Laplace-Beltrami operator

This worksheet explores the issue raised by this ask.sagemath question

In [1]:
version()
Out[1]:
'SageMath version 8.0.beta4, Release Date: 2017-04-27'
In [2]:
%display latex
In [3]:
Parallelism().set(nproc=1)

Manifold and coordinate charts

In [4]:
M = Manifold(2*3,'R^6',field='real',start_index=1)
In [5]:
m1, m2 = var('m1 m2', domain='real')
In [6]:
m_CM = m1+m2; mu1 = m1/m_CM; mu2 = m2/m_CM; mu = m1*m2/m_CM
In [7]:
c_Cart.<x1,y1,z1,x2,y2,z2> = M.chart()
c_Cart
Out[7]:
In [8]:
g = M.riemannian_metric('g')
g[1,1],g[2,2],g[3,3], g[4,4],g[5,5],g[6,6] = m1,m1,m1, m2,m2,m2; 
g.display()
Out[8]:
In [9]:
c_CM.<X,Y,Z,x,y,z> = M.chart()
c_CM
Out[9]:
In [10]:
ch_Cart_CM = c_Cart.transition_map(c_CM, [mu1*x1+mu2*x2,mu1*y1+mu2*y2,mu1*z1+mu2*z2, x1-x2,y1-y2,z1-z2],
                                   restrictions2 = [x!=0, y!=0, z!=0])
ch_Cart_CM.display()
Out[10]:
In [11]:
ch_Cart_CM.inverse().display()
Out[11]:
In [12]:
M.set_default_chart(c_CM)
M.set_default_frame(c_CM.frame())

The scalar field $\psi$

In [13]:
psiX = M.scalar_field({c_CM: function('Psi_X')(X,Y,Z)}, name='psiX',
                      latex_name='\Psi_X')
psiX.display()
Out[13]:
In [14]:
psix = M.scalar_field({c_CM: function('psi_x')(x,y,z)}, name='psix',
                      latex_name='\psi_x')
psix.display()
Out[14]:
In [15]:
psi = psiX*psix
psi.set_name('psi', latex_name='\psi')
psi.display()
Out[15]:

Laplace-Beltrami operator as $*\mathrm{d}*\mathrm{d}\psi$

In [16]:
%time dpsi = psi.exterior_derivative()
print(dpsi)
CPU times: user 1.8 s, sys: 72 ms, total: 1.87 s
Wall time: 1.81 s
1-form dpsi on the 6-dimensional differentiable manifold R^6
In [17]:
dpsi.display()
Out[17]:
In [18]:
%time sdpsi = dpsi.hodge_dual(g)
print(sdpsi)
CPU times: user 3min 21s, sys: 3.2 s, total: 3min 24s
Wall time: 3min 23s
5-form *dpsi on the 6-dimensional differentiable manifold R^6
In [19]:
%time dsdpsi = sdpsi.exterior_derivative()
print(dsdpsi)
CPU times: user 4.1 s, sys: 140 ms, total: 4.24 s
Wall time: 4.09 s
6-form d*dpsi on the 6-dimensional differentiable manifold R^6
In [20]:
dsdpsi.display()
Out[20]:
In [21]:
%time LBpsi = dsdpsi.hodge_dual(g)
print(LBpsi)
CPU times: user 7min 11s, sys: 3.55 s, total: 7min 15s
Wall time: 7min 11s
Scalar field *d*dpsi on the 6-dimensional differentiable manifold R^6

Laplace-Beltrami operator as $\nabla_\mu \nabla^\mu \psi$

In [22]:
nabla = g.connection()
print(nabla)
Levi-Civita connection nabla_g associated with the Riemannian metric g on the 6-dimensional differentiable manifold R^6
In [23]:
%time LBpsi1 = nabla(nabla(psi).up(g)).trace()
LBpsi1
CPU times: user 12.7 s, sys: 248 ms, total: 13 s
Wall time: 12.8 s
Out[23]:
In [24]:
LBpsi1.coord_function()
Out[24]:

Test:

In [25]:
(LBpsi1 - LBpsi).display()
Out[25]:
In [ ]: