws and Noether's Theorem

The two integrals of motion for the orbital example above can be stated as follows:

*First***: ** if the Lagrangian does not depend on
the variable that
is, it's *invariant under rotation*,* *meaning** **it has circular
symmetry, then

*angular momentum is conserved*.

*Second*: As stated earlier, if the Lagrangian is
independent of time, that is, it's *invariant under time translation*,
then *energy is conserved*. (This is nothing but the first integral of
the calculus of variations, recall that for an integrand function not explicitly
dependent on is constant.)

*Both these results link symmetries of the Lagrangian -- invariance
under rotation and time translation respectively -- with conserved quantities.*

This connection was first spelled out explicitly, and proved generally, by Emmy Noether, published in 1915. The essence of the theorem is that if the Lagrangian (which specifies the system completely) does not change when some continuous parameter is altered, then some function of the stays the same -- it is called a constant of the motion, or an integral of the motion.

To look further at this expression for energy, we take a closed system of particles interacting with each other, but "closed" means no interaction with the outside world (except possibly a time-independent potential).

The Lagrangian for the particles is, in Cartesian coordinates,

.

A set of general coordinates, by definition, uniquely specifies the system configuration, so the coordinate and velocity of a particular particleare given by

From this it is clear that the kinetic energy term is a homogeneous quadratic function of the 's (meaning every term is of degree two), so

This being of degree two in the time derivatives means

(If this isn't obvious to you, check it out with a couple of terms: )

Therefore, recalling from the time-independence of
the Lagrangian that is constant, we see that
this constant is simply , which as we
shall see, is the *Hamiltonian*, to be discussed later.

### Momentum

Another conservation law follows if the Lagrangian is unchanged by displacing the whole system through a distance This means, of course, that the system cannot be in some spatially varying external field -- it must be mechanically isolated.

It is natural to work in Cartesian coordinates to analyze this, each particle is moved the same distance , so

where the "differentiation by a vector" notation means differentiating with respect to each component, then adding the three terms. (I'm not crazy about this notation, but it's Landau's, so get used to it.)

For an isolated system, we must have on displacement, moving the whole thing through empty space in any directionchanges nothing, so it must be that the vector sum , so from the Cartesian Euler-Lagrange equations, writing

so, taking the system to be composed of particles of massand velocity,

the *momentum* of the system.

This vector conservation law is of course three separate directional
conservation laws, so even if there *is *an external field, if it doesn't
vary in a particular direction, the component of total momentum in that
direction will be conserved.

In the Newtonian picture, conservation of momentum in a closed system follows from Newton's third law. In fact, the above Lagrangian analysis is really Newton's third law in disguise. Since we're working in Cartesian coordinates, , the force on the th particle, and if there are no external fields, just means that if you add all the forces on all the particles, the sum is zero. For the Lagrangian of a two particle system to be invariant under translation through space, the potential must have the form , from which automatically .

### Center of Mass

If an inertial frame of referenceis moving at constant velocity relative to inertial framethe velocities of individual particles in the frames are related by so the total momenta are related by

If we choose then the system is "at
rest" in the frame Of
course, the individual particles might be moving, what is at rest in is the *center
of mass* defined by

(Check this by differentiating both sides with respect to time.)

The energy of a mechanical system in its rest frame is often
called its *internal energy*, we'll denote it by (This includes
kinetic and potential energies.) The total energy of a moving system is then

(*Exercise*: verify this.)

### Angular Momentum

Conservation of momentum followed from the invariance of the
Lagrangian on being displaced in arbitrary directions in space, the homogeneity
of space, angular momentum conservation is the consequence of the *isotropy*
of space -- there is no preferred direction.

So angular momentum of an isolated body in space is invariant even if the body is not symmetric itself.

The strategy is just as before, except now instead of an
infinitesimal displacement we make an infinitesimal *rotation*,

and of course the velocities will also be rotated:

We must have

Now by definition, and from Lagrange's equations so the isotropy of space implies that

Notice the second term is identically zero anyway, since two of the three vectors in the triple product are parallel:

That leaves the first term. The equation can be written:

Integrating, we find that

is a constant of motion, the *angular momentum*.

The angular momentum of a system is different about different origins. (Think of a single moving particle.) The angular momentum in the rest frame is often called the intrinsic angular momentum, the angular momentum in a frame in which the center of mass is at positionand moving with velocity is

(*Exercise*: check this.)

For a system of particles in a fixed external central field the system is
invariant with respect to rotations *about that point*, so angular
momentum about that point is conserved. For a field "cylindrically" invariant
for rotations about an axis, angular momentum about that axis is conserved.

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