From a6cde755f70147f01f3cc8b4767cfae15003c252 Mon Sep 17 00:00:00 2001
From: dboe <dboe@ipp.mpg.de>
Date: Wed, 7 Jul 2021 12:46:03 +0200
Subject: [PATCH] added theory and udo stuff
---
tfields/bases/udo/christoffelSymbols.py | 24 +
...rdinate_system_implementation_generator.py | 648 +
tfields/bases/udo/metric.py | 139 +
tfields/bases/udo/sympyGradient.py | 19 +
tfields/bases/udo/sympyTensors.py | 849 +
tfields/bases/udo/tensorTransformation.py | 68 +
tfields/bases/udo/testSympy.py | 10 +
theory/1901.05512.pdf | Bin 0 -> 1754784 bytes
theory/CatalogueOfSpacetimes.pdf | 19065 ++++++++++++++++
.../CurvilinearAnalysisInAEuclideanSpace.pdf | Bin 0 -> 1068495 bytes
...escribingTheWorldWithVectorsAndTensors.pdf | Bin 0 -> 242254 bytes
.../QuickOverviewOfCurvilinearCoordinates.pdf | Bin 0 -> 123470 bytes
12 files changed, 20822 insertions(+)
create mode 100644 tfields/bases/udo/christoffelSymbols.py
create mode 100644 tfields/bases/udo/coordinate_system_implementation_generator.py
create mode 100644 tfields/bases/udo/metric.py
create mode 100644 tfields/bases/udo/sympyGradient.py
create mode 100644 tfields/bases/udo/sympyTensors.py
create mode 100644 tfields/bases/udo/tensorTransformation.py
create mode 100644 tfields/bases/udo/testSympy.py
create mode 100644 theory/1901.05512.pdf
create mode 100644 theory/CatalogueOfSpacetimes.pdf
create mode 100644 theory/CurvilinearAnalysisInAEuclideanSpace.pdf
create mode 100644 theory/DescribingTheWorldWithVectorsAndTensors.pdf
create mode 100644 theory/QuickOverviewOfCurvilinearCoordinates.pdf
diff --git a/tfields/bases/udo/christoffelSymbols.py b/tfields/bases/udo/christoffelSymbols.py
new file mode 100644
index 0000000..66457d8
--- /dev/null
+++ b/tfields/bases/udo/christoffelSymbols.py
@@ -0,0 +1,24 @@
+class ChristoffelSymbol2ndKind(object):
+ """
+ Calculate Christoffel Gamma_ij^k symbols of metric g
+ """
+
+ def __init__(self, g, x):
+ self.g = g
+ self.x = x
+
+ def ddu(self, i, j, k):
+ g = self.g
+ x = self.x
+ chr = 0
+ for m in [0, 1, 2, 3]:
+ chr += (
+ g.uu(k, m)
+ / 2
+ * (
+ g.dd(m, i).diff(x[j])
+ + g.dd(m, j).diff(x[i])
+ - g.dd(i, j).diff(x[m])
+ )
+ )
+ return chr.simplify()
diff --git a/tfields/bases/udo/coordinate_system_implementation_generator.py b/tfields/bases/udo/coordinate_system_implementation_generator.py
new file mode 100644
index 0000000..a891ae4
--- /dev/null
+++ b/tfields/bases/udo/coordinate_system_implementation_generator.py
@@ -0,0 +1,648 @@
+import re
+
+import numpy as np
+import sympy as sym
+
+from collections import OrderedDict
+from metric import Metric
+
+
+class JavaCoordinateSystemCreator(object):
+
+ # TODOs:
+ # - limits for variables
+ # - document
+ # - to java code routine needs to be enhanced to handle all possible cases
+
+ def __init__(
+ self, java_class_name, Metric, axes, java_name=None, extra_parameters=None
+ ):
+ self.java_class_name = java_class_name
+ self.g = Metric
+ self.code = ""
+
+ if java_name == "" or java_name is None:
+ index = java_class_name.index("Coordinates")
+ if index == -1:
+ raise Exception("Cannot guess java_name, please supply explicitly")
+ self.java_name = java_class_name[:index]
+ else:
+ self.java_name = java_name
+
+ if axes is None:
+ raise Exception("Cannot create default axes!")
+ else:
+ self.axes = axes
+
+ # needs to be a dictionary with type info
+ self.extra_parameters = extra_parameters
+
+ self.unable_to_invert = False
+ self.inv_transformation = None
+
+ def __str__(self):
+ if self.code == "":
+ self.__build()
+ return self.code
+
+ def __build(self):
+ self.code = ""
+ self.__build_header()
+ self.__build_constructor()
+ self.__build_dimension()
+ self.__build_isOrthogonal()
+ self.__build_to_base_point()
+ self.__build_from_base_transformation()
+ self.__build_to_base_transformation_vector()
+ self.__build_from_base_transformation_vector()
+ self.__build_metric_coefficient()
+ self.__build_metric_tensor()
+ # self.__build_jacobian_determinant()
+ # self.__build_christoffel_symbols_2nd_kind()
+ # self.__build_christoffel_symbols_1st_kind()
+ # self.__build_riemann_tensor()
+ # self.__build_ricci_tensor()
+ # self.__build_ricci_scalar()
+
+ self.code += "}\n"
+
+ def __build_header(self):
+ """
+ Builds the header of the Java implementation of the eu.hoefel.coordinates.CoordinateSystem interface
+ and attaches it to the variable representing the Java code.
+ """
+ self.code += "import java.util.NavigableSet;\n"
+ if self.extra_parameters is not None:
+ self.code += "import java.util.Objects;\n"
+
+ self.code += "\n"
+ self.code += "import eu.hoefel.coordinates.axes.Axes;\n"
+ self.code += "import eu.hoefel.coordinates.axes.Axis;\n"
+ if self.extra_parameters is not None:
+ self.code += "import eu.hoefel.coordinates.CoordinateSystems;\n"
+
+ self.code += "\n"
+ self.code += "@CoordinateSystemSymbols({\"\"}) // TODO: add symbols representing the coordinate system, e.g. 'cartesian' and 'cart' for the cartesian coordinate system\n"
+ self.code += (
+ "public final record " + self.java_class_name + "(NavigableSet<Axes> axes"
+ )
+ if self.extra_parameters is not None:
+ for name, type in self.extra_parameters.items():
+ self.code += ", " + type + " " + name
+
+ self.code += ") implements CoordinateSystem {\n\n"
+
+ self.code += " public static final NavigableSet<Axis> DEFAULT_AXES = Axes.of("
+
+ axes = []
+ for dimension, unit in self.axes.items():
+ axes.append("new Axis(" + str(dimension) + ", ")
+ if "." in unit: # probably smth like SiBaseUnit.METER
+ axes[len(axes) - 1] += unit
+ else:
+ axes[len(axes) - 1] += 'Unit.of("' + unit + '")'
+ axes[len(axes) - 1] += ")"
+ self.code += ", ".join(axes) + ");\n\n"
+
+ if self.extra_parameters is None:
+ self.code += (
+ ' private static final Consumer<Object[]> ARG_CHECK = args -> Axes.DEFAULT_ARG_CHECK.accept("'
+ + self.java_name
+ + '", args);\n\n'
+ )
+
+ def __build_constructor(self):
+ if self.extra_parameters is None:
+ self.code += " public " + self.java_class_name + "(Object... args) {\n"
+ self.code += " this(Axes.fromArgs(DEFAULT_AXES, ARG_CHECK, args));\n"
+ self.code += " }\n\n"
+ else:
+ self.code += " public " + self.java_class_name + " {\n"
+ self.code += ' Objects.requireNonNull(axes, "Axes may not be null. "\n'
+ self.code += (
+ ' + "Use the DEFAULT_AXES or the constructor that just requires '
+ + str([*self.extra_parameters.keys()])
+ + ' to get a reasonable default.");\n'
+ )
+ # TODO add limits for parameters
+ self.code += " }\n\n"
+
+ self.code += " public " + self.java_class_name + "(Object... args) {\n"
+ self.code += " this(Axes.fromArgs(DEFAULT_AXES, args)"
+
+ doubleIndex = 0
+ for name, type in self.extra_parameters.items():
+ if type == "double":
+ self.code += (
+ ", CoordinateSystems.doubleFromArgs("
+ + str(doubleIndex)
+ + ", args)"
+ )
+ doubleIndex += 1
+ else:
+ raise Exception(
+ "Don't know how to handle "
+ + type
+ + " for variable '"
+ + name
+ + "'!"
