[tex]L=\frac{1}{2}m(\dot{q}_1-\dot{q}_2)^2-V(q_1,q_2)[/tex](adsbygoogle = window.adsbygoogle || []).push({});

Because if we put

[tex]p_1=\frac{\partial L}{\partial \dot{q}_1}[/tex]

[tex]p_2=\frac{\partial L}{\partial \dot{q}_2}[/tex]

we get

[tex]p_1=-p_2=m(\dot{q}_1-\dot{q}_2)[/tex]

We can't invert to get [tex]\dot{q_1}[/tex] in terms of the two momenta. We can still write down a Hamiltonian of sorts since

[tex]H=p_1\dot{q}_1+p_2\dot{q}_2-L[/tex]

[tex]=p_1(\dot{q}_1-\dot{q}_2)-L[/tex]

[tex]=\frac{p_1^2}{m}-L[/tex]

[tex]=\frac{p_1^2}{2m}+V(q_1,q_2)[/tex]

or equivalently

[tex]=\frac{p_2^2}{2m}+V(q_1,q_2)[/tex]

or

[tex]=\frac{p_1^2}{4m}+\frac{p_2^2}{4m}+V(q_1,q_2)[/tex]

The main thing then is that we can't get an equation of motion which looks like

[tex]\dot{q_j}=\frac{\partial H}{\partial p_j}[/tex]

What do we do with Lagrangians like this? Does the Hamiltonian method just fail?

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# Can a Hamiltonian be formed from this Lagrangian?

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