# Variational Calculus

## Sources

• Most of these notes comes from the Variational Calculus taught by Prof. José Figueroa O'Farrill, University of Edinburgh

## Notation

• denotes the space of possible paths (i.e. curves) between points and

## Introduction

• Shortest path between two given points on a surface
• Curve between two given points in the place that yields a surface of revolution of minimum area when revolved around a given axis
• Curve along which a bead will slide (under the effect of gravity) in the shortest time

Underpins much of modern mathematical physics, via Hamilton's principle of least action

Consider "standard" directional derivative of , with , at along some vector :

where is a critical point of , i.e.

(since we know that form a basis in , so by lin. indep. we have the above!)

## Stuff

Let be a continuous function which satisfies

for all with

Then .

Observe that if we only consider , then we can simply genearlize to arbitrary since integration is a linear operation.

Let which obeys

Assume, for the sake of contradiction, that , i.e.

Let, w.l.o.g., .

Since is continuous, there is some interval such that and some such that

Suppose for the moment that there exists some such that

1. for all outside

Then observe that

This is clearly a contradiction with our initial assumption, hence such that . Hence, by continuity,

Now we just prove that there exists such a function which satisfies the properties we described earlier.

Let

which is a smooth function. Then let

which is a smooth function, since it's a product of smooth functions.

To make this vanish outside of , we have

This function is clearly always positive, hence,

Hence, letting we get a function used in the proof above!

This concludes our proof of Fundamental Lemma of the Calculus of Variations

### General variations

Suppose we want te shortest path in between and a curve on .

We assume that with differentiable, such that .

Then observe that , then

Then

where we've used the fact that

We cannot just drop the endpoint-terms anymore, since these are now not necessarily vanishing, as we had in the previous case.

Then, by our earlier assumption, we have

which implies that

Hence,

must hold for all and and

In particular, , then

and

i.e. is normal to at the point where it intersects with .

## Euler-Lagrange equations

### Endpoint-fixed variations

Let be the space of curves with

The Lagrangian is defined

where and . Let be "sufficiently" differentiable (usually taken to be smooth in applications).

Then the function , called the action, is defined

A path is a critical point for the action if, for all endpoint-fixed variations , we have

Bringing the differentiation into to the integral, we have

### Properties

1. If , then the "energy" given by

is constant along extremals of the Lagrangian. Then observe that

where we've used the fact that in the thrid equality. Thus,

and,

Hence,

i.e time invariance! This is an instance of Noether's Theorem.

If , so that the lagrangian does not depend explicitly on , then the energy

is constant.

This is known as the Beltrami's idenity.

### Euler-Lagrange

Let be the space of curves with

Let

where and , be sufficiently differentiable (typically smooth in applications) and let us consider the function defined by

Then he extremals must satisfy the Euler-Lagrange equations:

## Newtonian mechanics

### Notation

• worldlines refer to the trajectory of a particle:

with

### Galilean relativity

• Affine transformations (don't assume a basis)
• Relativity group: group of transformations on the universe preserving whichever structure we've endowed the universe with

The subgroup of affine transformations of which leave invariant the time interval between events and the distance between simultaneous events is called the Galilean group.

That is, the Galilean group consists of affine transformations of the form

These transformations can be written uniquely as a composition of three elementary galilean transformations:

1. translations in space and time:

2. orthogonal transformations in space:

3. and galilean boosts:

Observe that if choose the action

which has Lagrangian

then we observe that the minimizing path should satisfy

which is

where denotes the force. Further,

which is the momentum! Then,

Hence we're left with Newton's 2nd law.

## Noether's Theorem

### Notation

• of functions called one-parameter subgroup of diffeomorphisms , which are differentiable wrt.
• and are defined by

and

### Stuff

We've seen the following continuous symmetries this far:

1. Momentum is conserved:

2. Energy is conserved:

We say that is a symmetry of the Lagrangian if

where

Equivalently, one says that is invariant under .

Let of functions, defined for all and depending differentiably on .

Moreover, let this family satisfies the following properties:

1. for all
2. for all

Then the family is called a one-parameter subgroup of diffeomorphisms on .

Let be an action for curves , and let be invariant under a one-parameter group of diffeomorphisms .

Then the Noether charge , defined by

is conserved; that is, along physical trajectories.

Consider functions which are defined by Lagrangians and such that they are invariant under one-parameter family of diffeomorphisms of such that

This means, in particular, that

where

Then the Noether charge

is conserved along extremals; that is, along curves which obey the Euler-Lagrange equation.