+ )
+ self.code += ");\n"
+ self.code += " }\n\n"
+
+ self.code += " public " + self.java_class_name + "("
+
+ parms = ""
+ for name, type in self.extra_parameters.items():
+ parms += ", " + type + " " + name
+
+ self.code += parms[2:]
+ self.code += ") {\n"
+ self.code += " this(DEFAULT_AXES, "
+
+ parms = ""
+ for name, type in self.extra_parameters.items():
+ parms += ", " + name
+
+ self.code += parms[2:]
+ self.code += ");\n"
+ self.code += " }\n\n"
+
+ def __build_to_base_point(self):
+ self.code += " @Override\n"
+ self.code += " public double[] toBasePoint(double[] position) {\n"
+ if self.g.to_base_point is None:
+ self.code += ' throw new UnsupportedOperationException("Transformation not implemented!")'
+ else:
+ self.code += (
+ " double[] pointInBase = new double[" + str(self.g.dim()) + "];\n"
+ )
+ for i in range(self.g.dim()):
+ self.code += (
+ " pointInBase["
+ + str(i)
+ + "] = "
+ + self.symbolicToJava(self.g.to_base_point[i], java_name="position")
+ + ";\n"
+ )
+ self.code += " return pointInBase;\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __build_from_base_transformation(self):
+ self.code += " @Override\n"
+ self.code += " public double[] fromBasePoint(double[] position) {\n"
+
+ if not self.unable_to_invert and self.inv_transformation is None:
+ (
+ self.unable_to_invert,
+ self.inv_transformation,
+ self.inv_symbols,
+ ) = self.__invert_forward_transformation()
+
+ if self.g.to_base_point is None or self.unable_to_invert:
+ self.code += ' throw new UnsupportedOperationException("Transformation not implemented!")\n'
+ else:
+ self.code += (
+ " double[] pointInCurrentSystem = new double["
+ + str(self.g.dim())
+ + "];\n"
+ )
+ for i in range(self.g.dim()):
+ self.code += (
+ " pointInCurrentSystem["
+ + str(i)
+ + "] = "
+ + self.symbolicToJava(
+ self.inv_transformation[i],
+ java_name="position",
+ symbols=self.inv_symbols,
+ )
+ + ";\n"
+ )
+ self.code += " return pointInCurrentSystem;\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __invert_forward_transformation(self):
+ """
+ Inverts the given to_base_point transformation, i.e. it finds the transformation for a point given
+ in base coordinates to the current coordinate system.
+
+ Returns
+ -------
+ unable_to_invert
+ True if there was any failure in inverting
+ inv_results
+ The resulting inverse transformation or None if not successful
+ inv_symbols
+ The symbols corresponding to the inverse transformation
+ """
+ unable_to_invert = False
+ try:
+ inv_symbols = [i.as_dummy() for i in self.g.symbols]
+ inv_results = sym.solve(
+ [t[0] - t[1] for t in zip(inv_symbols, self.g.to_base_point)],
+ list(self.g.symbols),
+ dict=True,
+ )[0]
+ inv_results = [
+ sym.trigsimp(inv_results[s].simplify()) for s in self.g.symbols
+ ]
+ except Exception as e:
+ unable_to_invert = True
+ inv_results = None
+ inv_symbols = None
+ print("inversion failed: " + str(e))
+ return unable_to_invert, inv_results, inv_symbols
+
+ def __build_to_base_transformation_vector(self):
+ self.code += " @Override\n"
+ self.code += (
+ " public double[] toBaseVector(double[] position, double[] vector) {\n"
+ )
+ if self.g.to_base_point is None:
+ self.code += ' throw new UnsupportedOperationException("Transformation not implemented!")'
+ else:
+ self.code += " double[] vectorInBaseSys = new double[vector.length];\n"
+
+ vec = []
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ expr = self.g.to_base_point[j].diff(self.g.symbols[i]).simplify()
+ if expr == 0:
+ continue
+
+ if expr == 1:
+ expr_as_string = "vector[" + str(j) + "]"
+ else:
+ expr_as_string = (
+ self.symbolicToJava(expr) + "*vector[" + str(j) + "]"
+ )
+
+ if len(vec) <= i:
+ vec.append(expr_as_string)
+ else:
+ vec[i] += "+" + expr_as_string
+ if vec[i] != 0:
+ self.code += " vectorInBaseSys[" + str(i) + "] = " + vec[i] + ";\n"
+ self.code += " return vectorInBaseSys;\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __build_from_base_transformation_vector(self):
+ self.code += " @Override\n"
+ self.code += (
+ " public double[] fromBaseVector(double[] position, double[] vector) {\n"
+ )
+
+ if not self.unable_to_invert and self.inv_transformation is None:
+ (
+ self.unable_to_invert,
+ self.inv_transformation,
+ self.inv_symbols,
+ ) = self.__invert_forward_transformation()
+
+ if self.g.to_base_point is None or self.unable_to_invert:
+ self.code += ' throw new UnsupportedOperationException("Transformation not implemented!")'
+ else:
+ self.code += " double[] vectorInCurrentSys = new double[vector.length];\n"
+
+ vec = []
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ expr = (
+ self.inv_transformation[j].diff(self.inv_symbols[i]).simplify()
+ )
+ if expr == 0:
+ continue
+
+ if expr == 1:
+ expr_as_string = "vector[" + str(j) + "]"
+ else:
+ expr_as_string = (
+ self.symbolicToJava(expr, symbols=self.inv_symbols)
+ + "*vector["
+ + str(j)
+ + "]"
+ )
+
+ if len(vec) <= i:
+ vec.append(expr_as_string)
+ else:
+ vec[i] += "+" + expr_as_string
+ if vec[i] != 0:
+ self.code += (
+ " vectorInCurrentSys[" + str(i) + "] = " + vec[i] + ";\n"
+ )
+ self.code += " return vectorInCurrentSys;\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __build_metric_tensor(self):
+ self.code += " @Override\n"
+ self.code += " public double[][] metricTensor(double[] position, TensorTransformation behavior) {\n"
+ self.code += " int dim = " + str(self.g.dim()) + ";\n"
+ self.code += " double[][] g = new double[dim][dim];\n"
+ self.code += "\n"
+ self.code += " // Note that we skip all elements that are zero anyways\n"
+
+ # check whether the same elements in covariant and covariant entries are nonnull
+ sameElementsAreNonNull = True
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ if (self.g.gdd[i, j] != 0 and self.g.guu[i, j] == 0) or (
+ self.g.gdd[i, j] == 0 and self.g.guu[i, j] != 0
+ ):
+ sameElementsAreNonNull = False
+ else:
+ # Continue if the inner loop wasn't broken.
