Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Consider the simple harmonic oscillator $\frac{d^2p}{dt^2}=-p$ as a Hamiltonian system with Hamiltonian given by $H=\frac{1}{2}p^2+\frac{1}{2}q^2$. The famous Liouville theorem for integrable systems then says this system can be integrated by ''quadrature.'' I have searched high and low on the internet for a decent explanation of what this means and can't really turn up much other than it is some process involving integrals of known functions and ''algebraic'' operations (Note: The quotations around quadrature and algebraic is directly from Arnold's book on classical mechanics.) My questions is the following: the harmonic oscillator is probably the first and easiest example of a Hamiltonian system then how do you arrive at the solution to the harmonic oscillator using ''quadrature.'' Or should I think of solvable by quadrature as an existence theorem to an ODE.

share|improve this question
add comment

1 Answer

up vote 4 down vote accepted

The Hamilton equations of motion are $$ \frac{dx}{dt} = p $$ $$ \frac{dp}{dt} = -x $$ The Hamiltonian $H = \frac{1}{2}(x^2 + p^2)$ is conserved, so along a solution it is some constant, say $E$. Then we can solve for $p$: $$ p = \sqrt{2E - x^2}$$ Then Hamilton's equations reduce to $$ \frac{dx}{dt} = \sqrt{2E - x^2}$$ which is a separable first order ODE. We obtain $$ \int \frac{dx}{\sqrt{2E - x^2}} = \int dt$$ which is a solution in "quadratures" (explicit integrals of known functions). The Liouville theorem for integrable systems should be thought of as a generalization of separable ODEs.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.