/* Copyright 2002-2024 CS GROUP
    * Licensed to CS GROUP (CS) under one or more
    * contributor license agreements.  See the NOTICE file distributed with
    * this work for additional information regarding copyright ownership.
    * CS licenses this file to You under the Apache License, Version 2.0
    * (the "License"); you may not use this file except in compliance with
    * the License.  You may obtain a copy of the License at
    *
    *   http://www.apache.org/licenses/LICENSE-2.0
   *
   * Unless required by applicable law or agreed to in writing, software
   * distributed under the License is distributed on an "AS IS" BASIS,
   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   * See the License for the specific language governing permissions and
   * limitations under the License.
   */
  package org.orekit.orbits;
  
  import org.hipparchus.Field;
  import org.hipparchus.CalculusFieldElement;
  import org.hipparchus.analysis.differentiation.FieldUnivariateDerivative1;
  import org.hipparchus.geometry.euclidean.threed.FieldVector3D;
  import org.hipparchus.util.FastMath;
  import org.hipparchus.util.FieldSinCos;
  import org.hipparchus.util.MathArrays;
  import org.orekit.errors.OrekitIllegalArgumentException;
  import org.orekit.errors.OrekitInternalError;
  import org.orekit.errors.OrekitMessages;
  import org.orekit.frames.FieldKinematicTransform;
  import org.orekit.frames.Frame;
  import org.orekit.time.FieldAbsoluteDate;
  import org.orekit.utils.FieldPVCoordinates;
  import org.orekit.utils.TimeStampedFieldPVCoordinates;
  
  
  /**
   * This class handles equinoctial orbital parameters, which can support both
   * circular and equatorial orbits.
   * <p>
   * The parameters used internally are the equinoctial elements which can be
   * related to Keplerian elements as follows:
   *   <pre>
   *     a
   *     ex = e cos(ω + Ω)
   *     ey = e sin(ω + Ω)
   *     hx = tan(i/2) cos(Ω)
   *     hy = tan(i/2) sin(Ω)
   *     lv = v + ω + Ω
   *   </pre>
   * where ω stands for the Perigee Argument and Ω stands for the
   * Right Ascension of the Ascending Node.
   * <p>
   * The conversion equations from and to Keplerian elements given above hold only
   * when both sides are unambiguously defined, i.e. when orbit is neither equatorial
   * nor circular. When orbit is either equatorial or circular, the equinoctial
   * parameters are still unambiguously defined whereas some Keplerian elements
   * (more precisely ω and Ω) become ambiguous. For this reason, equinoctial
   * parameters are the recommended way to represent orbits. Note however than
   * the present implementation does not handle non-elliptical cases.
   * </p>
   * <p>
   * The instance <code>EquinoctialOrbit</code> is guaranteed to be immutable.
   * </p>
   * @see    Orbit
   * @see    KeplerianOrbit
   * @see    CircularOrbit
   * @see    CartesianOrbit
   * @author Mathieu Rom&eacute;ro
   * @author Luc Maisonobe
   * @author Guylaine Prat
   * @author Fabien Maussion
   * @author V&eacute;ronique Pommier-Maurussane
   * @since 9.0
   * @param <T> type of the field elements
   */
  public class FieldEquinoctialOrbit<T extends CalculusFieldElement<T>> extends FieldOrbit<T>
          implements PositionAngleBased {
  
      /** Semi-major axis (m). */
      private final T a;
  
      /** First component of the eccentricity vector. */
      private final T ex;
  
      /** Second component of the eccentricity vector. */
      private final T ey;
  
      /** First component of the inclination vector. */
      private final T hx;
  
      /** Second component of the inclination vector. */
      private final T hy;
  
      /** Cached longitude argument (rad). */
      private final T cachedL;
  
      /** Cache type of position angle (longitude argument). */
      private final PositionAngleType cachedPositionAngleType;
  
     /** Semi-major axis derivative (m/s). */
     private final T aDot;
 
     /** First component of the eccentricity vector derivative. */
     private final T exDot;
 
     /** Second component of the eccentricity vector derivative. */
     private final T eyDot;
 
     /** First component of the inclination vector derivative. */
     private final T hxDot;
 
     /** Second component of the inclination vector derivative. */
     private final T hyDot;
 
     /** Derivative of cached longitude argument (rad/s). */
     private final T cachedLDot;
 
     /** Partial Cartesian coordinates (position and velocity are valid, acceleration may be missing). */
     private FieldPVCoordinates<T> partialPV;
 