+ continue
+ # Inner loop was broken, break the outer.
+ break
+
+ if self.g.gdd != self.g.guu and not sameElementsAreNonNull:
+ self.code += " return switch (behavior) {\n"
+ self.code += " case COVARIANT -> {\n"
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ if self.g.gdd[i, j] == 0:
+ continue
+ self.code += (
+ " g["
+ + str(i)
+ + "]["
+ + str(j)
+ + "] = metricCoefficient(position, behavior, "
+ + str(i)
+ + ", "
+ + str(j)
+ + ");\n"
+ )
+ self.code += " return g;\n"
+ self.code += " }\n"
+
+ self.code += " case CONTRAVARIANT -> {\n"
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ if self.g.guu[i, j] == 0:
+ continue
+ self.code += (
+ " g["
+ + str(i)
+ + "]["
+ + str(j)
+ + "] = metricCoefficient(position, behavior, "
+ + str(i)
+ + ", "
+ + str(j)
+ + ");\n"
+ )
+ self.code += " return g;\n"
+ self.code += " }\n"
+ self.code += " };\n"
+ self.code += " }\n"
+ else:
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ if self.g.gdd[i, j] == 0:
+ continue
+ self.code += (
+ " g["
+ + str(i)
+ + "]["
+ + str(j)
+ + "] = metricCoefficient(position, behavior, "
+ + str(i)
+ + ", "
+ + str(j)
+ + ");\n"
+ )
+ self.code += " return g;\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __build_metric_coefficient(self):
+ self.code += " @Override\n"
+ self.code += " public double metricCoefficient(double[] position, TensorTransformation behavior, int i, int j) {\n"
+
+ self.code += " if (i < 0 || j < 0) {\n"
+ self.code += ' throw new IllegalArgumentException(("Metric coefficient not available for i=%d, j=%d "\n'
+ self.code += (
+ ' + "(too low dimension, only '
+ + str(self.g.dim())
+ + " dimensions "
+ + str([*range(self.g.dim())]).replace(" ", "")
+ + " are supported for "
+ + self.java_name.lower()
+ + ' coordinates)")\n'
+ )
+ self.code += " .formatted(i, j));\n"
+ self.code += (
+ " } else if (i > "
+ + str(self.g.dim() - 1)
+ + " || j > "
+ + str(self.g.dim() - 1)
+ + ") {\n"
+ )
+ self.code += ' throw new IllegalArgumentException(("Metric coefficient not available for i=%d, j=%d "\n'
+ self.code += (
+ ' + "(too high dimension, only '
+ + str(self.g.dim())
+ + " dimensions "
+ + str([*range(self.g.dim())]).replace(" ", "")
+ + " are supported for "
+ + self.java_name.lower()
+ + ' coordinates)")\n'
+ )
+ self.code += " .formatted(i, j));\n"
+ self.code += " }\n\n"
+
+ if self.g.gdd != self.g.guu:
+ self.code += " return switch (behavior) {\n"
+ self.code += " case COVARIANT -> {\n"
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ if self.g.gdd[i, j] == 0:
+ continue
+ self.code += (
+ " if (i == "
+ + str(i)
+ + " && j == "
+ + str(j)
+ + ") return "
+ + self.symbolicToJava(self.g.gdd[i, j])
+ + ";\n"
+ )
+ self.code += " return 0;\n"
+ self.code += " }\n"
+
+ self.code += " case CONTRAVARIANT -> {\n"
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ if self.g.guu[i, j] == 0:
+ continue
+ self.code += (
+ " if (i == "
+ + str(i)
+ + " && j == "
+ + str(j)
+ + ") return "
+ + self.symbolicToJava(self.g.guu[i, j])
+ + ";\n"
+ )
+ self.code += " return 0;\n"
+ self.code += " }\n"
+ self.code += " };\n"
+ self.code += " }\n"
+ else:
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ if self.g.gdd[i, j] == 0:
+ continue
+ self.code += (
+ " if (i == "
+ + str(i)
+ + " && j == "
+ + str(j)
+ + ") return "
+ + self.symbolicToJava(self.g.gdd[i, j])
+ + ";\n"
+ )
+ self.code += " return 0;\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __build_dimension(self):
+ self.code += " @Override\n"
+ self.code += " public int dimension() {\n"
+ self.code += " return " + str(self.g.dim()) + ";\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __build_isOrthogonal(self):
+ self.code += " @Override\n"
+ self.code += " public boolean isOrthogonal() {\n"
+
+ isOrthogonal = True
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ if i == j:
+ continue
+ if self.g.gdd[i, j] != 0:
+ isOrthogonal = False
+ break
+ else:
+ # Continue if the inner loop wasn't broken.
+ continue
+ # Inner loop was broken, break the outer.
+ break
+
+ self.code += " return " + str(isOrthogonal).lower() + ";\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def symbolicToJava(self, expression, java_name="position", symbols=None):
+ if symbols is None:
+ symbols = self.g.symbols
+
+ expression = str(expression)
+ mapping = {}
+ for i in range(len(symbols)):
+ mapping[str(symbols[i])] = java_name + "[" + str(i) + "]"
+
+ # add math symbols
+ mapping["atan"] = "Math.atan"
+ mapping["sin"] = "Math.sin"
+ mapping["cos"] = "Math.cos"
+ mapping["tan"] = "Math.tan"
+ mapping["abs"] = "Math.abs"
+ mapping["Abs"] = "Math.abs"
+ mapping["sqrt"] = "Math.sqrt"
+ ## TODO and many more
+
+ for key in OrderedDict(
+ sorted(mapping.items(), reverse=True, key=lambda t: len(str(t[0])))
+ ):
+ if key == "tan":
+ # we need to be a bit careful, as it may find Math.atan...
