First Integrals

Question 1

Newton’s Equation

Answer 1

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

Question 2

First Integral

Definition

Answer 2

-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

Question 3

Time Independent First Integral

Definition

Answer 3

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

∂Ф/∂t = 0

Question 4

Geometric Meaning of First Integrals

Answer 4

-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

Question 5

Level Sets and Integral Curves

Answer 5

• 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

Question 6

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

Answer 6

• 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

Question 7

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

Answer 7

• 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

Question 8

First Integrals and Differentiable Functions

Answer 8

-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

Question 9

Counting First Integrals

Answer 9

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

Question 10

Functionally Independent

Definition

Answer 10

-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

Question 11

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

Answer 11

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

Question 12

Existence Theorem

Answer 12

-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

Question 13

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

Answer 13

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

Question 14

What do we use reduction of order to do?

Answer 14

• 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

Question 15

How to reduce order?

Answer 15

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