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MATH2391 Non-Linear Differential Equations > First Integrals > Flashcards

Flashcards in First Integrals Deck (15)
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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


First Integral

-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
dФ/dt = ∂Ф/∂t + Σ dxn/dt ∂Ф/∂xn
= ∂Ф/∂t + |F(|x) . ∇Ф = 0


Time Independent First Integral

-we say that a first integral is time independent if:
∂Ф/∂t = 0


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


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


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


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


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


Counting First Integrals

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


Functionally Independent

-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


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


Existence Theorem

-let |x' = |f(|x), |xϵℝ^n and |f(|x) be a smooth vector field
in a small vicinity of any point |xo, there exist N functionally independent (time dependent) first integrals


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

-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


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


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