Find a suitable library that implements coordinate vectors, so you don't have to implement them as arrays of doubles. Ideally your struct Particle should look like:

struct Particle {
    double m;   // mass
    vec3 x;     // position
    vec3 v;     // velocity
    vec3 F;     // force
    vec3 F_old; // force past time step
};

And a suitable library will provide functions and operator overloads to make working with these types very easy. You should be able to write something like:

void Nbody::update_position(Particle *p) {
    double a = dt * 0.5 / p->m;
    p->x += dt * (p->v + a * p->F);
    p->F_old = p->F;
}

There are many libraries available. I am partial to GLM myself. For a discussion of possible libraries, see https://stackoverflow.com/questions/1380371/what-are-the-most-widely-used-c-vector-matrix-math-linear-algebra-libraries-a.

Use a vector math library

Find a suitable library that implements coordinate vectors, so you don't have to implement them as arrays of doubles. Ideally your struct Particle should look like:

struct Particle {
    double m;   // mass
    vec3 x;     // position
    vec3 v;     // velocity
    vec3 F;     // force
    vec3 F_old; // force past time step
};

And a suitable library will provide functions and operator overloads to make working with these types very easy. You should be able to write something like:

void Nbody::update_position(Particle *p) {
    double a = dt * 0.5 / p->m;
    p->x += dt * (p->v + a * p->F);
    p->F_old = p->F;
}

There are many libraries available. I am partial to GLM myself. For a discussion of possible libraries, see https://stackoverflow.com/questions/1380371/what-are-the-most-widely-used-c-vector-matrix-math-linear-algebra-libraries-a.

Make function manipulating Particles member functions of Particle

You have a lot of functions that mainly manipulate a particle's state, but they are not part of struct Particle itself. For example, update_position() is something that apart from the timestep dt only manipulates a Particle's member variables. If you make it a member function of Particle, it becomes a much cleaner looking function:

struct Particle {
    ...
    void update_position(double dt);
};

void Particle::update_position(double dt) {
    double a = dt * 0.5 / m;
    x += dt * (v + a * F);
    F_old = F;
}

And you call it like so:

void Nbody::comp_position() {
    for (auto &p: particles) {
        p.update_position(dt);
    }
}

You can do the same for update_velocity(), and even force().