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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Solar System
This demo simulates the gravitational pull between multiple massive bodies.
We will use Euler steps in time which do not preserve energy.
Φ-Flow
&nbsp;&nbsp;&nbsp; Documentation
&nbsp;&nbsp;&nbsp; API
&nbsp;&nbsp;&nbsp; Demos
Step1:  We use the convenience import of Φ<sub>Flow</sub> which imports the core submodules, such as math and vis.
Step2:  Let's define the initial positions for our planets.
We stack the position vectors of the planets a long a new instance dimension we call planets.
Φ<sub>Flow</sub> also allows us to name the individual elements along that dimension. These names are part of the shape and will be present on all tensors derived from x that have this dimension.
Step3:  We can plot these positions by constructing a PointCloud, using the bounds parameter to specify the axis range. Since the universe we are simulating is infinite, the bounds have no physical meaning in this case.
Step4:  Next, let's give the planets initial velocities so that they circle the sun.
For a nice plot, we pass the vector-valued velocities as the values of our PointCloud.
Step5:  Next, we define the masses of our planets which will determine the gravitational pull.
These also allow us to make our plot prettier by passing a Sphere as the elements of the PointCloud.
Step6:  To Simulate our system, we define a simple update step and run it repeatedly, collecting all intermediate states in the lists xs and vs.
For the pair-wise distances between our planets, we use math.rename_dims to rename our planets dimension to others. Our previous tensors are constant along others  and the renamed tensor is constant along planets.
When combining tensors with different dimensions in an operation, they will be reshaped automatically.
Step7:  Finally, let's plot the system trajectory as an animation!
First, we need to define a dimension which will list our animation frames.
Step8:  Now we can specify this dimension via the animate argument.
Using vis.overlay additionally allows us to plot multiple fields in one figure.
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Python Code:
# !pip install phiflow
Explanation: Solar System
This demo simulates the gravitational pull between multiple massive bodies.
We will use Euler steps in time which do not preserve energy.
Φ-Flow
&nbsp;&nbsp;&nbsp; Documentation
&nbsp;&nbsp;&nbsp; API
&nbsp;&nbsp;&nbsp; Demos
End of explanation
from phi.flow import *
Explanation: We use the convenience import of Φ<sub>Flow</sub> which imports the core submodules, such as math and vis.
End of explanation
x = math.stack({'Sun': (0, 0), 'Earth': (10, 0), 'Mars': (0, 12)}, instance('planets'))
x
Explanation: Let's define the initial positions for our planets.
We stack the position vectors of the planets a long a new instance dimension we call planets.
Φ<sub>Flow</sub> also allows us to name the individual elements along that dimension. These names are part of the shape and will be present on all tensors derived from x that have this dimension.
End of explanation
vis.plot(PointCloud(x, bounds=Box(x=(-2, 12), y=(-1, 13))))
Explanation: We can plot these positions by constructing a PointCloud, using the bounds parameter to specify the axis range. Since the universe we are simulating is infinite, the bounds have no physical meaning in this case.
End of explanation
v = math.rotate_vector(x, PI/2)
v = math.divide_no_nan(v, math.vec_length(v))
vis.plot(PointCloud(x, values=v, bounds=Box(x=(-2, 12), y=(-1, 13))))
Explanation: Next, let's give the planets initial velocities so that they circle the sun.
For a nice plot, we pass the vector-valued velocities as the values of our PointCloud.
End of explanation
masses = math.stack({'Sun': 1000, 'Earth': 10, 'Mars': 10}, instance('planets'))
vis.plot(PointCloud(Sphere(x, radius=masses**0.333 * .2), bounds=Box(x=(-2, 12), y=(-1, 13))))
Explanation: Next, we define the masses of our planets which will determine the gravitational pull.
These also allow us to make our plot prettier by passing a Sphere as the elements of the PointCloud.
End of explanation
def simulate(x, v, dt=.5):
    dx = x - math.rename_dims(x, 'planets', 'others')
    a = - .01 * math.sum(math.divide_no_nan(math.rename_dims(masses, 'planets', 'others') * dx, math.vec_squared(dx) ** 1.5), 'others')
    return x + v * dt, v + a * dt
xs, vs = [x], [v]
for i in range(100):
    x, v = simulate(x, v)
    xs.append(x)
    vs.append(v)
Explanation: To Simulate our system, we define a simple update step and run it repeatedly, collecting all intermediate states in the lists xs and vs.
For the pair-wise distances between our planets, we use math.rename_dims to rename our planets dimension to others. Our previous tensors are constant along others  and the renamed tensor is constant along planets.
When combining tensors with different dimensions in an operation, they will be reshaped automatically.
End of explanation
xs = math.stack(xs, batch('time'))
vs = math.stack(vs, batch('time'))
xs
Explanation: Finally, let's plot the system trajectory as an animation!
First, we need to define a dimension which will list our animation frames.
End of explanation
vis.plot(vis.overlay(
    PointCloud(Sphere(xs, radius=masses ** 0.333 / 5), bounds=Box(x=(-20, 20), y=(-20, 20))),
    PointCloud(xs, vs * 2, bounds=Box(x=(-20, 20), y=(-20, 20)))), animate='time', frame_time=100)
Explanation: Now we can specify this dimension via the animate argument.
Using vis.overlay additionally allows us to plot multiple fields in one figure.
End of explanation
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