SYSTEMS OF DIFFERENTIAL EQUATIONS, EULER S FORMULA. Uniqueness for solutions of ifferential equations. We consier the system of ifferential equations given by x = v( x), () t with a given initial conition x() = x. Here x R n an v is a function that maps R n into R n. We shall assume that for any two vectors x, x we v( x ) ( x ) L x x where L is some constant, usually calle the Lipschitz constant. An example is ( x) = A x where A is a constant real n n matrix. IWe compute A x A x = A( x x ) = ( x x ) A T A( x x ) λ ( x x ) where λ is the largest eigenvalue of A T A. The following is relatively easy to prove. Theorem.. The ifferential equation () has at most one solution that satisfies the given initial conition. Proof. Suppose there are two solutions x (t) an x (t) both satisfying x () = x () = x. Integrating we see that both solutions satisfy the equation x i (t) = x + v( x i (τ))τ, i =,. Hence, noting that the initial conition rops out, we get x (t) x (t) = v( x (τ))τ v( x (τ))τ = [ v( x (τ)) v( x (τ))]τ Using the Minkowski inequality which is essentially the triangle inequality we get x (t) x (t) an using the Lipschitz conition x (t) x (t) L v( x (τ)) v( x (τ)) τ x (τ)) x (τ) τ. an this hols for all t as long as the solutions exist. If t < T we have that x (t) x (t) L This inequality implies that for all t T that x (τ)) x (τ) τ L x (t) x (t) LT M(T ) T x (τ)) x (τ) τ
SYSTEMS OF DIFFERENTIAL EQUATIONS, EULER S FORMULA where we set M(T ) = max [,T ] x (t) x (t). Hence we also have that M(T ) LT M(T ) an if we choose T such that LT < it follows that M(T ) =. Hence the two solution coincie on the time interval [, T ]. Choosing x(t ) as the new initial conition the solution must coincie on the interval [T, T ] also an so on. We can argue the same way that for negative times the solutions have to coincie.. Some remarks about the e At Recall that we efine the exponential of a matrix e At by e At A n t n =. n! Here are some facts Theorem.. We have for all s, t R. n= e At e As = e A(t+s) Proof. Pick any initial conition x. The function x(t) = e A(t+s) x is a solution of the equation x = A x. This follows from t ea(t+s) = Ae A(t+s). Further the function y(t) = e At e As x is also a solution of the equation ẋ = A x. moreover, for t = we have that x() = e As x = y(). By uniqueness x((t) = y(t) an thus e At e As x = e A(t+s) x for all x. Since x is arbitrary this proves the theorem. An interesting consequence of this theorem is that e At is invertible for all t. e At e A( t) = e A(t t) = I. 3. One parameter families of matrices We say that a family of n n matrices P (t) is a one parameter family if an for all t, s R, P () = I P (t)p (s) = P (t + s). We shall only consier one parameter families that are ifferentiable. A particularly useful iea is to consier one parameter families of rotations R(φ). These are matrices that satisfy R(φ) T R(φ) = I. First we compute the erivative R(φ + ε) R(φ) R(φ) = lim φ ε ε R(ε) I = lim R(φ) = ΩR(φ) ε ε
where we enote SYSTEMS OF DIFFERENTIAL EQUATIONS, EULER S FORMULA 3 R(ε) I Ω = lim ε ε The matrix Ω is not arbitrary. Inee, ifferentiating = φ R(). φ IRT (φ)r(φ) = φ I = an bu the prouct rule IR T (φ)r(φ) = Ω T + Ω φ φ= an we learn that Ω must be a skew symmetric matrix, Ω T = Ω. So far this worke in arbitrary imensions. We specialize to three imension an write the general skew symmetric matrix as Ω = ω 3 ω ω 3 ω ω ω note the interesting fact that Ω x = ω x. We also note that Ω ω =. Recall that we have the equation R (φ) = ΩR(φ) an this allows us to compute R(φ) explicitly. We shall assume that the vector ω is normalize. We have to compute e Ωφ Ω n φ n = n! Here are some computations: which can be written as Ω = n= ω ω 3 ω ω ω ω 3 ω ω ω 3 ω ω ω 3 ω 3 ω ω 3 ω ω ω 3 Ω = I + ω ω T. Here we use that ω is a unit vector. Thus we can start a little table: Thus it makes sense to split Ω, Ω = I + ω ω T, Ω 3 = Ω, Ω 4 = Ω... e Ωφ = m= into even an o powers. We have that Ω m φ m (m)! + m= Ω m+ = ( ) m Ω Ω m+ φ m+ (m + )!