## Hamilton's canonical formalism

### Notation

• Considering

for curves

• Differentiable function
• denotes the Hamiltonian

### Canonical form of Euler-Lagrange equation

• Can convert E-L into equivalent first-order ODE:

is equivalent to system

for the variables .

• Consider

Then letting

the above system becomes

• Hamiltonian becomes

which has the property

known as Hamilton's equations.

• Observe that

where the coefficient matrix, let's call it , can be thought of as a bilinear form on , which defines a symplectic structure.

• In general case, existence of solution set to the equations is guaranteed by the implicit function theorem in the case where the Hessian is invertible
• is then said to be regular (or non-degenerate)

General case:

• Hamiltonian

• Total derivative of (or as we recognize, the exterior derivative of a continuous function)

where we have used that .

• Give us

• First-order version of Euler-Lagrange equations in canonical (or hamiltonian) form:

which we call Hamilton's equations.

In general, the Hamiltonian is given by

Taking the total derivative, we're left with

where we've used .

This gives us

### Conserved quantity using Poisson brackets

Consider energy conserved, i.e. , and a differentiable function :

where we have introduced the Poisson bracket

for any two differentiable functions of .

Hence

If depend explicity on , then the same calculation as above would show that

In this case could still be conserved if

Given two conserved quantities and , i.e. Poisson-commute with the hamiltonian .

Then we can generate new conserved quantities from old using the Jacobi identity:

Therefore is a conserved quantity.

#### Can we associate some * to an conserved charge?

Consider a conserved quantity which satisfy

Then

defines a vector field on the phase space which we may integrate to find through every point a solution to the differential equation

Then, by existence and uniqueness of solutions to IVPs on some open interval as given above, gives us the unique solutions and for some initial values and .

Therefore,

Thus we have a continuous symmetry of Hamilton's equations, which takes solutions to solutions.

That is, these solutions which are in some sense generated from and contain symmetries!

E.g. solution to the system of ODEs above could for example be a linear combination of and , in which case we would then understand that "symmetry" generated by is rotational symmetry.

The caveat is that this may not actually extend to a one-parameter family of diffeomorphisms, as we used earlier in Noether's theorem as the invariant functions or symmetries.

• Lagrangian:

• Exists

### TODO Integrability

A set of functions which Poisson-commute among themselves are said to be in involution.

Liouville's theorem says that if a hamiltonian on admits independent conserved quantities in involution, then there is a canonical transformation to so called action / angle variables such that Hamilton's equations imply that

Such a system is said to be integrable.

## Constrained problems

### Isoperimetric problems

#### Notation

• Typically we talk about the constrained optimization problem:

for .

#### Stuff

• Extremise a functional subject to a functional constraint

Consider a closed loop enclosing some area .

The Dido's problem or (original) isoperimetric problem is the problem of maximizing the area of while keeping the length of constant.

Consider for with initial conditions

Using Green's theorem, we have

and length

The problem is then that we want to extremise wrt. the constraint .

More generally, given functions for

being endpoint fixed, e.g. and for some .

Given the functionals

we want to extremise subject to .

Let and let be an extremum of subject to .

If (i.e. is not a critical point), then (called a Lagrange multiplier) s.t. is a critical point of the function defined by

Supose is an extremal of in the space of with . Then

for small , where .

This might constrain

and prevent use of the Fundamental Lemma of Calculus of Variations.

Idea:

• Consider and for all near , defined .
• Then express one of the variations as the other, allowing us to eliminate one.

Let be defined

Assume now that and .

If we specifically consider (wlog) , then IFT implies that

for "small" . Therefore,

is a

Let

be functionals of functions subject to BCs

Suppose that is an extremal of subject to the isoperimetric constraint . Then if is not an extremal of , there is a constant so that is an extremal of . That is,

#### Method of Lagrange multipliers for functionals

Suppose that we wish to extremise a functional

for .

Then the method consists of the following steps:

1. Ensure that has no extremals satisfying .
2. Solve EL-equation for

which is second-order ODE for .

3. Fix constants of integration from the BCs
4. Fix the value of using .

#### The catenary

##### Notation
• denotes height as a function of arclength
##### Catenary
• Uniform chain og length hangs under its own weight from two poles of height a distance apart
• Potential energy is given by , which we can parametrize

giving us

• Observation:
• All extremals of arclength are straight lines
• Extremal to constrained problem has non-zero gradient of the constraint
• → straight lines are not the solutions
• Consider lagrangian

• Using Beltrami's identity:

rewritten to

which, given the BCs we get the solution

which follows from taking the derivative of both sides and then solving.