+ expression = re.sub("(?<!Math.a)tan", mapping[key], expression)
+ else:
+ expression = expression.replace(key, mapping[key])
+
+ # handle exponents
+ while expression.find("**") != -1:
+ index = expression.find("**")
+ numBrackets = 0
+ for i in range(index - 1, -1, -1):
+ if expression[i] == ")":
+ numBrackets += 1
+ if expression[i] == "(":
+ numBrackets -= 1
+
+ if numBrackets <= 0 and (
+ np.any(
+ np.array([" ", "*", "+", "-", "/", "%", "("]) == expression[i]
+ )
+ or i == 0
+ ):
+ indexForBracketClose = len(expression)
+ bracketForExponent = False
+ for j, c in enumerate(expression[index + len("**") :]):
+ if j == 0 and c == "(":
+ bracketForExponent = True
+ continue
+
+ if not c.isdigit() and c != ".":
+ indexForBracketClose = (
+ len(expression[: index + len("**")]) + j
+ )
+ break
+ if bracketForExponent:
+ expression = (
+ expression[: index + len("**")]
+ + expression[index + len("**") + len("(") :]
+ )
+ else:
+ expression = (
+ expression[:indexForBracketClose]
+ + ")"
+ + expression[indexForBracketClose:]
+ )
+ expression = (
+ expression[:index] + ", " + expression[index + len("**") :]
+ )
+
+ if i != 0:
+ i += 1
+ expression = expression[:i] + "Math.pow(" + expression[i:]
+ break
+
+ return expression
+
+
+if __name__ == "__main__":
+ # TODO vector trafo
+ # TODO parameters
+
+ r = sym.symbols("r", real=True, positive=True)
+ theta = sym.symbols("theta", real=True)
+ syms = sym.symbols("r theta z", real=True)
+
+ axes = {0: "m", 1: "rad", 2: "rad"}
+ extra_parms = {"a": "double"}
+
+ m = Metric.fromTransformation(
+ syms,
+ to_base_point=[syms[0] * sym.cos(syms[1]), syms[0] * sym.sin(syms[1]), syms[2]],
+ )
+
+ code = JavaCoordinateSystemCreator(
+ "CylindricalCoordinates", m, axes, extra_parameters=extra_parms
+ )
+ print(code)
+ print("----")
+
+ m = Metric.fromTransformation(syms, to_base_point=[syms[0], syms[1], syms[2]])
+ print(JavaCoordinateSystemCreator("CartesianCoordinates", m, axes))
+ print("----")
diff --git a/tfields/bases/udo/metric.py b/tfields/bases/udo/metric.py
new file mode 100644
index 0000000..2d83f96
--- /dev/null
+++ b/tfields/bases/udo/metric.py
@@ -0,0 +1,139 @@
+import numpy as np
+import sympy as sym
+
+from sympy.matrices import dotprodsimp
+
+
+class Metric(object):
+ """
+ Represents a metric.
+
+ Parameters
+ ----------
+ symbols : numpy array or array_like of sympy.core.symbol.Symbol
+ The symbols representing the different dimensions, e.g. `sympy.symbols('r theta z', real=True)`.
+ Note that it is good to provide as much information on the symbols as possible, e.g. in the
+ aforementioned example that 'r' is positive
+ m : Matrix
+ The (covariant) metric represented by a Matrix
+ to_base_point : numpy array or array_like, optional
+ The transformation equation to get from the current coordinate system into the base coordinate system.
+ Should be expressed using the `symbols` and sympy functions for, e.g., the cosine.
+ Note that not providing this may limit the scope on subsequent operations with this metric.
+
+ Returns
+ -------
+ Metric
+ the (analytical) metric as derived from the transformation for a point into the base coordinate system
+ """
+
+ def __init__(self, symbols, m: sym.Matrix, to_base_point=None):
+ self.gdd = m # covariant
+ print("Inverting metric...")
+
+ # necessary due to a regression in sympy
+ with dotprodsimp(False):
+ self.guu = m.inv() # contravariant
+
+ print("Metric successfully inverted")
+ self.symbols = symbols
+ self.to_base_point = to_base_point
+
+ @staticmethod
+ def fromTransformation(symbols, to_base_point):
+ """
+ Creates a new metric from the given to_base_point transformation.
+
+ Parameters
+ ----------
+ symbols : numpy array or array_like of sympy.core.symbol.Symbol
+ The symbols representing the different dimensions, e.g. `sympy.symbols('r theta z', real=True)`
+ to_base_point : numpy array or array_like
+ The transformation equation to get from the current coordinate system into the base coordinate system.
+ Should be expressed using the `symbols` and sympy functions for, e.g., the cosine.
+
+ Returns
+ -------
+ Metric
+ the (analytical) metric as derived from the transformation for a point into the base coordinate system
+ """
+ if to_base_point is None:
+ raise Exception("You need to specify the point-to-base-transformation!")
+
+ print("Reconstructing metric from point-to-base-transformation...")
+ dim = len(to_base_point)
+ partial_derivatives = [[0 for i in range(dim)] for i in range(dim)]
+
+ for i in range(dim):
+ for j in range(dim):
+ partial_derivatives[i][j] = sym.diff(to_base_point[j], symbols[i])
+
+ g = [[0 for i in range(dim)] for i in range(dim)]
+ for i in range(dim):
+ for j in range(dim):
+ for k in range(dim):
+ g[i][j] += partial_derivatives[i][k] * partial_derivatives[j][k]
+ g[j][i] = g[i][j] = g[i][j].simplify()
+
+ print("Metric successfully reconstructed!")
+
+ return Metric(symbols, sym.Matrix(g), to_base_point)
+
+ def __str__(self):
+ """
+ Creates a representative string of the (covariant) metric.
+
+ Returns
+ -------
+ str
+ the analytic metric wth covariant components represented by a string
+ """
+ return "g_dd = " + str(self.gdd)
+
+ def dd(self, i, j):
+ """
+ Gets the covariant metric coefficient at the specified position.
+
+ Parameters
+ ----------
+ i : int
+ The row index in the metric
+ j : int
+ The column index in the metric
+
+ Returns
+ -------
+ sympy expression
+ the (analytical) covariant metric coefficient g_ij
+ """
+ return self.gdd[i, j]
+
+ def uu(self, i, j):
+ """
+ Gets the contravariant metric coefficient at the specified position.
+
+ Parameters
+ ----------
+ i : int
+ The row index in the metric
+ j : int
+ The column index in the metric
+
+ Returns
+ -------
+ sympy expression
+ the (analytical) contravariant metric coefficient g^ij
+ """
+ return self.guu[i, j]
+
+ def dim(self) -> int:
+ """
+ Gets the dimensionality of the metric.
+ For example, a metric with size 3x3 will be of dimensionality 3.
+
+ Returns
+ -------
+ int
+ the dimension of the metric
+ """
+ return int(np.sqrt(len(self.gdd)))
diff --git a/tfields/bases/udo/sympyGradient.py b/tfields/bases/udo/sympyGradient.py
new file mode 100644
index 0000000..037d1f1
--- /dev/null
+++ b/tfields/bases/udo/sympyGradient.py
@@ -0,0 +1,19 @@
+import sys
+from sympy import *
+import numpy as np
+from collections import OrderedDict
+import re
+
+from sympy.matrices import dotprodsimp
+from sympyTensors import Metric
+
+r = symbols("r", real=True, positive=True)
+theta = symbols("theta", real=True)
+z = symbols("z", real=True)
+
+from sympy.vector import gradient
+from sympy.vector import CoordSys3D
+
+c = CoordSys3D("c", transformation="cylindrical", variable_names=("r", "theta", "z"))
+
+gradient(c.r)
diff --git a/tfields/bases/udo/sympyTensors.py b/tfields/bases/udo/sympyTensors.py
new file mode 100644
index 0000000..c48a3bd
--- /dev/null
+++ b/tfields/bases/udo/sympyTensors.py
@@ -0,0 +1,849 @@
+import sys
+from sympy import *
+import numpy as np
+from collections import OrderedDict
+import re
+
+from sympy.matrices import dotprodsimp
+
+
+class Metric(object):
+ """
+ Turn matrix into upper and lower metric
+
+ m: covariant matrix
+ """
+
+ def __init__(self, symbols, m: Matrix, to_base=None):
+ self.gdd = m # covariant
+ print("Inverting metric...")
