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MATH2391 Non-Linear Differential Equations > First Integrals > Flashcards

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

-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

-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?

-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?

-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

-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

-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

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

-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?

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?

-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

How to reduce order?

1) each first integral is equal to a constant, rearrange the first integrals to write the variables in terms of these constants and one other variable e.g. y
3) substitute these expressions for the variables into the original system of differentials
4) you should now have obtained a separable first order equation in terms of y