     /** Creates a new instance.
      * @param a  semi-major axis (m)
      * @param ex e cos(ω + Ω), first component of eccentricity vector
      * @param ey e sin(ω + Ω), second component of eccentricity vector
      * @param hx tan(i/2) cos(Ω), first component of inclination vector
      * @param hy tan(i/2) sin(Ω), second component of inclination vector
      * @param l  (M or E or v) + ω + Ω, mean, eccentric or true longitude argument (rad)
      * @param type type of longitude argument
      * @param cachedPositionAngleType type of cached longitude argument
      * @param frame the frame in which the parameters are defined
      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
      * @param date date of the orbital parameters
      * @param mu central attraction coefficient (m³/s²)
      * @exception IllegalArgumentException if eccentricity is equal to 1 or larger or
      * if frame is not a {@link Frame#isPseudoInertial pseudo-inertial frame}
      * @since 12.1
      */
     public FieldEquinoctialOrbit(final T a, final T ex, final T ey,
                                  final T hx, final T hy, final T l,
                                  final PositionAngleType type, final PositionAngleType cachedPositionAngleType,
                                  final Frame frame, final FieldAbsoluteDate<T> date, final T mu)
         throws IllegalArgumentException {
         this(a, ex, ey, hx, hy, l,
              a.getField().getZero(), a.getField().getZero(), a.getField().getZero(), a.getField().getZero(), a.getField().getZero(),
              computeKeplerianLDot(type, a, ex, ey, mu, l, type), type, cachedPositionAngleType, frame, date, mu);
     }
 
     /** Creates a new instance.
      * @param a  semi-major axis (m)
      * @param ex e cos(ω + Ω), first component of eccentricity vector
      * @param ey e sin(ω + Ω), second component of eccentricity vector
      * @param hx tan(i/2) cos(Ω), first component of inclination vector
      * @param hy tan(i/2) sin(Ω), second component of inclination vector
      * @param l  (M or E or v) + ω + Ω, mean, eccentric or true longitude argument (rad)
      * @param type type of longitude argument
      * @param frame the frame in which the parameters are defined
      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
      * @param date date of the orbital parameters
      * @param mu central attraction coefficient (m³/s²)
      * @exception IllegalArgumentException if eccentricity is equal to 1 or larger or
      * if frame is not a {@link Frame#isPseudoInertial pseudo-inertial frame}
      */
     public FieldEquinoctialOrbit(final T a, final T ex, final T ey,
                                  final T hx, final T hy, final T l,
                                  final PositionAngleType type,
                                  final Frame frame, final FieldAbsoluteDate<T> date, final T mu)
             throws IllegalArgumentException {
         this(a, ex, ey, hx, hy, l,
                 a.getField().getZero(), a.getField().getZero(), a.getField().getZero(), a.getField().getZero(), a.getField().getZero(),
                 computeKeplerianLDot(type, a, ex, ey, mu, l, type), type, type, frame, date, mu);
     }
 
     /** Creates a new instance.
      * @param a  semi-major axis (m)
      * @param ex e cos(ω + Ω), first component of eccentricity vector
      * @param ey e sin(ω + Ω), second component of eccentricity vector
      * @param hx tan(i/2) cos(Ω), first component of inclination vector
      * @param hy tan(i/2) sin(Ω), second component of inclination vector
      * @param l  (M or E or v) + ω + Ω, mean, eccentric or true longitude argument (rad)
      * @param aDot  semi-major axis derivative (m/s)
      * @param exDot d(e cos(ω + Ω))/dt, first component of eccentricity vector derivative
      * @param eyDot d(e sin(ω + Ω))/dt, second component of eccentricity vector derivative
      * @param hxDot d(tan(i/2) cos(Ω))/dt, first component of inclination vector derivative
      * @param hyDot d(tan(i/2) sin(Ω))/dt, second component of inclination vector derivative
      * @param lDot  d(M or E or v) + ω + Ω)/dr, mean, eccentric or true longitude argument  derivative (rad/s)
      * @param type type of longitude argument
      * @param cachedPositionAngleType of cached longitude argument
      * @param frame the frame in which the parameters are defined
      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
      * @param date date of the orbital parameters
      * @param mu central attraction coefficient (m³/s²)
      * @exception IllegalArgumentException if eccentricity is equal to 1 or larger or
      * if frame is not a {@link Frame#isPseudoInertial pseudo-inertial frame}
      * @since 12.1
      */
     public FieldEquinoctialOrbit(final T a, final T ex, final T ey,
                                  final T hx, final T hy, final T l,
                                  final T aDot, final T exDot, final T eyDot,
                                  final T hxDot, final T hyDot, final T lDot,
                                  final PositionAngleType type, final PositionAngleType cachedPositionAngleType,
                                  final Frame frame, final FieldAbsoluteDate<T> date, final T mu)
         throws IllegalArgumentException {
         super(frame, date, mu);
 
         if (ex.getReal() * ex.getReal() + ey.getReal() * ey.getReal() >= 1.0) {
             throw new OrekitIllegalArgumentException(OrekitMessages.HYPERBOLIC_ORBIT_NOT_HANDLED_AS,
                                                      getClass().getName());
         }
         this.cachedPositionAngleType = cachedPositionAngleType;
         this.a     = a;
         this.aDot  = aDot;
         this.ex    = ex;
         this.exDot = exDot;
         this.ey    = ey;
         this.eyDot = eyDot;
         this.hx    = hx;
         this.hxDot = hxDot;
         this.hy    = hy;
         this.hyDot = hyDot;
 
         final FieldUnivariateDerivative1<T> lUD = initializeCachedL(l, lDot, type);
         this.cachedL = lUD.getValue();
         this.cachedLDot = lUD.getFirstDerivative();
 
         this.partialPV = null;
 