• Impose the isoperimetric condition:

• Introducing , we find the following transcendental equation

for which

• is the trivial solution
• small, together with the condition ensures that
• But for the exponential term dominates

by continuity.

### Holonomic and nonholonomic constraints

We say a constraint is scleronomic if the constraint does not depend explicitly on , and rheonomic if it does.

We say a constraint is holonomic if it does not depend explicitly on , and nonholonomic if it does.

In the case where nonholonomic constraints are at most linear in , then we say that the constraints are pfaffian constraints.

#### Typical usecases

• Finding geodesics on a surface as defined as the zero locus of a function, which are scleronomic and holonomic.
• Reducing higher-order lagrangians to first-order lagriangians
• Mechanical problems: e.g. "rolling without sliding", which are typically nonholonomic

#### Holonomic constraints

Let be a smooth extremal for

with for all .

Then there exists such that obeys the EL equations of

is the gradient wrt. for all , NOT .

Let be admissible variation of .

Then the constraint requires

Let

Then the contraint above implies that

i.e. if then is tangent to the implicitly defined surface .

Consider , then the above gives us

We therefore suppose that

The implicit function then implies that we can solve for one component. Then the above gives us

i.e. is arbitrary and is fully determined by in this "small" neighborhood .

Then

Since is arbitrary, FTCV implies that

Which is jus the E-L equation for a lagrangian given by

Above we use the implicit function theorem to prove the existence of such extremals, but one can actually prove this using something called "smooth partion of unity".

In that case we will basically do the above for a bunch of different neighborhoods, and the sum them together to give us (apparently) the same answer!

#### Examples

##### (Non-holonomic) Higher order lagrangians

can be replaced by

which are scleronomic and nonholonomic.

So we consider the Lagriangian

Then

So the E-L equations gives us

which gives us

where we've used

## Variational PDEs

### Notation

• be a function on the set
• Lagrangian over a surface with corresponding action

where and denote collectively the partial derivatives and for

• BCs are given by

where is given

• Variations are functions such that

### Stuff

Then

where we've used integration by parts in the last equality.

The Divergence (or rather, Stokes') theorem allows us to rewrite the last integral as

where is the arclength. And since this vanishes.

Generalisation of the FLCV the first integral term must vanish and so we get

We can generalise this to more than just 2D!

### Multidimensional Euler-Lagrange equations

Let

• be a bounded region with (piecewise) smooth boundary.
• denote the coordinates for
• be the Lagrangian for maps where denotes collectively the partial derivatives

Then the general Euler-Lagrange equations are given by

using Einstein summation.

Notice that we are treating as a function of the x's and differentiate wrt. keeping all other x's fixed!

(This is really Stokes' theorem)

Let

• be bounded open set with (piecewise) smooth boundary
• be a smooth vector field defined on
• be the unit outward-pointing normal of

Then,

(using Einstein summation) where is the volume element in and is the area element in and denotes the Euclidean inner product in .

Let

• be bounded open with (piecewise) smooth boundary
• be a continuous function which obeys

for all functions vanishing on .

Then .

### Noether's theorem for multidimesional Lagrangians

#### Notation

• Lagrangian

where

• Use the notation

• Conserved now refers to "divergenceless", that is, is a conserved quantity if

where we're using Einstein summation.

• for denotes a one-parameter group of diffeomorphisms
• is defined

### Stuff

• We say it's a conserved "current" because

where denotes the normal to the boundary

Consider

And let

such that

We suppose that the action is invariant, so that the Lagrangian obeys

or equivalently, one can (usually more easily) check that the following is true

The Noether current is then given by

since the RHS is independent of . The LHS on the other hand is given by

where .

Now we observe the following:

and

where we've simply taking the derivatives of the Taylor expansions. Hence, we are left with

Now we need to evaluate and it's derivative wrt. . First we notice that

We now compute . We first have

Finally using the fact that if a we have some matrix given by

we have

Finally we need to compute , which one will find to be

AND I NEED TO PRACTICE FOR MY EXAM INSTEAD OF DOING THIS. General ideas are the above, and then just find an expression for the missing part. Then, you do some nice manipulation, botain an expression which vanish due to the EL equations being satisfied by the non-transformed , and you end up with the Noether's current for the multi-dimensional case.

