The equations dx/dt=v at evolution page, tell EJS use the current value v to calculate x (t+dt) from x(t).

It does not have any other effect.

The left hand side will change according to value at the right hand side of the equation.

It is not the reversed order.

In computer language: x=x+1; does not mean "x equal to x+1" which is always false.

It mean get value from x, add 1 and save the value back to x.

I do not understand what do you mean by

i got the time t to increment by dt = 0.05, but the v does not vary as i change the position of circle coordinate (x,y)

What is your expectation?

I do not understand what you really want to do. Please explain it in more detail.

The EJS simulation use the equation at evolution pages to calculate new value at t+dt from value at t.

And it does not remember previous values unless you add an array to store those values.

If you want to use mouse action to change the time , you need to

save values for all the variables

or

add code to move the time either forward or backward.