     }
 
     /** Creates a new instance.
      * @param a  semi-major axis (m)
      * @param ex e cos(ω + Ω), first component of eccentricity vector
      * @param ey e sin(ω + Ω), second component of eccentricity vector
      * @param hx tan(i/2) cos(Ω), first component of inclination vector
      * @param hy tan(i/2) sin(Ω), second component of inclination vector
      * @param l  (M or E or v) + ω + Ω, mean, eccentric or true longitude argument (rad)
      * @param aDot  semi-major axis derivative (m/s)
      * @param exDot d(e cos(ω + Ω))/dt, first component of eccentricity vector derivative
      * @param eyDot d(e sin(ω + Ω))/dt, second component of eccentricity vector derivative
      * @param hxDot d(tan(i/2) cos(Ω))/dt, first component of inclination vector derivative
      * @param hyDot d(tan(i/2) sin(Ω))/dt, second component of inclination vector derivative
      * @param lDot  d(M or E or v) + ω + Ω)/dr, mean, eccentric or true longitude argument  derivative (rad/s)
      * @param type type of longitude argument
      * @param frame the frame in which the parameters are defined
      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
      * @param date date of the orbital parameters
      * @param mu central attraction coefficient (m³/s²)
      * @exception IllegalArgumentException if eccentricity is equal to 1 or larger or
      * if frame is not a {@link Frame#isPseudoInertial pseudo-inertial frame}
      * @since 12.1
      */
     public FieldEquinoctialOrbit(final T a, final T ex, final T ey,
                                  final T hx, final T hy, final T l,
                                  final T aDot, final T exDot, final T eyDot,
                                  final T hxDot, final T hyDot, final T lDot,
                                  final PositionAngleType type,
                                  final Frame frame, final FieldAbsoluteDate<T> date, final T mu)
             throws IllegalArgumentException {
         this(a, ex, ey, hx, hy, l, aDot, exDot, eyDot, hxDot, hyDot, lDot, type, type, frame, date, mu);
     }
 
     /** Constructor from Cartesian parameters.
      *
      * <p> The acceleration provided in {@code pvCoordinates} is accessible using
      * {@link #getPVCoordinates()} and {@link #getPVCoordinates(Frame)}. All other methods
      * use {@code mu} and the position to compute the acceleration, including
      * {@link #shiftedBy(CalculusFieldElement)} and {@link #getPVCoordinates(FieldAbsoluteDate, Frame)}.
      *
      * @param pvCoordinates the position, velocity and acceleration
      * @param frame the frame in which are defined the {@link FieldPVCoordinates}
      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
      * @param mu central attraction coefficient (m³/s²)
      * @exception IllegalArgumentException if eccentricity is equal to 1 or larger or
      * if frame is not a {@link Frame#isPseudoInertial pseudo-inertial frame}
      */
     public FieldEquinoctialOrbit(final TimeStampedFieldPVCoordinates<T> pvCoordinates,
                                  final Frame frame, final T mu)
         throws IllegalArgumentException {
         super(pvCoordinates, frame, mu);
 
         //  compute semi-major axis
         final FieldVector3D<T> pvP = pvCoordinates.getPosition();
         final FieldVector3D<T> pvV = pvCoordinates.getVelocity();
         final FieldVector3D<T> pvA = pvCoordinates.getAcceleration();
         final T r2 = pvP.getNormSq();
         final T r  = r2.sqrt();
         final T V2 = pvV.getNormSq();
         final T rV2OnMu = r.multiply(V2).divide(mu);
 
         // compute semi-major axis
         a = r.divide(rV2OnMu.negate().add(2));
 
         if (!isElliptical()) {
             throw new OrekitIllegalArgumentException(OrekitMessages.HYPERBOLIC_ORBIT_NOT_HANDLED_AS,
                                                      getClass().getName());
         }
 
         // compute inclination vector
         final FieldVector3D<T> w = pvCoordinates.getMomentum().normalize();
         final T d = getOne().divide(getOne().add(w.getZ()));
         hx =  d.negate().multiply(w.getY());
         hy =  d.multiply(w.getX());
 
         // compute true longitude argument
         cachedPositionAngleType = PositionAngleType.TRUE;
         final T cLv = (pvP.getX().subtract(d.multiply(pvP.getZ()).multiply(w.getX()))).divide(r);
         final T sLv = (pvP.getY().subtract(d.multiply(pvP.getZ()).multiply(w.getY()))).divide(r);
         cachedL = sLv.atan2(cLv);
 
