By Jerry Marsden

ISBN-10: 091409811X

ISBN-13: 9780914098119

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20), which is t x(t) = x0 e−γ t + e−γ t 0 t eγ s f (s)ds = x0 e−γ t + 0 e−γ (t−s) f (s)ds. 5 Solving vector linear differential equations 21 We can just as easily solve a linear equation when the coefﬁcient γ is a function of time. 25) 0 and the solution is x(t) = x0 e− (t) + e− t (t) e (s) f (s)ds. 27) by transforming to a new set of variables, y = U x, where U is a matrix chosen so that the equations for the new variables are decoupled from each other. 28) where D is a diagonal matrix. For many square matrices A, there exists a matrix U so that D is diagonal.

29), it is straightforward to verify that U U † = I and that D = U AU † . It is also true, though not as obvious, that U † U = I . Exercises 1. The equation for the damped harmonic oscillator is: m d 2x dx + 2γ m + kx = 0. 47) √ Assume that γ < k/m, in the which the resonator is said to be “under-damped”. Show that x(t) = e−γ t cos(ωt)x(0) is a solution to Eq. 47), and ﬁnd the expression for the frequency ω in terms of m, k and γ . 2. Write Eq. 47) as a ﬁrst-order vector differential equation. 3.

17) and when α is less than one we have lim V (x(T )) = T α lim N (1−α) → ∞. 18) Neither of these make sense for the purposes of obtaining a stochastic differential equation that describes real systems driven by noise. Thus we are forced to choose α = 1 and hence V ( Wn ) ∝ t. When we are working in the continuum limit the Gaussian increments, dW , are referred to as being inﬁnitesimal. A general SDE for a single variable x(t) is then written as dx = f (x, t)dt + g(x, t)dW. 19) Since the variance of dW must be proportional to dt, and since any constant of proportionality can always be absorbed into g(x, t), the variance of dW is deﬁned to be equal to dt.

### Applications of global analysis in mathematical physics by Jerry Marsden

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