+ print(type(symbols))
+ print(type(symbols[0]))
+ # necessary due to a regression in sympy
+ with dotprodsimp(False):
+ self.guu = m.inv() # contravariant
+
+ print("Metric successfully inverted")
+ self.symbols = symbols
+ self.to_base = to_base
+
+ """
+ Creates a new metric from the given to_base_point transformation.
+ """
+
+ @staticmethod
+ def fromTransformation(symbols, to_base_point):
+ if to_base_point is None:
+ raise Exception(
+ "You need to specify either the to-base-transformation or the from-base-transformation"
+ )
+
+ print("Reconstructing metric from to-base-transformation")
+ dim = len(to_base_point)
+ partial_derivatives = [[0 for i in range(dim)] for i in range(dim)]
+
+ for i in range(dim):
+ for j in range(dim):
+ partial_derivatives[i][j] = diff(to_base_point[j], symbols[i])
+
+ g = [[0 for i in range(dim)] for i in range(dim)]
+ for i in range(dim):
+ for j in range(dim):
+ for k in range(dim):
+ g[i][j] += partial_derivatives[i][k] * partial_derivatives[j][k]
+ g[j][i] = g[i][j] = g[i][j].simplify()
+ print("Metric successfully reconstructed")
+ return Metric(symbols, Matrix(g), to_base_point)
+
+ def __str__(self):
+ return "g_dd = \n" + str(self.gdd)
+
+ def dd(self, i, j):
+ return self.gdd[i, j]
+
+ def uu(self, i, j):
+ return self.guu[i, j]
+
+ def dim(self):
+ return int(np.sqrt(len(self.gdd)))
+
+
+class Gamma(object):
+ """
+ Calculate Christoffel Gamma_ij^k symbols of metric g
+ """
+
+ def __init__(self, g: Metric, x):
+ self.g = g
+ self.x = x
+
+ """
+
+ """
+
+ def ddu(self, m, i, j):
+ g = self.g
+ x = self.x
+ chr = 0
+ for k in range(self.g.dim()):
+ chr += (
+ g.uu(m, k)
+ / 2
+ * (
+ g.dd(k, i).diff(x[j])
+ + g.dd(k, j).diff(x[i])
+ - g.dd(i, j).diff(x[k])
+ )
+ )
+ return chr.simplify()
+ # return chr
+
+
+class Riemann(object):
+ """
+ Calculate Riemann tensor R^mu_nu,rho,sigma
+ """
+
+ def __init__(self, g: Metric, G: Gamma, x):
+ self.g = g
+ self.G = G
+ self.x = x
+
+ def uddd(self, mu, nu, rho, sigma):
+ G = self.G
+ x = self.x
+ R = G.ddu(mu, nu, sigma).diff(x[rho]) - G.ddu(mu, nu, rho).diff(x[sigma])
+ for lam in range(self.g.dim()):
+ R += G.ddu(mu, rho, lam) * G.ddu(lam, nu, sigma) - G.ddu(
+ mu, sigma, lam
+ ) * G.ddu(lam, nu, rho)
+ return R.simplify()
+
+ def dddd(self, mu, nu, rho, sigma):
+ g = self.g
+ R = 0
+ for lam in range(self.g.dim()):
+ R += g.dd(mu, lam) * self.uddd(lam, nu, rho, sigma)
+ return R.simplify()
+
+
+class Ricci(object):
+ """
+ Calculate Ricci tensor from Riemann tensor
+ """
+
+ def __init__(self, R: Riemann):
+ self.R = R
+
+ def dd(self, mu, nu):
+ R = self.R
+ Ric = 0
+ for rho in range(self.R.g.dim()):
+ Ric += R.uddd(rho, mu, rho, nu)
+ return Ric.simplify()
+
+
+class RicciScalar(object):
+ """
+ Calculate Ricci scalar from Ricci tensor
+ """
+
+ def __init__(self, Ric, g):
+ self.Ric = Ric
+ self.g = g
+
+ def value(self):
+ Ric = self.Ric
+ g = self.g
+ RS = 0
+ for mu in range(self.g.dim()):
+ for nu in range(self.g.dim()):
+ RS += g.uu(mu, nu) * Ric.dd(mu, nu)
+ return RS.simplify()
+
+
+class JavaCoordinateSystemCreator(object):
+ def __init__(self, java_class_name, Metric, java_name=None, extra_parameters=None):
+ self.java_class_name = java_class_name
+ self.g = Metric
+ self.code = ""
+
+ if java_name == "" or java_name is None:
+ index = java_class_name.index("Coordinates")
+ if index == -1:
+ raise Exception("Cannot guess java_name, please supply explicitly")
+ self.java_name = java_class_name[:index]
+ else:
+ self.java_name = java_name
+
+ # TODO handle extra parameters
+
+ def __str__(self):
+ self.__build()
+ return self.code
+
+ def __build(self):
+ self.__build_header()
+ self.__build_dimension()
+ self.__build_to_base_transformation()
+ self.__build_from_base_transformation()
+ self.__build_to_base_transformation_vector()
+ self.__build_from_base_transformation_vector()
+ self.__build_metric_coefficient()
+ self.__build_metric_tensor()
+ self.__build_jacobian_determinant()
+ self.__build_isOrthogonal()
+ self.__build_christoffel_symbols_2nd_kind()
+ self.__build_christoffel_symbols_1st_kind()
+ self.__build_riemann_tensor()
+ self.__build_ricci_tensor()
+ self.__build_ricci_scalar()
+
+ self.code += "}\n"
+
+ def __build_header(self):
+ self.code += "@CoordinateSystemSymbols({\"\"}) // TODO: add symbols representing the coordinate system, e.g. 'cartesian' and 'cart' for the cartesian coordinate system\n"
+ self.code += (
+ "public final record "
+ + self.java_class_name
+ + "(NavigableSet<Axes> axes) implements CoordinateSystem {\n\n"
+ )
+
+ def __build_to_base_transformation(self):
+ self.code += " @Override\n"
+ self.code += " public Function<double[],double[]> toBaseTransformation() {\n"
+ if self.g.to_base is None:
+ self.code += (
+ ' throw new IllegalStateException("Transformation not implemented!")'
+ )
+ else:
+ self.code += " return pos -> {\n"
+ self.code += (
+ " double[] pointInBase = new double[" + str(self.g.dim()) + "];\n"
+ )
+ for i in range(self.g.dim()):
+ self.code += (
+ " pointInBase["
+ + str(i)
+ + "] = "
+ + self.symbolicToJava(self.g.to_base[i], java_name="pos")
+ + ";\n"
+ )
+ self.code += " return pointInBase;\n"
+ self.code += " };\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __build_from_base_transformation(self):
+ self.code += " @Override\n"
+ self.code += " public Function<double[],double[]> fromBaseTransformation() {\n"
+
+ unable_to_invert = False
+ try:
+ inv_syms = [i.as_dummy() for i in self.g.symbols]
+ inv_results = solve(
+ [t[0] - t[1] for t in zip(inv_syms, self.g.to_base)],
+ list(self.g.symbols),
+ dict=True,
+ )[0]
+ inv_results = [inv_results[s] for s in self.g.symbols]
+ except:
+ unable_to_invert = True
+
+ if self.g.to_base is None or unable_to_invert:
+ self.code += (
+ ' throw new IllegalStateException("Transformation not implemented!")'
