SamplePublisher`GrassmannCalculus`
FDt  |
|
| | ||||
|
| | ||||
Details and Options
Examples
(1)
Basic Examples
(1)
In[1]:=
<<GrassmannCalculus`
In[2]:=
SetGrassmannNSpace[2,{x,y},"Vector"]
[]
 | GrassmannSymbolsPalette |
The default time variable in is .
EstablishFluxionNotation
t
In[3]:=
| EstablishFluxionNotation |
 | DtConstants |
e
x
e
y
The first and second time derivatives of a coordinate and then converted to a regular partial derivative.
In[4]:=
| FDt |
| FDt |
| FluxionToDerivatives |
Out[4]=
x
Out[4]=
¨
x
Out[4]=
′′
x
The first and second time derivatives of a typical expression.
In[5]:=
a+bx+Sin[xy]+⋀
[%]
e
x
e
y
y
x
e
x
e
y
| FDt |
Out[5]=
a+bx+Sin[xy]+⋀
e
x
e
y
y
x
e
x
e
y
Out[5]=
b+Cos[xy](y+x)++Log[x]⋀
e
x
x
e
y
x
y
y
x
y
x
x
y
e
x
e
y
We can take a total derivative with respect to a Fluxion or a variable to generate Newton's force law from a Lagrangian. We use the GrassmannCalculus routine to give formatting of total derivatives in . Note the dot above the left hand side in the Lagrangian expression.
FormatTotalDerivative
StandardForm
In[6]:=
FormatTotalDerivative[True]
FDt[Dt[L, FDt[x]]] Dt[L, x]
% /. L 1/2 m FDt[x]^2 - V[x] /. DtIndependentVariables[{x, FDt[x], m}]
% /. FluxionToDerivatives
FormatTotalDerivative[True]
FDt[Dt[L, FDt[x]]] Dt[L, x]
% /. L 1/2 m FDt[x]^2 - V[x] /. DtIndependentVariables[{x, FDt[x], m}]
% /. FluxionToDerivatives
FormatTotalDerivative[True]
Out[6]=
L
x
L
x
Out[6]=
m-[x]
¨
x
′
V
Out[6]=
m[t]-[x[t]]
′′
x
′
V
A basis vector can be made differentiable by defining a Symbol synonym for it (here eOy for the orthonormal y basis vector) and not including it in the constants list. This has restricted usage, but it could be used to solve for the rates of change of basis vectors in rotating systems, in terms of constant basis vectors, say.
In[7]:=
Format[eOy]=
[xeOy]
e
y
| FDt |
Out[7]=
e
y
Out[7]=
x+
e
y
e
y
x
 |
|
 |
|
""