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quasar987

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I must solve the following two coupled EDOs in the context of a Lagrangian mechanics problem (a rigid pendulum of lenght l attached to a mass sliding w/o friction on the x axis). The problem statement does not mention that we can make small angle approximation. It says "find the equations of motion and solve them for the following initial conditions:...". Is this feasable?

[tex](m_1+m_2)\ddot{x}+m_2l\ddot{\theta}\cos(\theta)-m_2l\dot{\theta}^2\sin(\theta)=0[/tex]

[tex]l\ddot{\theta}+\ddot{x}\cos(\theta)+g\sin(\theta)=0[/tex]

They can be uncoupled but there remains a second order non-linear ODE to solve.

Is this doable analytically?

And an annexed question (perhaps this one is more of a physical nature): why can we say that [itex]\dot{\theta}\approx 0[/itex] in the small angle approximation? The angle can be small and nevertheless vary furiously fast. What indicates that if theta is small, the so is its derivative?

[tex](m_1+m_2)\ddot{x}+m_2l\ddot{\theta}\cos(\theta)-m_2l\dot{\theta}^2\sin(\theta)=0[/tex]

[tex]l\ddot{\theta}+\ddot{x}\cos(\theta)+g\sin(\theta)=0[/tex]

They can be uncoupled but there remains a second order non-linear ODE to solve.

Is this doable analytically?

And an annexed question (perhaps this one is more of a physical nature): why can we say that [itex]\dot{\theta}\approx 0[/itex] in the small angle approximation? The angle can be small and nevertheless vary furiously fast. What indicates that if theta is small, the so is its derivative?

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