User_54

As far as you know what the energy is, maybe you can propose a couple of equations conserving energy?

This situation is similar to a situation in the classical electrodynamics where a charge acceleration creates electromagnetic waves. Maybe the electrodynamical example can be helpful here?

So, no fruitful ideas on fixing the system of equations?

Yes, to a great extent: Eq. (2) is a driven oscillator equation. The damping force is simply negligible here, I guess. Well, I was given two equations written above and without much explanation about the total system. I am free to modify the system equations and give it a "proper' explanation.

An external force can be x1-dependent and time dependent. I gave two examples: a standing wall and a moving particle, and a moving wall and an initially still particle. In both cases the external force accelerates particle-1 and changes its velocity.

We can multiply Eq. (2) by M2 to have it, if you like. If an external force is time dependent (a moving wall), then F(t) acts during a short period and transfers some energy to M1. This energy does not depend on presence or absence of oscillator in this set (1), (2). On the orher hand, the oscillator also gains energy due to acceleration of particle-1. Does it look like energy is conserved?

I described a situation followed form the Eqs. (1) and (2). Where am I wrong?

Let us consider a potential wall. Upon hitting the wall the particle reflects with the same energy, but the oscillator acquires energy too. Where does the latter come from?

Apparently the total energy is not conserved (in a potential force F). Is it possible to modify these equation in such a way that tha total energy is conserved? If so, what are the modified equations?

If I add [latex]-\epsilon \ddot{x}_1[/latex] to the right-hand side of Eq. (1), then I will decrease acceleration of particle-1. On the other hand, if the oscillator phase is such that it "helps" accelerate particle-1, then this fact is not contained in the modified Eq (1). So something different is needed.

I would like to obtain a self-consistent desctiprion. Particle-1 does not currently "know" about oscillator. Should it konow, and if so, how to take the energy loss into account?

No, they are just different scalar variables. I would edit my post to avoid confusion, but it is not possible anymore. Momentum cannot be conserved because of an external force [latex]F(x_1,t)[/latex] acting on the particle [latex]M_1[/latex]. My problem is to "guess" or "derive" the interaction between 1 and 2.

I have two equations: [latex]M_1\ddot{x}_1=F,\qquad (1)[/latex] and [latex]\ddot{x}_2+\omega^2\cdot x_2=\epsilon\cdot\ddot{x}_1.\qquad (2)[/latex] The first equation says that particle [latex]M_1[/latex] can be affected with some force [latex]F[/latex] and the second one describes the oscillation amplitude variations if the first particle is affected by a force. Oscillations take some energy. How to complete the first equation to take into account this energy loss? Thanks.