#### Examples

##### Minimal surface

Let be a twice differentiable function.

The grahp defines a surface . The area of this surface is the functional of given by

If is an extremal of this function, we say that is a minimal surface.

In this case the Lagrangian is

with the EL-equations

where

Therefore,

and similarily for . When combined, and multiplied by since the combination equal zero anyways, we're left with

where we've used the fact that .

This is then the equation which must be satisfied by a minimal surface.

##### model

then

Then

Then the Noether's current is

which explicitly in this case is

Then

## Classical Field Theory

### Notation

• denotes a field written
• Concerned with action functionals of the form

where is called a Lagrangian density and , i.e.

• for for some and

• denotes a "cylindrical" region

• is the outward normal to the boundary

### Stuff

Lagrangian density is just used to refer to the fact that we are now looking at a variation of the form

So it's like the is the Lagrangian now, and the "inner" functional is the a Lagrangian density.

Klein-Gordon equation in (i.e. ) is given by

where is called the mass.

If , then this is the wave equation, whence the Klein-Gordon equation is a sort of massive wave equation.

More succinctly, introducing the matrix

then the Klein-Gordon equation can be written

Note: you sometimes might see this written

where they use the notation so we have sort of "summed out" the .

### Noether's Theorem for improper actions

• Consider action function for a classical field which is invariant under continuous one-parameter symmetry with Noether current
• Integrate (zero) divergence of the current on a "cylindrical region" and apply the Divergence Theorem

• consists of "sides" of the "cylinder", where
• is the m-sphere of radius
• top cap
• bottom cap
• Can rewrite the above as

using the fact that points outward at the bottom cap → negative axis

• Last term vanishes due to BCs on the field implies on as
• Since arbitrary, we have

is conserved, i.e. we have a Noether's charge for the improper case!

### Maxwell equations

#### Notation

• is the magnetic field
• is the electric field
• is the electric charge density
• is the electric current density
• is the magnetic potential
• Let

#### Stuff

• Maxwell's equations

• Observe that can be solved by writing

• Does not determine uniquely since

leaves unchanged (since ), and is called a gauge transformation

• Substituting into Maxwell's equations:

• Thus there exists a function (again since ), called the electric potential, such that

• Performing gauge transformation → changes and unless also transform

• In summary, two of Maxwell's equations can be solved by

where and are defined up to gauge transformations

for some function

• We can fix the "gauge freedom" (i.e. limit the space of functions ) by imposing restrictions on , which often referred to as a choice of gauge, e.g. Lorenz gauge

The ambiguity in the definition of and in the Maxwell's equations can be exploited to impose the Lorenz gauge condition:

In which case the remaining two Maxwell equations become wave equations with "sources":

From these wave-equations we get electromagnetic waves!

#### Maxwell's equations are variational

• Let and at first
• Consider Lagrangian density

as functions of and , i.e.

• Observe that

• does not depend explicitly on or , only on their derivatives, so E-L are

and

which are precisely the two remaining Maxwell equations when and .

We can obtain the Maxwell equations with and nonzero by modifying :

We can rewrite by introducing the electromagnetic 4-potential

with so that .

The electromagnetic 4-current is defined

so that .

We define the fieldstrength

which obeys .

We can think of as entries of the antisymmetric matrix

where we have used that

In the terms of the fieldstrength we can write Maxwell's equations as

where we have used the "raised indices" of with as follows:

The Euler-Lagrange equations of are given by

and the gauge transformations are

under which are invariant.

In the absence of sources, so when , is gauge invariant.

Let denote the action corresponding to , then

where

and

Therefore,

Substituing this into our E-L equations from above, we (apparently) get

In the absence of sources, so when , is gauge invariant. This is seen by only considering the second-order of the transformation

1. First show that is invariant under the following

Consider

where

and

then

1. Find the Nother currents and

## Examples

### The Kepler Problem

• Illustrates Noether's Theorem and some techniques for the calculation of Poisson brackets
• Will set up problem both from a Lagrangian and a Hamiltonian point of view and show how to solve the system by exploiting conserved quantities

#### Notation

• Two particles of masses and moving in , with and denoting the corresponding positions
• Assuming particles cannot occupy same position at same time, i.e. for all .
• We then have the total kinetic energy of the system given by

and potential energy

#### Lagrangian description

• Lagrangian is, as usual given by

• is invariant under the diagonal action of the Euclidean group of on the configuration space, i.e. if is an orthonormal transformation and , then

leaves the Lagrangian invariant.