Flashcards in First Integrals Deck (15)

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1

## Newton's Equation

###
m x'' = F(x)

-where the force F(x) depends only on the position of the point, but does not depend on time t or velocity v=dx/dt

2

##
First Integral

Definition

###
-a non-constant differentiable function |Ф = |Ф(|x,t) is called a first integral of the system:

d|x/dt = |F(|x) , |x, |F(|x)ϵℝ^n

IF

dФ/dt = ∂Ф/∂t + Σ dxn/dt ∂Ф/∂xn

= ∂Ф/∂t + |F(|x) . ∇Ф = 0

3

##
Time Independent First Integral

Definition

###
-we say that a first integral is time independent if:

∂Ф/∂t = 0

4

## Geometric Meaning of First Integrals

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-time independent first integrals Ф(|x) have simple geometrical interpretation, a trajectory passing through the point |xo stays on the surface:

V = {|xϵℝ^n | Ф(x)=Ф(xo)}

-which is a level set of the first integral

5

## Level Sets and Integral Curves

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-a level set of time dependent first integrals gives a surface in the extended phase space

-if one point of an integral curve belongs to this surface, then the whole integral curve belongs to it

6

## What is the dimension of the level sets of a first integral for a system of N equations?

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-for a system of N equations, the elements of the level set of a first integral are N-1 dimensional surfaces

-a trajectory of the system belongs to a level set

7

## Where are the trajectories in systems with more than one first integral?

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-a trajectory must simultaneously belong to the surface levels of each first integral

-i.e. it is a line of the intersection of the surfaces

8

## First Integrals and Differentiable Functions

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-if Ф=Ф(|x,t) is a first integral, then any differentiable function f(Ф) is a first integral since:

df(Ф)/dt = dФ/dt * df(Ф)/dФ = 0 since dФ/dt=0 by definition

-if a system has two or more first integrals then any differentiable function of these integrals is a first integral;

df(Ф1,Ф2,..,Фn)/dt

= dФ1/dt df/dФ1 + ... + dФn/dt df/dФn = 0

9

## Counting First Integrals

### -counting first integrals we should count only functionally independent first integrals

10

##
Functionally Independent

Definition

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-if Ф = f(Ф1,...,Фn) then the gradients of Ф and Ф1,...,Фn are linearly dependent:

∇Ф = ∇ f(Ф1,...,Фn)

= df/dФ1 ∇Ф1 + ... + df/dФn ∇Фn

-note that the coefficients can be functions of dynamical variables x1,...,xn

-in other words, if gradients ∇Ф1, ... , ∇Фn are linearly independent, then the first integrals Ф1,...,Фn are functionally independent

11

## How to show that first integrals Ф1,...,Фm are functionally indepenent

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1) compute their gradients

2) form an NxM matrix:

J(Ф1,..,Фm) = first row ∇Ф1, second row ∇Ф2, ... , mth row ∇Фm

3) reduce to REF to determine the rank of the matrix

4) if rank(J) = M then the first integrals Ф1,...,Фm are functionally independent

12

## Existence Theorem

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-let |x' = |f(|x), |xϵℝ^n and |f(|x) be a smooth vector field

THEN

in a small vicinity of any point |xo, there exist N functionally independent (time dependent) first integrals

13

## How many time independent first integrals can be found for a particular system?

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N-1

-since you can express t in terms of first integral Ф1 and x1, x2, ..., xn and then sub that expression for t into the other N-1 first integrals Ф2,Ф3,...,ФN

14

## What do we use reduction of order to do?

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-having one time independent first integral, we can reduce the order of the system by one

-having k functionally independent and time independent first integrals, we can reduce the order of the system by k

15