Action and Response functions.

[jorgext]

Dear colleagues,
I have the following question in my head:
Give two canonical coordinates $$P$$, $$Q$$, and a corresponding set of equations of motions, not necessary solved.
Is it possible to find analytically the action $$S$$ and response $$R$$functions of the system, which satisfy the relations:
$$\dfrac{\delta S}{\delta Q}=\dfrac{\delta R}{\delta \dot Q}$$
$$\dfrac{\delta S}{\delta P}=\dfrac{\delta P}{\delta \dot P}$$
Or this is just imagination?
Best Regards,
Jorge.

 

[jorgext]

Any comments?