# Variational Calculus

## Table of Contents

## Sources

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

## Overview

## Notation

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

## Introduction

Precise analytical techniques to answer:

- 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

- 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

### Notation

- or denotes the Lagrangian

### 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

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:

translations in space and time:

orthogonal transformations in space:

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:

Momentum is conserved:

Energy is conserved:

We say that is a **symmetry** of the Lagrangian if

where

- is a diffeomorphism

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:

- for all
- 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**)

- is then said to be

**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*.

#### Example:

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:

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

which is second-order ODE for .

- Fix constants of integration from the BCs
- Fix the value of using .

#### Classical isoperimetric problem

#### 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

- All extremals of arclength are
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*equationfor which

- is the
*trivial*solution -
*small*, together with the condition ensures thatBut for the exponential term dominates

by

*continuity*.

- is the

### Holonomic and nonholonomic constraints

- Constraints are simply
*functions*, i.e. for some- Instead of
*functionals*as we saw for isoperimetric problems

- Instead of

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*can be replaced by

which are scleronomic and nonholonomic.

*Mechanical problems*: e.g. "rolling without sliding", which are typically nonholonomic

#### Holonomic constraints

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!

#### Nonholonomic constraints

#### 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

#### Noether's current for multidimensional Lagrangian

## 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 .

### Calculus of variations with improper integrals

is

*unbounded*, and so we need to considerwhere

i.e. the

*closed*ball of radiusVary the action

where we have omitted the

*boundary term*which is seen by applying the Divergence Theorem and using the BCs on the variation

Using Fundamental Lemma of Calculus of Variations we obtain the E-L equations

### 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*sinceleaves 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 thatPerforming

*gauge transformation*→ changes and*unless*also transformIn summary,

*two*of Maxwell's equations can be solved bywhere 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

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

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

- First show that is
*invariant*under the following

Consider

where

and

then

- 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 byand

*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 , thenleaves the Lagrangian

*invariant*.