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I'm reading Thornton, Marion, Classical Dynamics, section 12.2 and stuck at undersating some point :

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My question is, how can we take a general solution of (12.1) as of the form (12.9) (underlined statement), from (12.2) ?

I feel that I am unfamilier to the system of ordinary differential equations.

Can anyone explain more friendly or help?

Plantation
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    The key is linearity. (12.2) means you are looking for simple solutions called stationary (same time sinusoidal dependence for every oscillator). Since your system is linear, a linear combination of such simple solutions is again a solution of the problem. It turns out (spectral theorem) that these stationary solutions are enough to obtain all solutions by linear combinations hence (12.9) – LPZ Aug 10 '22 at 10:27
  • O.K. Thanks for pointing out. Perhaps, can you indicate me where can I find associated reference about your explanation ? – Plantation Aug 10 '22 at 11:59
  • C.f. : I found in Friedberg, Linear Algebra (fourth edition), p.137~139 that 1) Corollary : The solution space of any $n$-th order homogeneous linear differential equation with constant coefficients is an $n$-dimensional subspace of $C^{\infty}$ (his book, p.130 definition) – Plantation Aug 10 '22 at 11:59
  • And 2) Theorem 2.34 : Given a homogeneous linear differential equation with constant coefficients and auxillary polynomial $(t-c_1)^{n_1}(t-c_2)^{n_2}\cdots (t-c_k)^{n_k}$, where $n_1, n_2 , \cdots , n_k$ are positive integers and $c_1, c_2, \cdots c_k$ are distinct complex number, the following set is a basis for the solution space of the equation : ${e^{c_1t}, te^{c_1t}, \cdots , t^{n_1 -1}e^{c_1t}, \cdots , e^{c_kt}, te^{c_kt}, \cdots , t^{n_k-1}e^{c_kt} }$. – Plantation Aug 10 '22 at 11:59
  • Is there an analogue for 'system of ordinary differential equations'? e.g., The set of solutions of two-system of second order homogeous linear differential equations with constant coefficients forms a 4-dimensional linear space? – Plantation Aug 10 '22 at 11:59
  • Please don't post images in place of text and mathematical equations. – Frobenius Aug 10 '22 at 15:31
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    Related : Eigenvalue equation for kinetic and potential energy. – Frobenius Aug 10 '22 at 15:33
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    For a general linear system, yes you just need to apply Jordan’s reduction. More details can be found in V Arnold’s book on ODE’s. However, usually your linear system is conservative, in which case your matrix symmetric. In this case, it is more relevant to apply the spectral theorem. Again, more details can be found in Arnold’s book on ODE’s or his Mathematical Methods for Classical Mechanics. – LPZ Aug 10 '22 at 22:22
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    Btw, the idea that you can always reduce the high order ode’s to a system of first order ode’s is the concept of “phase space” – LPZ Aug 10 '22 at 22:49
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    @Frobenius : Yes. Thanks for link~ – Plantation Aug 11 '22 at 01:09
  • @Ipz : O.K. I'll refer. Thanks for guide~~ – Plantation Aug 11 '22 at 01:09

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