         // compute eccentricity vector
         final T eSE = FieldVector3D.dotProduct(pvP, pvV).divide(a.multiply(mu).sqrt());
         final T eCE = rV2OnMu.subtract(1);
         final T e2  = eCE.square().add(eSE.square());
         final T f   = eCE.subtract(e2);
         final T g   = e2.negate().add(1).sqrt().multiply(eSE);
         ex = a.multiply(f.multiply(cLv).add( g.multiply(sLv))).divide(r);
         ey = a.multiply(f.multiply(sLv).subtract(g.multiply(cLv))).divide(r);
 
         partialPV = pvCoordinates;
 
         if (hasNonKeplerianAcceleration(pvCoordinates, mu)) {
             // we have a relevant acceleration, we can compute derivatives
 
             final T[][] jacobian = MathArrays.buildArray(a.getField(), 6, 6);
             getJacobianWrtCartesian(PositionAngleType.MEAN, jacobian);
 
             final FieldVector3D<T> keplerianAcceleration    = new FieldVector3D<>(r.multiply(r2).reciprocal().multiply(mu.negate()), pvP);
             final FieldVector3D<T> nonKeplerianAcceleration = pvA.subtract(keplerianAcceleration);
             final T   aX                       = nonKeplerianAcceleration.getX();
             final T   aY                       = nonKeplerianAcceleration.getY();
             final T   aZ                       = nonKeplerianAcceleration.getZ();
             aDot  = jacobian[0][3].multiply(aX).add(jacobian[0][4].multiply(aY)).add(jacobian[0][5].multiply(aZ));
             exDot = jacobian[1][3].multiply(aX).add(jacobian[1][4].multiply(aY)).add(jacobian[1][5].multiply(aZ));
             eyDot = jacobian[2][3].multiply(aX).add(jacobian[2][4].multiply(aY)).add(jacobian[2][5].multiply(aZ));
             hxDot = jacobian[3][3].multiply(aX).add(jacobian[3][4].multiply(aY)).add(jacobian[3][5].multiply(aZ));
             hyDot = jacobian[4][3].multiply(aX).add(jacobian[4][4].multiply(aY)).add(jacobian[4][5].multiply(aZ));
 
             // in order to compute true longitude argument derivative, we must compute
             // mean longitude argument derivative including Keplerian motion and convert to true anomaly
             final T lMDot = getKeplerianMeanMotion().
                             add(jacobian[5][3].multiply(aX)).add(jacobian[5][4].multiply(aY)).add(jacobian[5][5].multiply(aZ));
             final FieldUnivariateDerivative1<T> exUD = new FieldUnivariateDerivative1<>(ex, exDot);
             final FieldUnivariateDerivative1<T> eyUD = new FieldUnivariateDerivative1<>(ey, eyDot);
             final FieldUnivariateDerivative1<T> lMUD = new FieldUnivariateDerivative1<>(getLM(), lMDot);
             final FieldUnivariateDerivative1<T> lvUD = FieldEquinoctialLongitudeArgumentUtility.meanToTrue(exUD, eyUD, lMUD);
             cachedLDot = lvUD.getFirstDerivative();
 
         } else {
             // acceleration is either almost zero or NaN,
             // we assume acceleration was not known
             // we don't set up derivatives
             aDot  = getZero();
             exDot = getZero();
             eyDot = getZero();
             hxDot = getZero();
             hyDot = getZero();
             cachedLDot = computeKeplerianLDot(cachedPositionAngleType, a, ex, ey, mu, cachedL, cachedPositionAngleType);
         }
 
     }
 
     /** Constructor from Cartesian parameters.
      *
      * <p> The acceleration provided in {@code pvCoordinates} is accessible using
      * {@link #getPVCoordinates()} and {@link #getPVCoordinates(Frame)}. All other methods
      * use {@code mu} and the position to compute the acceleration, including
      * {@link #shiftedBy(CalculusFieldElement)} and {@link #getPVCoordinates(FieldAbsoluteDate, Frame)}.
      *
      * @param pvCoordinates the position end velocity
      * @param frame the frame in which are defined the {@link FieldPVCoordinates}
      * (<em>must</em> be a {@link Frame#isPseudoInertial pseudo-inertial frame})
      * @param date date of the orbital parameters
      * @param mu central attraction coefficient (m³/s²)
      * @exception IllegalArgumentException if eccentricity is equal to 1 or larger or
      * if frame is not a {@link Frame#isPseudoInertial pseudo-inertial frame}
      */
     public FieldEquinoctialOrbit(final FieldPVCoordinates<T> pvCoordinates, final Frame frame,
                             final FieldAbsoluteDate<T> date, final T mu)
         throws IllegalArgumentException {
         this(new TimeStampedFieldPVCoordinates<>(date, pvCoordinates), frame, mu);
     }
 