+ )
+ else:
+ self.code += " return pos -> {\n"
+ self.code += (
+ " double[] pointInCurrentSystem = new double["
+ + str(self.g.dim())
+ + "];\n"
+ )
+ for i in range(self.g.dim()):
+ self.code += (
+ " pointInCurrentSystem["
+ + str(i)
+ + "] = "
+ + self.symbolicToJava(
+ trigsimp(inv_results[i].simplify()),
+ java_name="pos",
+ symbols=inv_syms,
+ )
+ + ";\n"
+ )
+ self.code += " return pointInCurrentSystem;\n"
+ self.code += " };\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __build_to_base_transformation_vector(self):
+ self.code += " @Override\n"
+ self.code += (
+ " public double[] toBaseVector(double[] position, double[] vector) {\n"
+ )
+ if self.g.to_base is None:
+ self.code += (
+ ' throw new IllegalStateException("Transformation not implemented!")'
+ )
+ else:
+ self.code += " double[] vectorInBaseSys = new double[vector.length];\n"
+
+ vec = []
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ expr = self.g.to_base[j].diff(self.g.symbols[i]).simplify()
+ if expr == 0:
+ continue
+
+ if expr == 1:
+ expr_as_string = "vector[" + str(j) + "]"
+ else:
+ expr_as_string = (
+ self.symbolicToJava(expr) + "*vector[" + str(j) + "]"
+ )
+
+ if len(vec) <= i:
+ vec.append(expr_as_string)
+ else:
+ vec[i] += "+" + expr_as_string
+ if vec[i] != 0:
+ self.code += " vectorInBaseSys[" + str(i) + "] = " + vec[i] + ";\n"
+ self.code += " return vectorInBaseSys;\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __build_from_base_transformation_vector(self):
+ self.code += " @Override\n"
+ self.code += (
+ " public double[] fromBaseVector(double[] position, double[] vector) {\n"
+ )
+
+ unable_to_invert = False
+ try:
+ inv_syms = [i.as_dummy() for i in self.g.symbols]
+ inv_results = solve(
+ [t[0] - t[1] for t in zip(inv_syms, self.g.to_base)],
+ list(self.g.symbols),
+ dict=True,
+ )[0]
+ inv_results = [trigsimp(inv_results[s].simplify()) for s in self.g.symbols]
+ except:
+ unable_to_invert = True
+
+ if self.g.to_base is None or unable_to_invert:
+ self.code += (
+ ' throw new IllegalStateException("Transformation not implemented!")'
+ )
+ else:
+ self.code += " double[] vectorInCurrentSys = new double[vector.length];\n"
+
+ vec = []
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ expr = inv_results[j].diff(inv_syms[i]).simplify()
+ if expr == 0:
+ continue
+
+ if expr == 1:
+ expr_as_string = "vector[" + str(j) + "]"
+ else:
+ expr_as_string = (
+ self.symbolicToJava(expr, symbols=inv_syms)
+ + "*vector["
+ + str(j)
+ + "]"
+ )
+
+ if len(vec) <= i:
+ vec.append(expr_as_string)
+ else:
+ vec[i] += "+" + expr_as_string
+ if vec[i] != 0:
+ self.code += (
+ " vectorInCurrentSys[" + str(i) + "] = " + vec[i] + ";\n"
+ )
+ self.code += " return vectorInCurrentSys;\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __build_metric_tensor(self):
+ self.code += " @Override\n"
+ self.code += " public double[][] metricTensor(double[] position, BasisChangeBehavior behavior) {\n"
+ self.code += " int dim = " + str(self.g.dim()) + ";\n"
+ self.code += " double[][] g = new double[dim][dim];\n"
+ self.code += "\n"
+ self.code += " // Note that we skip all elements that are zero anyways\n"
+
+ # check whether the same elements in covariant and covariant entries are nonnull
+ sameElementsAreNonNull = True
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ if (self.g.gdd[i, j] != 0 and self.g.guu[i, j] == 0) or (
+ self.g.gdd[i, j] == 0 and self.g.guu[i, j] != 0
+ ):
+ sameElementsAreNonNull = False
+ else:
+ # Continue if the inner loop wasn't broken.
+ continue
+ # Inner loop was broken, break the outer.
+ break
+
+ if self.g.gdd != self.g.guu and not sameElementsAreNonNull:
+ self.code += " return switch (behavior) {\n"
+ self.code += " case COVARIANT -> {\n"
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ if self.g.gdd[i, j] == 0:
+ continue
+ self.code += (
+ " g["
+ + str(i)
+ + "]["
+ + str(j)
+ + "] = metricCoefficient(position, behavior, "
+ + str(i)
+ + ", "
+ + str(j)
+ + ");\n"
+ )
+ self.code += " return g;\n"
+ self.code += " }\n"
+
+ self.code += " case CONTRAVARIANT -> {\n"
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ if self.g.guu[i, j] == 0:
+ continue
+ self.code += (
+ " g["
+ + str(i)
+ + "]["
+ + str(j)
+ + "] = metricCoefficient(position, behavior, "
+ + str(i)
+ + ", "
+ + str(j)
+ + ");\n"
+ )
+ self.code += " return g;\n"
+ self.code += " }\n"
+ self.code += " };\n"
+ self.code += " }\n"
+ else:
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ if self.g.gdd[i, j] == 0:
+ continue
+ self.code += (
+ " g["
+ + str(i)
+ + "]["
+ + str(j)
+ + "] = metricCoefficient(position, behavior, "
+ + str(i)
+ + ", "
+ + str(j)
+ + ");\n"
+ )
+ self.code += " return g;\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __build_metric_coefficient(self):
+ self.code += " @Override\n"
+ self.code += " public double metricCoefficient(double[] position, BasisChangeBehavior behavior, int i, int j) {\n"
+
+ self.code += " if (i < 0 || j < 0) {\n"
+ self.code += ' throw new IllegalArgumentException(("Metric coefficient not available for i=%d, j=%d "\n'
+ self.code += (
+ ' + "(too low dimension, only '
+ + str(self.g.dim())
+ + " dimensions "
+ + str([*range(self.g.dim())]).replace(" ", "")
+ + " are supported for "
+ + self.java_name.lower()
+ + ' coordinates)")\n'
+ )
+ self.code += " .formatted(i, j));\n"
+ self.code += (
+ " } else if (i > "
+ + str(self.g.dim() - 1)
+ + " || j > "
+ + str(self.g.dim() - 1)
+ + ") {\n"
+ )
+ self.code += ' throw new IllegalArgumentException(("Metric coefficient not available for i=%d, j=%d "\n'
+ self.code += (
+ ' + "(too high dimension, only '
+ + str(self.g.dim())
+ + " dimensions "
+ + str([*range(self.g.dim())]).replace(" ", "")
+ + " are supported for "
+ + self.java_name.lower()
+ + ' coordinates)")\n'
+ )
+ self.code += " .formatted(i, j));\n"
+ self.code += " }\n\n"
+
+ if self.g.gdd != self.g.guu:
+ self.code += " return switch (behavior) {\n"
+ self.code += " case COVARIANT -> {\n"
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ if self.g.gdd[i, j] == 0:
+ continue
+ self.code += (
+ " if (i == "
+ + str(i)
+ + " && j == "
+ + str(j)
+ + ") return "
+ + self.symbolicToJava(self.g.gdd[i, j])
+ + ";\n"
+ )
+ self.code += " return 0;\n"
+ self.code += " }\n"
+
+ self.code += " case CONTRAVARIANT -> {\n"
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ if self.g.guu[i, j] == 0:
+ continue
+ self.code += (
+ " if (i == "
+ + str(i)
+ + " && j == "
+ + str(j)
+ + ") return "
+ + self.symbolicToJava(self.g.guu[i, j])
+ + ";\n"
+ )
+ self.code += " return 0;\n"
+ self.code += " }\n"
+ self.code += " };\n"
+ self.code += " }\n"
+ else:
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ if self.g.gdd[i, j] == 0:
+ continue
+ self.code += (
+ " if (i == "
+ + str(i)
+ + " && j == "
+ + str(j)
+ + ") return "
+ + self.symbolicToJava(self.g.gdd[i, j])
+ + ";\n"
+ )
+ self.code += " return 0;\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __build_dimension(self):
+ self.code += " @Override\n"
+ self.code += " public int dimension() {\n"
+ self.code += " return " + str(self.g.dim()) + ";\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __build_isOrthogonal(self):
+ self.code += " @Override\n"
+ self.code += " public boolean isOrthogonal() {\n"
+
+ isOrthogonal = True
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ if i == j:
+ continue
+ if self.g.gdd[i, j] != 0:
+ isOrthogonal = False
+ break
+ else:
+ # Continue if the inner loop wasn't broken.
