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There is a problem from a Russian textbook in classical mechanics.

Consider a scalar equation $$\ddot x=F(t,x,\dot x),\quad x\in\mathbb{R}.$$ Show that this equation can be multiplied by a function ##\mu(t,x,\dot x)\ne 0## such that the resulting equation

$$\mu\ddot x=\mu F(t,x,\dot x)$$ has the Lagrangian form

$$\frac{d}{dt}\frac{\partial L}{\partial \dot x}-\frac{\partial L}{\partial x}=0.$$

I can only say that by the Cauchy-Kowalewski theorem it can be done locally provided ##F## is an analytic function.

But some relatively elementary way is supposed. Sure it is a local assertion.

What do you think?

Consider a scalar equation $$\ddot x=F(t,x,\dot x),\quad x\in\mathbb{R}.$$ Show that this equation can be multiplied by a function ##\mu(t,x,\dot x)\ne 0## such that the resulting equation

$$\mu\ddot x=\mu F(t,x,\dot x)$$ has the Lagrangian form

$$\frac{d}{dt}\frac{\partial L}{\partial \dot x}-\frac{\partial L}{\partial x}=0.$$

I can only say that by the Cauchy-Kowalewski theorem it can be done locally provided ##F## is an analytic function.

But some relatively elementary way is supposed. Sure it is a local assertion.

What do you think?

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