4 SYSTEMS OF DIFFERENTIAL EQUATIONS, EULER S FORMULA an hence the secon sum reuces to Ω m+ φ m+ = Ω (m + )! m= m= ( ) m φ m+ (m + )! For the even sum have to be careful noting that for m =,,... Ω m = ( ) m (I ω ω T ). For m = we have the ientity which we write an get that m= Ω m φ m (m)! I = I ω ω T + ω ω T = ω ω T + (I ω ω T ) which equals ω ω T + (I ω ω T ) cos φ. To summarize, we have shown that m= = Ω sin φ. ( ) m φ m (m)! e Ωφ = cos φi + ω ω T ( cos φ) + Ω sin φ Let s note a few things: The vector ω is an eigenvector for this matrix with eigenvalue. This is the axis of rotation. Take ω = i.e, the z axis. Then we get the matrix cos φ cos φ + sin φ = cos φ sin φ sin φ cos φ which is precisely a rotation in the positive irection by an angle φ. To summarize: Theorem 3.. The rotation about the ω axis by an angle φ is given by R(φ) = cos φi + ( cos φ) ω ω T + Ω sin φ, in particular R(φ) x = cos φ x + ( cos φ)( ω x) ω + sin φ( ω x). This is Euler s formula. Because Ω + I = ω ω T Euler s formula canals be written in the form R(φ) = cos φi + ( cos φ)(ω + I) + Ω sin φ = I + ( cos φ)ω + sin φω Note that the angle is any value between an π. If φ < we may replace φ by φ which keeps the sign of the cosine function fixe but changes the sign of the sign function. Thus if, aitionally we reverse the irection of ω we get back the same rotation. Neeless to say that the rotation by an angle φ = or φ = π is the ientity. Also note that in terms of R(φ) we have that [R(φ) + R(φ)T ] = cos φi + ( cos φ) ω ω T
an SYSTEMS OF DIFFERENTIAL EQUATIONS, EULER S FORMULA 5 [R(φ) R(φ)T ] = Ω sin φ 4. A purely algebraic erivation of Euler s formula Our previous result concerns solution of the ifferential equation R (φ) = ΩR(φ). Suppose now that you are given an arbitrary rotation M. Can we fin φ an Ω so that M = I + ( cos φ)ω + sin φω? To be more specific we have the following theorem. Theorem 4.. Let M be a 3 3 rotation. Define an provie that φ, π, π. Then cos φ = TrM Ω = sin φ [M M T ] M = M = I + ( cos φ)ω + sin φω. For φ =, π we have that M = I an for φ = π M = I + Ω, an hence, Euler s formula hols in these cases as well. Recall that a 3 3 matrix M is a rotation if it satisfies M T M = I an etm = +. We woul like to show that there exist a unit vector ω an an angle φ, φ π such that As usual M = cos φi + ( cos φ) ω ω T + Ω sin φ. Ω = We first start with a simple Lemma: ω 3 ω ω 3 ω ω ω Lemma 4.. Let M be a rotation in three space, i.e., M T M = I an etm = +. Then the matrix M must have the eigenvalue. Moreover, the other two eigenvalues must be of the form e ±iφ for some φ π. Proof. To see this consier.. et(m I) = etm T et(m I) = etm T (M I) = et(i M T ) = et(i M) T = et(i M) = et(m I). Hence et(m I) = an is an eigenvalue. If we enote the other two eigenvalues by λ an λ we must have that λ + λ + = TrM an λ λ = (Why?) Hence λ + λ = TrM, λ λ =. The best way to solve these equations is to note that 3 TrM 3 (Why?) Hence we may efine cos φ = TrM,
6 SYSTEMS OF DIFFERENTIAL EQUATIONS, EULER S FORMULA an we have to solve the equations λ + λ = cos φ, λ λ =. We easily fin that λ = e iφ an λ = e iφ. Thus we have the eigenvalues e iφ, e iφ,. Let us assume that φ, π, π. These cases we eal with later. Recall that cos φ = TrM, an efine Ω = sin φ [M M T ] Note that this suggests itself from Euler s formula (Why?). We have to check that Cayley s theorem tells us that an eveloping the proucts yiels Now M = I + ( cos φ)ω + sin φω =: R (M I)(M e iφ I)(M e iφ I) = M 3 ( + cos φ)m + ( + cos φ)m I =. I + ( cos φ)ω + sin φω = I + cos φ 4 sin φ [M M T ] + sin φ sin φ [M M T ] We further have that an by Cayley s theorem = I + 4( + cos φ) [M M T ] + [M M T ]. [M M T ] = M + M T I M = ( + cos φ)m ( + cos φ)i + M T, M T = ( + cos φ)m T ( + cos φ)i + M so that Thus, M + M T I = ( + cos φ)[m + M T ] 4( + cos φ)i R = [M + M T ] + [M M T ] = M. The remaining cases are easily ealt with. Assume that φ = or π. Then TrM = 3. Now the matrix M is of the form [ u, u, u 3 ] all of them being unit vectors. The trace, therefore is u + u + u 33 = 3 since each of these numbers is between an they all must be equals to. This means that the rotation matrix must be the ientity matrix. The case φ = π implies that must be a two fol eigenvalue. From this we get three facts: M = I an hence M = M T an M + I has a two imensional null space. Set P = M + I an note that P = P, P T = P
SYSTEMS OF DIFFERENTIAL EQUATIONS, EULER S FORMULA 7 Hence P projects the three imensional space onto a one imensional space an therefore it must be of the form P = ω ω T for some unit vector ω. Thus, which is what we wante to show. M = I + ω ω T = I + Ω