+ continue
+ # Inner loop was broken, break the outer.
+ break
+
+ self.code += " return " + str(isOrthogonal).lower() + ";\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __build_jacobian_determinant(self):
+ self.code += " @Override\n"
+ self.code += " public double jacobianDeterminant(double[] position) {\n"
+
+ det = sqrt(self.g.gdd.det()).simplify()
+
+ self.code += " return " + self.symbolicToJava(det) + ";\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __build_christoffel_symbols_2nd_kind(self):
+ self.code += " @Override\n"
+ self.code += " public double christoffelSymbol2ndKind(double[] position, int m, int i, int j) {\n"
+
+ gamma = Gamma(self.g, self.g.symbols)
+
+ for m in range(self.g.dim()):
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ if gamma.ddu(m, i, j) == 0:
+ continue
+ self.code += (
+ " if (m == "
+ + str(m)
+ + " && i == "
+ + str(i)
+ + " && j == "
+ + str(j)
+ + ") return "
+ + self.symbolicToJava(gamma.ddu(m, i, j))
+ + ";\n"
+ )
+ self.code += " return 0;\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __build_christoffel_symbols_1st_kind(self):
+ self.code += " @Override\n"
+ self.code += " public double christoffelSymbol1stKind(double[] position, int i, int j, int k) {\n"
+
+ gamma = Gamma(self.g, self.g.symbols)
+
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ for k in range(self.g.dim()):
+ christoffelSymbol1stKind = 0
+ for u in range(self.g.dim()):
+ christoffelSymbol1stKind += self.g.gdd[u, k] * gamma.ddu(
+ u, i, j
+ )
+ christoffelSymbol1stKind = christoffelSymbol1stKind.simplify()
+
+ if christoffelSymbol1stKind == 0:
+ continue
+ self.code += (
+ " if (i == "
+ + str(i)
+ + " && j == "
+ + str(j)
+ + " && k == "
+ + str(k)
+ + ") return "
+ + self.symbolicToJava(christoffelSymbol1stKind)
+ + ";\n"
+ )
+ self.code += " return 0;\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __build_riemann_tensor(self):
+ self.code += " @Override\n"
+ self.code += " public double riemannTensor(double[] position, int i, int j, int k, int l) {\n"
+
+ gamma = Gamma(self.g, self.g.symbols)
+ riem = Riemann(self.g, gamma, self.g.symbols)
+
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ for k in range(self.g.dim()):
+ for l in range(self.g.dim()):
+ if riem.dddd(i, j, k, l) == 0:
+ continue
+ self.code += (
+ " if (i == "
+ + str(i)
+ + " && j == "
+ + str(j)
+ + " && k == "
+ + str(k)
+ + " && l == "
+ + str(l)
+ + ") return "
+ + self.symbolicToJava(riem.dddd(i, j, k, l))
+ + ";\n"
+ )
+ self.code += " return 0;\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __build_ricci_tensor(self):
+ self.code += " @Override\n"
+ self.code += " public double ricciTensor(double[] position, int i, int j) {\n"
+
+ gamma = Gamma(self.g, self.g.symbols)
+ riem = Riemann(self.g, gamma, self.g.symbols)
+ ric = Ricci(riem)
+
+ for i in range(self.g.dim()):
+ for j in range(self.g.dim()):
+ if ric.dd(i, j) == 0:
+ continue
+ self.code += (
+ " if (i == "
+ + str(i)
+ + " && j == "
+ + str(j)
+ + ") return "
+ + self.symbolicToJava(ric.dd(i, j))
+ + ";\n"
+ )
+ self.code += " return 0;\n"
+ self.code += " }\n"
+ self.code += "\n"
+
+ def __build_ricci_scalar(self):
+ self.code += " @Override\n"
+ self.code += " public double ricciScalar(double[] position) {\n"
+
+ gamma = Gamma(self.g, self.g.symbols)
+ riem = Riemann(self.g, gamma, self.g.symbols)
+ ric = Ricci(riem)
+ ric_scalar = RicciScalar(ric, self.g)
+
+ self.code += " return " + self.symbolicToJava(ric_scalar.value()) + ";\n"
+ self.code += " }\n"
+
+ def symbolicToJava(self, expression, java_name="position", symbols=None):
+ if symbols is None:
+ symbols = self.g.symbols
+
+ expression = str(expression)
+ mapping = {}
+ for i in range(len(symbols)):
+ mapping[str(symbols[i])] = java_name + "[" + str(i) + "]"
+
+ # add math symbols
+ mapping["atan"] = "Math.atan"
+ mapping["sin"] = "Math.sin"
+ mapping["cos"] = "Math.cos"
+ mapping["tan"] = "Math.tan"
+ mapping["abs"] = "Math.abs"
+ mapping["Abs"] = "Math.abs"
+ mapping["sqrt"] = "Math.sqrt"
+ ## TODO and many more
+
+ for key in OrderedDict(
+ sorted(mapping.items(), reverse=True, key=lambda t: len(str(t[0])))
+ ):
+ if key == "tan":
+ # we need to be a bit careful, as it may find Math.atan...
