- #1
Jazradel
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Homework Statement
Consider a mechanical system describe by the conservative 2nd-order ODE
[itex]\frac{\partial^{2}x}{\partial t^{2}}=f(x)[/itex]
(which could be non linear). If the potential energy is [itex]V(x)=-\int^{x}_{0} f(\xi) d \xi[/itex], show that the system satisfies conservation of energy [itex]\frac{1}{2}x^{2}+V(x)=E[/itex] (E is a constant).
Homework Equations
As above.
The Attempt at a Solution
I've missed almost everything we've done on ODEs, so I don't really have any idea how to being. Even knowing what to call the problem, or a link to some notes/worked examples/relevant textbook would be great. I think the start is to define:
[itex]x(t)=\begin{array}{c}
x_{1}(t) \\
x_{2}(t) \\
\end{array}[/itex]
Then sub into the first equation:
[itex]\frac{\partial^{2} x_{1}}{\partial t^{2}}=f_{1}(x_{1},x_{2})[/itex]
[itex]\frac{\partial^{2} x_{2}}{\partial t^{2}}=f_{2}(x_{1},x_{2})[/itex]
Now I think I should use the chain rule, and integrate equation 2, but I can't see how.
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