choreo.ActionSym#

class ActionSym(Py_ssize_t[::1] BodyPerm, double[:, ::1] SpaceRot, Py_ssize_t TimeRev, Py_ssize_t TimeShiftNum, Py_ssize_t TimeShiftDen)#

This class defines the symmetries in a N-body system.

A symmetry \(\sigma\) is a transformation of paths that leaves the physics of a N-body system invariant.

\[x_{\text{LoopTarget}}(t) = \text{SpaceRot} \cdot x_{\text{LoopSource}} (\text{TimeRev} * (t - \text{TimeShift}))\]

Where SpaceRot is assumed orthogonal and TimeShift is defined as a rational fraction.

Useful to detect loops and constraints.

cf Palais’ principle of symmetric criticality in [1]

Cited:

[1]

Richard S Palais. The principle of symmetric criticality. Communications in Mathematical Physics, 69(1):19–30, 1979.

Class Constructor

__init__()

Specifies a symmetry of the action functional.

Parameters:

BodyPerm (numpy.ndarray(shape = (nbody), dtype = np.intp)) – Permutation of the bodies.
SpaceRot (numpy.ndarray(shape = (geodim, geodim), dtype = np.float64)) – Isometry of space.
TimeRev (int) – A value of -1 denotes time reversal, and a value of 1 denotes no time reversal.
TimeShiftNum (int) – Numerator of the rational time shift.
TimeShiftDen (int) – Denominator of the rational time shift.

Methods Summary

`ApplyT`	Returns a rational transformed time instant given an input rational time instant.
`ApplyTInv`	Returns a rational inverse transformed time instant given an input rational time instant.
`ApplyTInvSegm`	Returns a rational inverse transformed time segment given an input rational time segment.
`ApplyTSegm`	Returns a rational transformed time segment given an input rational time segment.
`BuildCayleyGraph`	Builds the Cayley graph of a list of group generators.
`Compose`	Returns the composition of two transformations.
`Conjugate`	Returns the conjugation of a transformation with respect to another transformation.
`FromDict`	Returns a new `choreo.ActionSym` from a `dict`.
`Identity`	Returns the identity transformation.
`Inverse`	Returns the inverse of a symmetry transformation.
`InvolutivePermutations`	Generates all involutive permutations of size `n`.
`IsIdentity`	Returns `True` if the transformation is within `atol` of the identity.
`IsIdentityPerm`	Returns `True` if the body permutation part of the transformation is the identity permutation.
`IsIdentityPermAndRot`	Returns `True` if parts of the transformation are within `atol` of the identity.
`IsIdentityPermAndRotAndTimeRev`	Returns `True` if parts of the transformation are within `atol` of the identity.
`IsIdentityRot`	Returns `True` if the space isometry part of the transformation is within `atol` of the identity.
`IsIdentityRotAndTime`	Returns `True` if parts of the transformation are within `atol` of the identity.
`IsIdentityRotAndTimeRev`	Returns `True` if parts of the transformation are within `atol` of the identity.
`IsIdentityTimeRev`	Returns `True` if the transformation does not reverse time.
`IsIdentityTimeShift`	Returns `True` if the transformation does not shift time.
`IsSame`	Returns `True` if the two transformations are within `atol` of each other.
`IsSamePerm`	Returns `True` if the two transformations have identical body permutations.
`IsSameRot`	Returns `True` if the two transformations have space isometries within `atol` of each other.
`IsSameRotAndTime`	Returns `True` if the two transformations have properties within `atol` of each other.
`IsSameRotAndTimeRev`	Returns `True` if the two transformations have properties within `atol` of each other.
`IsSameTimeRev`	Returns `True` if the two transformations have identical time reversal.
`IsSameTimeShift`	Returns `True` if the two transformations have identical time shifts.
`IsWellFormed`	Returns `True` if the transformation is well-formed.
`Random`	Returns a random transformation.
`SurjectiveDirectSpaceRot`	Surjective parametrization of direct isometries.
`TimeDerivative`	Returns the time derivative of a symmetry transformation.
`TimeShifts`	Generates all rational fractions in \([0,1[\) with denominator lower or equal to `max_den`.
`TransformPos`	Computes a transformed position by the space isometry.
`TransformSegment`	Computes a transformed segment of positions by the time and space isometry.
`__init__`	Specifies a symmetry of the action functional.
`to_dict`	Returns a `dict` containing the transformation informations.

Attributes

`BodyPerm`	`numpy.ndarray(shape = (nbody), dtype = np.intp)` Permutation of the bodies.
`SpaceRot`	`numpy.ndarray(shape = (nbody), dtype = np.intp)` Isometry of space.
`TimeRev`	`int` Time reversal
`TimeShiftDen`	`int` Denominator of the rational time shift.
`TimeShiftNum`	`int` Numerator of the rational time shift.