+ expression = re.sub("(?<!Math.a)tan", mapping[key], expression)
+ else:
+ expression = expression.replace(key, mapping[key])
+
+ # handle exponents
+ while expression.find("**") != -1:
+ index = expression.find("**")
+ numBrackets = 0
+ for i in range(index - 1, -1, -1):
+ if expression[i] == ")":
+ numBrackets += 1
+ if expression[i] == "(":
+ numBrackets -= 1
+
+ if numBrackets <= 0 and (
+ np.any(
+ np.array([" ", "*", "+", "-", "/", "%", "("]) == expression[i]
+ )
+ or i == 0
+ ):
+ indexForBracketClose = len(expression)
+ bracketForExponent = False
+ for j, c in enumerate(expression[index + len("**") :]):
+ if j == 0 and c == "(":
+ bracketForExponent = True
+ continue
+
+ if not c.isdigit() and c != ".":
+ indexForBracketClose = (
+ len(expression[: index + len("**")]) + j
+ )
+ break
+ if bracketForExponent:
+ expression = (
+ expression[: index + len("**")]
+ + expression[index + len("**") + len("(") :]
+ )
+ else:
+ expression = (
+ expression[:indexForBracketClose]
+ + ")"
+ + expression[indexForBracketClose:]
+ )
+ expression = (
+ expression[:index] + ", " + expression[index + len("**") :]
+ )
+
+ if i != 0:
+ i += 1
+ expression = expression[:i] + "Math.pow(" + expression[i:]
+ break
+
+ return expression
+
+
+if __name__ == "__main__":
+ # TODO vector trafo
+ # TODO parameters
+
+ r = symbols("r", real=True, positive=True)
+ theta = symbols("theta", real=True)
+ syms = symbols("r theta z", real=True)
+
+ print(type(syms[0] * cos(syms[1]) / 2))
+ m = Metric.fromTransformation(
+ syms, to_base_point=[syms[0] * cos(syms[1]), syms[0] * sin(syms[1]), syms[2]]
+ )
+
+ print(JavaCoordinateSystemCreator("CylindricalCoordinates", m))
+ print("----")
+
+ m = Metric.fromTransformation(syms, to_base_point=[syms[0], syms[1], syms[2]])
+ print(JavaCoordinateSystemCreator("CartesianCoordinates", m))
+ print("----")
+
+ a = symbols("a", real=True, constant=True)
+ sigma = symbols("sigma", real=True)
+ tau = symbols("tau", real=True, positive=True)
+ theta = symbols("theta", real=True, positive=True)
+ syms = [sigma, tau, theta]
+
+ to_base_point = [
+ a * sinh(syms[1]) * cos(syms[2]) / (cosh(syms[1]) - cos(syms[0])),
+ a * sinh(syms[1]) * sin(syms[2]) / (cosh(syms[1]) - cos(syms[0])),
+ a * sign(syms[0]) * abs(sin(syms[0]) / (cosh(syms[1]) - cos(syms[0]))),
+ ]
+
+ # m = Metric.fromTransformation(syms, to_base_point=to_base_point)
+ # print(JavaCoordinateSystemCreator("ToroidalCoordinates", m))
+ # print("----")
+
+ # jcsc = JavaCoordinateSystemCreator("CylindricalCoordinates", m)
+ # print(jcsc.symbolicToJava("r**2+z**2"))
+
+ # syms = symbols('r theta z')
+ ## define the metric for 3d cylindrical coordinates
+ # m = [[0 for i in range(3)] for i in range(3)]
+ # m[0][0] = 1
+ # m[1][1] = syms[0]**2
+ # m[2][2] = 1
+ #
+ # metric = Metric(Matrix(m))
+ #
+ #
+ # gamma = Gamma(metric, syms)
+ # riem = Riemann(metric, gamma, syms)
+ # ric = Ricci(riem)
+ # ric_scalar = RicciScalar(ric,metric)
+ #
+ # codeprint_christoffels(gamma)
+ # print("")
+ # codeprint_riem(riem)
+ # print("")
+ # codeprint_ricci(ric)
+ # print("")
+ # print(ric_scalar.value())
diff --git a/tfields/bases/udo/tensorTransformation.py b/tfields/bases/udo/tensorTransformation.py
new file mode 100644
index 0000000..a42ee90
--- /dev/null
+++ b/tfields/bases/udo/tensorTransformation.py
@@ -0,0 +1,68 @@
+import sys
+from sympy import *
+import numpy as np
+from collections import OrderedDict
+import re
+
+from sympy.matrices import dotprodsimp
+from sympyTensors import Metric
+
+r = symbols("r", real=True, positive=True)
+theta = symbols("theta", real=True)
+z = symbols("z", real=True)
+
+g = [[1, 0, 0], [0, r ** 2, 0], [0, 0, 1]]
+# g = [[1,0,0],[0,1,0],[0,0,1]]
+
+syms = [r, theta, z]
+
+metric = Metric(symbols, Matrix(g))
+
+# sigma = [[1,2,3],[1.5,3,4.5],[2,4,6]]
+sigma = [[2, 3, 4], [3, 4.5, 6], [4, 6, 8]]
+
+result = [[0, 0, 0], [0, 0, 0], [0, 0, 0]]
+for i in range(3):
+ for j in range(3):
+ result[i][j] += (
+ sigma[0][j] * metric.guu[0, i]
+ + sigma[1][j] * metric.guu[1, i]
+ + sigma[2][j] * metric.guu[2, i]
+ )
+print("sigma^i_j " + str(result))
+
+sigma = result
+result = [[0, 0, 0], [0, 0, 0], [0, 0, 0]]
+for i in range(3):
+ for j in range(3):
+ result[i][j] += (
+ sigma[i][0] * metric.guu[0, j]
+ + sigma[i][1] * metric.guu[1, j]
+ + sigma[i][2] * metric.guu[2, j]
+ )
+print("sigma^ij " + str(result))
+
+sigma = result
+result = [[0, 0, 0], [0, 0, 0], [0, 0, 0]]
+for i in range(3):
+ for j in range(3):
+ result[i][j] += (
+ sigma[0][j] * metric.gdd[0, i]
+ + sigma[1][j] * metric.gdd[1, i]
+ + sigma[2][j] * metric.gdd[2, i]
+ )
+print("sigma_i^j " + str(result))
+
+sigma = result
+result = [[0, 0, 0], [0, 0, 0], [0, 0, 0]]
+for i in range(3):
+ for j in range(3):
+ result[i][j] += (
+ sigma[i][0] * metric.gdd[0, j]
+ + sigma[i][1] * metric.gdd[1, j]
+ + sigma[i][2] * metric.gdd[2, j]
+ )
+print("sigma_ij " + str(result))
+
+
+print("Done")
diff --git a/tfields/bases/udo/testSympy.py b/tfields/bases/udo/testSympy.py
new file mode 100644
index 0000000..28731b5
--- /dev/null
+++ b/tfields/bases/udo/testSympy.py
@@ -0,0 +1,10 @@
+import matplotlib.pyplot as plt
+import numpy as np
+import sympy as sym
+
+r = np.linspace(-1.6, np.pi, 1000)
+theta = np.linspace(-np.pi, np.pi, len(r))
+
+plt.plot(r, np.arctan2(theta, r))
+plt.plot(r, -2 * np.arctan((r - np.sqrt(r ** 2 + theta ** 2)) / theta))
+plt.show()
diff --git a/theory/1901.05512.pdf b/theory/1901.05512.pdf
new file mode 100644
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--
GitLab