Equation of motion for a particle in a potential V(x)

[LagrangeEuler]

If I have particle moving in the potential $V(x)$, when I write equation of motion
$\frac{dx}{dt}=-V'(x)+q(t)$
and when I integrate this equation do I need to look $V'(x)$ as function of time, or I just could write
$x(t)=-V'(x)t+\int^t_0q(t)dt$
Thanks for your answer!

 

[Einj]

It depends on how V is defined. According to what you wrote V=V(x) is just a function of space and hence you are correct.

Anyway, what's q(t)? And how did you obtain that equation of motion?

 

[LagrangeEuler]

Yes but $x=x(t)$ and $V=V(x)$. So I am confused. But because $V(x)$ is potential I think that I write equation in correct form. This is potential in which particle moves.

 

[ShayanJ]

It seems you're using kind of a "constant of differentiation"(like a constant of integration) which is mathematically wrong!
You should add a constant only when you integrate something, not when you differentiate something!

 

[Orodruin]

No, you are not allowed to integrate the equations of motion like that. Even if V does not depend explicitly on time, it does so implicitly through x.

I also agree with earlier answers that your EoM looks weird.

 

[LagrangeEuler]

Yes but you know. Particle is moving in some potential $V(x)$. In certain moment $t$ it has coordinate $x(t)$. $q(t)$ is certain pulse. How do you write down this solution?

 

[ShayanJ]

Yes but you know. Particle is moving in some potential $V(x)$. In certain moment $t$ it has coordinate $x(t)$. $q(t)$ is certain pulse. How do you write down this solution?

Well...You can't write the solution without knowing what is q(t)!

 

[ShayanJ]

Yes but you know. Particle is moving in some potential $V(x)$. In certain moment $t$ it has coordinate $x(t)$. $q(t)$ is certain pulse. How do you write down this solution?

Well...You can't write the solution without knowing what is q(t)!

Also the following is wrong.

$x(t)=-V'(x)t+\int^t_0q(t)dt$

Because x is an unknown function of time so you can't integrate. That's called a differential equation and it has its own methods.