     /** Constructor from any kind of orbital parameters.
      * @param op orbital parameters to copy
      */
     public FieldEquinoctialOrbit(final FieldOrbit<T> op) {
         super(op.getFrame(), op.getDate(), op.getMu());
 
         a     = op.getA();
         ex    = op.getEquinoctialEx();
         ey    = op.getEquinoctialEy();
         hx    = op.getHx();
         hy    = op.getHy();
         cachedPositionAngleType = PositionAngleType.TRUE;
         cachedL    = op.getLv();
 
         aDot  = op.getADot();
         exDot = op.getEquinoctialExDot();
         eyDot = op.getEquinoctialEyDot();
         hxDot = op.getHxDot();
         hyDot = op.getHyDot();
         cachedLDot = op.getLvDot();
     }
 
     /** Constructor from Field and EquinoctialOrbit.
      * <p>Build a FieldEquinoctialOrbit from non-Field EquinoctialOrbit.</p>
      * @param field CalculusField to base object on
      * @param op non-field orbit with only "constant" terms
      * @since 12.0
      */
     public FieldEquinoctialOrbit(final Field<T> field, final EquinoctialOrbit op) {
         super(op.getFrame(), new FieldAbsoluteDate<>(field, op.getDate()), field.getZero().newInstance(op.getMu()));
 
         a     = getZero().newInstance(op.getA());
         ex    = getZero().newInstance(op.getEquinoctialEx());
         ey    = getZero().newInstance(op.getEquinoctialEy());
         hx    = getZero().newInstance(op.getHx());
         hy    = getZero().newInstance(op.getHy());
         cachedPositionAngleType = op.getCachedPositionAngleType();
         cachedL    = getZero().newInstance(op.getL(cachedPositionAngleType));
 
         aDot  = getZero().newInstance(op.getADot());
         exDot = getZero().newInstance(op.getEquinoctialExDot());
         eyDot = getZero().newInstance(op.getEquinoctialEyDot());
         hxDot = getZero().newInstance(op.getHxDot());
         hyDot = getZero().newInstance(op.getHyDot());
         cachedLDot = getZero().newInstance(op.getLDot(cachedPositionAngleType));
     }
 
     /** Constructor from Field and Orbit.
      * <p>Build a FieldEquinoctialOrbit from any non-Field Orbit.</p>
      * @param field CalculusField to base object on
      * @param op non-field orbit with only "constant" terms
      * @since 12.0
      */
     public FieldEquinoctialOrbit(final Field<T> field, final Orbit op) {
         this(field, (EquinoctialOrbit) OrbitType.EQUINOCTIAL.convertType(op));
     }
 
     /** {@inheritDoc} */
     @Override
     public OrbitType getType() {
         return OrbitType.EQUINOCTIAL;
     }
 
     /** {@inheritDoc} */
     @Override
     public T getA() {
         return a;
     }
 
     /** {@inheritDoc} */
     @Override
     public T getADot() {
         return aDot;
     }
 
     /** {@inheritDoc} */
     @Override
     public T getEquinoctialEx() {
         return ex;
     }
 
     /** {@inheritDoc} */
     @Override
     public T getEquinoctialExDot() {
         return exDot;
     }
 
     /** {@inheritDoc} */
     @Override
     public T getEquinoctialEy() {
         return ey;
     }
 
     /** {@inheritDoc} */
     @Override
     public T getEquinoctialEyDot() {
         return eyDot;
     }
 
     /** {@inheritDoc} */
     @Override
     public T getHx() {
         return hx;
     }
 
     /** {@inheritDoc} */
     @Override
     public T getHxDot() {
         return hxDot;
     }
 
     /** {@inheritDoc} */
     @Override
     public T getHy() {
         return hy;
     }
 
     /** {@inheritDoc} */
     @Override
     public T getHyDot() {
         return hyDot;
     }
 
     /** {@inheritDoc} */
     @Override
     public T getLv() {
         switch (cachedPositionAngleType) {
             case TRUE:
                 return cachedL;
 
             case ECCENTRIC:
                 return FieldEquinoctialLongitudeArgumentUtility.eccentricToTrue(ex, ey, cachedL);
 
             case MEAN:
                 return FieldEquinoctialLongitudeArgumentUtility.meanToTrue(ex, ey, cachedL);
 
             default:
                 throw new OrekitInternalError(null);
         }
     }
 
     /** {@inheritDoc} */
     @Override
     public T getLvDot() {
         switch (cachedPositionAngleType) {
             case ECCENTRIC:
                 final FieldUnivariateDerivative1<T> lEUD = new FieldUnivariateDerivative1<>(cachedL, cachedLDot);
                 final FieldUnivariateDerivative1<T> exUD     = new FieldUnivariateDerivative1<>(ex,     exDot);
                 final FieldUnivariateDerivative1<T> eyUD     = new FieldUnivariateDerivative1<>(ey,     eyDot);
                 final FieldUnivariateDerivative1<T> lvUD = FieldEquinoctialLongitudeArgumentUtility.eccentricToTrue(exUD, eyUD,
                         lEUD);
                 return lvUD.getFirstDerivative();
 
             case TRUE:
                 return cachedLDot;
 
