First Integrals Flashcards

1
Q

Newton’s Equation

A

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

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2
Q

First Integral

Definition

A
-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
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3
Q

Time Independent First Integral

Definition

A

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

∂Ф/∂t = 0

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4
Q

Geometric Meaning of First Integrals

A

-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

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5
Q

Level Sets and Integral Curves

A
  • 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
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6
Q

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

A
  • 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
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7
Q

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

A
  • 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
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8
Q

First Integrals and Differentiable Functions

A

-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

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9
Q

Counting First Integrals

A

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

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10
Q

Functionally Independent

Definition

A

-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

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11
Q

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

A

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

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12
Q

Existence Theorem

A

-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

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13
Q

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

A

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

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14
Q

What do we use reduction of order to do?

A
  • 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
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15
Q

How to reduce order?

A

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

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