# Find the minimum of a energy conservation equation.

I have an equation for two particle collisions (the equation is just energy-momentum conservation):

$$k_{A,x}+k_{B,x} = p_{1,x}+p_{2,x}$$ $$k_{A,y}+k_{B,y} = p_{1,y}+p_{2,y}$$ $$k_{A,z}+k_{B,z} = p_{1,z}+p_{2,z}$$ $$\sqrt{m_{1}^{2}+\mathbf{k}_{A}^{2}}+\sqrt{m_{2}^{2}+\mathbf{k}_{B}^{2}} = \sqrt{m_{1}^{2}+\mathbf{p}_{1}^{2}}+\sqrt{m_{2}^{2}+\mathbf{p}_{2}^{2}}.$$

where, $$k_{A,x}\in[p_{c,x}-3\sigma,p_{c,x}+3\sigma]$$, $$k_{B,x}=-k_{A,x}$$ and $$k_{A,y},k_{B,y},k_{A,z},k_{B,z}\in[-3\sigma,3\sigma]$$, $$p_{c,x}$$ and $$\sigma$$ is a positive number, and $$p_{c,x} > 3\sigma$$.

The first three equations will give $$\mathbf{p}_{2}$$ in terms of $$\mathbf{k}_{A},\mathbf{k}_{B},\mathbf{p}_{1}$$.

I would like to know the minimum and maximum value of $$|\mathbf{p}_{1}|$$ if we already know $$|\mathbf{k}_{A}|_{min},|\mathbf{k}_{A}|_{max},|\mathbf{k}_{B}|_{min},|\mathbf{k}_{B}|_{max}$$. Or the boundary for $$|\mathbf{p}_{1}|$$ in terms of $$p_{c,x}$$ and $$\sigma$$,i.e., I would like to know the minimum and maximum value of $$|\mathbf{p}_{1}|$$ in terms of $$p_{c,x}$$ and $$\sigma$$.

Here is a solution I found for this question, which I can not persuade myself, but it may help clear the question.

Since $$\mathbf{k}_{A}$$ and $$\mathbf{k}_{B}$$ denoting the input momentum of two incoming particles, if $$|\mathbf{k}_{A}|$$ and $$|\mathbf{k}_{B}|$$ all take the minimum values, i.e., $$\mathbf{k}_{A}=(p_{c,x}-3\sigma,0,0)$$ and $$\mathbf{k}_{B}=(-p_{c,x}+3\sigma,0,0)$$ hence $$\mathbf{k}_{A}+\mathbf{k}_{B}=\mathbf{p}_{1}+\mathbf{p}_{2}$$, the outcoming particles would have the minimum values. Thus we have $$\sqrt{m_{1}^{2}+\mathbf{k}_{A}^{2}}+\sqrt{m_{2}^{2}+\mathbf{k}_{A}^{2}} = \sqrt{m_{1}^{2}+\mathbf{p}_{1}^{2}}+\sqrt{m_{2}^{2}+\mathbf{p}_{1}^{2}}.$$ Thus one can easily see that the minimum value for $$|\mathbf{p}_{1}|$$ is $$|\mathbf{k}_{A}|_{min}$$

This is somewhat reluctant.

Does anyone have once encountered such problem or how to make my proof more convinsing?

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Or equivalently, if $$\delta_{1},\delta_{2},\delta_{3},\delta_{4}\in[-3\sigma,3\sigma]$$ and $$p_{c,x}$$ and $$\sigma$$ are positive numbers, and $$p_{c,x} > 3\sigma$$., how to prove the following inequality to be wrong: $$\sqrt{m_{1}^{2}+(-p_{c,x}+\delta_{1})^{2}+\delta_{2}^{2}}+\sqrt{m_{2}^{2}+(p_{c,x}+\delta_{3})^{2}+\delta_{4}^{2}} < \sqrt{m_{1}^{2}+(p_{c,x}-3\sigma)^{2}}+\sqrt{m_{2}^{2}+(p_{c,x}-3\sigma)^{2}+(\delta_{1}+\delta_{3})^{2}+(\delta_{2}+\delta_{4})^{2}}$$

• It would help to know the following: (1) the meaning of the equations you give (just cite a link or name or something), and (2) the function you are trying to minimize or maximize. . . also the meaning of the variables would help a little as well.
– user648059
Mar 1 '19 at 14:15
• @se2018, I have updated the question. Mar 1 '19 at 14:47