             case MEAN:
                 final FieldUnivariateDerivative1<T> lMUD = new FieldUnivariateDerivative1<>(cachedL, cachedLDot);
                 final FieldUnivariateDerivative1<T> exUD2    = new FieldUnivariateDerivative1<>(ex,     exDot);
                 final FieldUnivariateDerivative1<T> eyUD2    = new FieldUnivariateDerivative1<>(ey,     eyDot);
                 final FieldUnivariateDerivative1<T> lvUD2 = FieldEquinoctialLongitudeArgumentUtility.meanToTrue(exUD2,
                         eyUD2, lMUD);
                 return lvUD2.getFirstDerivative();
 
             default:
                 throw new OrekitInternalError(null);
         }
     }
 
     /** {@inheritDoc} */
     @Override
     public T getLE() {
         switch (cachedPositionAngleType) {
             case TRUE:
                 return FieldEquinoctialLongitudeArgumentUtility.trueToEccentric(ex, ey, cachedL);
 
             case ECCENTRIC:
                 return cachedL;
 
             case MEAN:
                 return FieldEquinoctialLongitudeArgumentUtility.meanToEccentric(ex, ey, cachedL);
 
             default:
                 throw new OrekitInternalError(null);
         }
     }
 
     /** {@inheritDoc} */
     @Override
     public T getLEDot() {
 
         switch (cachedPositionAngleType) {
             case TRUE:
                 final FieldUnivariateDerivative1<T> lvUD = new FieldUnivariateDerivative1<>(cachedL, cachedLDot);
                 final FieldUnivariateDerivative1<T> exUD     = new FieldUnivariateDerivative1<>(ex,     exDot);
                 final FieldUnivariateDerivative1<T> eyUD     = new FieldUnivariateDerivative1<>(ey,     eyDot);
                 final FieldUnivariateDerivative1<T> lEUD = FieldEquinoctialLongitudeArgumentUtility.trueToEccentric(exUD, eyUD,
                         lvUD);
                 return lEUD.getFirstDerivative();
 
             case ECCENTRIC:
                 return cachedLDot;
 
             case MEAN:
                 final FieldUnivariateDerivative1<T> lMUD = new FieldUnivariateDerivative1<>(cachedL, cachedLDot);
                 final FieldUnivariateDerivative1<T> exUD2    = new FieldUnivariateDerivative1<>(ex,     exDot);
                 final FieldUnivariateDerivative1<T> eyUD2    = new FieldUnivariateDerivative1<>(ey,     eyDot);
                 final FieldUnivariateDerivative1<T> lEUD2 = FieldEquinoctialLongitudeArgumentUtility.meanToEccentric(exUD2,
                         eyUD2, lMUD);
                 return lEUD2.getFirstDerivative();
 
             default:
                 throw new OrekitInternalError(null);
         }
     }
 
     /** {@inheritDoc} */
     @Override
     public T getLM() {
         switch (cachedPositionAngleType) {
             case TRUE:
                 return FieldEquinoctialLongitudeArgumentUtility.trueToMean(ex, ey, cachedL);
 
             case MEAN:
                 return cachedL;
 
             case ECCENTRIC:
                 return FieldEquinoctialLongitudeArgumentUtility.eccentricToMean(ex, ey, cachedL);
 
             default:
                 throw new OrekitInternalError(null);
         }
     }
 
     /** {@inheritDoc} */
     @Override
     public T getLMDot() {
 
         switch (cachedPositionAngleType) {
             case TRUE:
                 final FieldUnivariateDerivative1<T> lvUD = new FieldUnivariateDerivative1<>(cachedL, cachedLDot);
                 final FieldUnivariateDerivative1<T> exUD     = new FieldUnivariateDerivative1<>(ex,     exDot);
                 final FieldUnivariateDerivative1<T> eyUD     = new FieldUnivariateDerivative1<>(ey,     eyDot);
                 final FieldUnivariateDerivative1<T> lMUD = FieldEquinoctialLongitudeArgumentUtility.trueToMean(exUD, eyUD, lvUD);
                 return lMUD.getFirstDerivative();
 
             case MEAN:
                 return cachedLDot;
 
             case ECCENTRIC:
                 final FieldUnivariateDerivative1<T> lEUD = new FieldUnivariateDerivative1<>(cachedL, cachedLDot);
                 final FieldUnivariateDerivative1<T> exUD2    = new FieldUnivariateDerivative1<>(ex,     exDot);
                 final FieldUnivariateDerivative1<T> eyUD2    = new FieldUnivariateDerivative1<>(ey,     eyDot);
                 final FieldUnivariateDerivative1<T> lMUD2 = FieldEquinoctialLongitudeArgumentUtility.eccentricToMean(exUD2,
                         eyUD2, lEUD);
                 return lMUD2.getFirstDerivative();
 
             default:
                 throw new OrekitInternalError(null);
         }
     }
 
     /** Get the longitude argument.
      * @param type type of the angle
      * @return longitude argument (rad)
      */
     public T getL(final PositionAngleType type) {
         return (type == PositionAngleType.MEAN) ? getLM() :
                                               ((type == PositionAngleType.ECCENTRIC) ? getLE() :
                                                                                    getLv());
     }
 
     /** Get the longitude argument derivative.
      * @param type type of the angle
      * @return longitude argument derivative (rad/s)
      */
     public T getLDot(final PositionAngleType type) {
         return (type == PositionAngleType.MEAN) ? getLMDot() :
                                               ((type == PositionAngleType.ECCENTRIC) ? getLEDot() :
                                                                                    getLvDot());
     }
 
     /** {@inheritDoc} */
     @Override
     public boolean hasNonKeplerianAcceleration() {
         return aDot.getReal() != 0. || exDot.getReal() != 0 || hxDot.getReal() != 0. || eyDot.getReal() != 0. || hyDot.getReal() != 0. ||
                 FastMath.abs(cachedLDot.subtract(computeKeplerianLDot(cachedPositionAngleType, a, ex, ey, getMu(), cachedL, cachedPositionAngleType)).getReal()) > TOLERANCE_POSITION_ANGLE_RATE;
     }
 
     /** {@inheritDoc} */
     @Override
     public T getE() {
         return ex.square().add(ey.square()).sqrt();
     }
 
     /** {@inheritDoc} */
     @Override
     public T getEDot() {
         if (!hasNonKeplerianRates()) {
             return getZero();
         }
         return ex.multiply(exDot).add(ey.multiply(eyDot)).divide(ex.square().add(ey.square()).sqrt());
 
     }
 
     /** {@inheritDoc} */
     @Override
     public T getI() {
         return hx.square().add(hy.square()).sqrt().atan().multiply(2);
     }
 
     /** {@inheritDoc} */
     @Override
     public T getIDot() {
         if (!hasNonKeplerianRates()) {
             return getZero();
         }
         final T h2 = hx.square().add(hy.square());
         final T h  = h2.sqrt();
         return hx.multiply(hxDot).add(hy.multiply(hyDot)).multiply(2).divide(h.multiply(h2.add(1)));
 
     }
 
     /** Compute position and velocity but not acceleration.
      */
     private void computePVWithoutA() {
 
         if (partialPV != null) {
             // already computed
             return;
         }
 
         // get equinoctial parameters
         final T lE = getLE();
 
         // inclination-related intermediate parameters
         final T hx2   = hx.square();
         final T hy2   = hy.square();
         final T factH = getOne().divide(hx2.add(1.0).add(hy2));
 
         // reference axes defining the orbital plane
         final T ux = hx2.add(1.0).subtract(hy2).multiply(factH);
         final T uy = hx.multiply(hy).multiply(factH).multiply(2);
         final T uz = hy.multiply(-2).multiply(factH);
 
         final T vx = uy;
         final T vy = (hy2.subtract(hx2).add(1)).multiply(factH);
         final T vz =  hx.multiply(factH).multiply(2);
 
         // eccentricity-related intermediate parameters
         final T ex2  = ex.square();
         final T exey = ex.multiply(ey);
         final T ey2  = ey.square();
         final T e2   = ex2.add(ey2);
         final T eta  = getOne().subtract(e2).sqrt().add(1);
         final T beta = getOne().divide(eta);
 
         // eccentric longitude argument
         final FieldSinCos<T> scLe = FastMath.sinCos(lE);
         final T cLe    = scLe.cos();
         final T sLe    = scLe.sin();
         final T exCeyS = ex.multiply(cLe).add(ey.multiply(sLe));
 
         // coordinates of position and velocity in the orbital plane
         final T x      = a.multiply(getOne().subtract(beta.multiply(ey2)).multiply(cLe).add(beta.multiply(exey).multiply(sLe)).subtract(ex));
         final T y      = a.multiply(getOne().subtract(beta.multiply(ex2)).multiply(sLe).add(beta .multiply(exey).multiply(cLe)).subtract(ey));
 
         final T factor = getMu().divide(a).sqrt().divide(getOne().subtract(exCeyS));
         final T xdot   = factor.multiply(sLe.negate().add(beta.multiply(ey).multiply(exCeyS)));
         final T ydot   = factor.multiply(cLe.subtract(beta.multiply(ex).multiply(exCeyS)));
 
         final FieldVector3D<T> position =
                         new FieldVector3D<>(x.multiply(ux).add(y.multiply(vx)),
                                             x.multiply(uy).add(y.multiply(vy)),
                                             x.multiply(uz).add(y.multiply(vz)));
         final FieldVector3D<T> velocity =
                         new FieldVector3D<>(xdot.multiply(ux).add(ydot.multiply(vx)), xdot.multiply(uy).add(ydot.multiply(vy)), xdot.multiply(uz).add(ydot.multiply(vz)));
 
         partialPV = new FieldPVCoordinates<>(position, velocity);
 
     }
 
     /** Initialize cached argument of longitude with rate.
      * @param l input argument of longitude
      * @param lDot rate of input argument of longitude
      * @param inputType position angle type passed as input
      * @return argument of longitude to cache with rate
      * @since 12.1
      */
     private FieldUnivariateDerivative1<T> initializeCachedL(final T l, final T lDot,
                                                             final PositionAngleType inputType) {
         if (cachedPositionAngleType == inputType) {
             return new FieldUnivariateDerivative1<>(l, lDot);
 
         } else {
             final FieldUnivariateDerivative1<T> exUD = new FieldUnivariateDerivative1<>(ex, exDot);
             final FieldUnivariateDerivative1<T> eyUD = new FieldUnivariateDerivative1<>(ey, eyDot);
             final FieldUnivariateDerivative1<T> lUD = new FieldUnivariateDerivative1<>(l, lDot);
 
             switch (cachedPositionAngleType) {
 
                 case ECCENTRIC:
                     if (inputType == PositionAngleType.MEAN) {
                         return FieldEquinoctialLongitudeArgumentUtility.meanToEccentric(exUD, eyUD, lUD);
                     } else {
                         return FieldEquinoctialLongitudeArgumentUtility.trueToEccentric(exUD, eyUD, lUD);
                     }
 
                 case TRUE:
                     if (inputType == PositionAngleType.MEAN) {
                         return FieldEquinoctialLongitudeArgumentUtility.meanToTrue(exUD, eyUD, lUD);
                     } else {
                         return FieldEquinoctialLongitudeArgumentUtility.eccentricToTrue(exUD, eyUD, lUD);
                     }
 
                 case MEAN:
                     if (inputType == PositionAngleType.TRUE) {
                         return FieldEquinoctialLongitudeArgumentUtility.trueToMean(exUD, eyUD, lUD);
                     } else {
                         return FieldEquinoctialLongitudeArgumentUtility.eccentricToMean(exUD, eyUD, lUD);
                     }
 
                 default:
                     throw new OrekitInternalError(null);
 
             }
 
         }
 
     }
 
     /** Initialize cached argument of longitude.
      * @param l input argument of longitude
      * @param positionAngleType position angle type passed as input
      * @return argument of longitude to cache
      * @since 12.1
      */
     private T initializeCachedL(final T l, final PositionAngleType positionAngleType) {
         return FieldEquinoctialLongitudeArgumentUtility.convertL(positionAngleType, l, ex, ey, cachedPositionAngleType);
     }
 
     /** Compute non-Keplerian part of the acceleration from first time derivatives.
      * @return non-Keplerian part of the acceleration
      */
     private FieldVector3D<T> nonKeplerianAcceleration() {
 
         final T[][] dCdP = MathArrays.buildArray(a.getField(), 6, 6);
         getJacobianWrtParameters(PositionAngleType.MEAN, dCdP);
 
         final T nonKeplerianMeanMotion = getLMDot().subtract(getKeplerianMeanMotion());
         final T nonKeplerianAx =     dCdP[3][0].multiply(aDot).
                                  add(dCdP[3][1].multiply(exDot)).
                                  add(dCdP[3][2].multiply(eyDot)).
                                  add(dCdP[3][3].multiply(hxDot)).
                                  add(dCdP[3][4].multiply(hyDot)).
                                  add(dCdP[3][5].multiply(nonKeplerianMeanMotion));
         final T nonKeplerianAy =     dCdP[4][0].multiply(aDot).
                                  add(dCdP[4][1].multiply(exDot)).
                                  add(dCdP[4][2].multiply(eyDot)).
                                  add(dCdP[4][3].multiply(hxDot)).
                                  add(dCdP[4][4].multiply(hyDot)).
                                  add(dCdP[4][5].multiply(nonKeplerianMeanMotion));
         final T nonKeplerianAz =     dCdP[5][0].multiply(aDot).
                                  add(dCdP[5][1].multiply(exDot)).
                                  add(dCdP[5][2].multiply(eyDot)).
                                  add(dCdP[5][3].multiply(hxDot)).
                                  add(dCdP[5][4].multiply(hyDot)).
                                  add(dCdP[5][5].multiply(nonKeplerianMeanMotion));
 
         return new FieldVector3D<>(nonKeplerianAx, nonKeplerianAy, nonKeplerianAz);
 
     }
 
     /** {@inheritDoc} */
     @Override
     protected FieldVector3D<T> initPosition() {
 
         // get equinoctial parameters
         final T lE = getLE();
 
         // inclination-related intermediate parameters
         final T hx2   = hx.square();
         final T hy2   = hy.square();
         final T factH = getOne().divide(hx2.add(1.0).add(hy2));
 
         // reference axes defining the orbital plane
         final T ux = hx2.add(1.0).subtract(hy2).multiply(factH);
         final T uy = hx.multiply(hy).multiply(factH).multiply(2);
         final T uz = hy.multiply(-2).multiply(factH);
 
         final T vx = uy;
         final T vy = (hy2.subtract(hx2).add(1)).multiply(factH);
         final T vz =  hx.multiply(factH).multiply(2);
 
         // eccentricity-related intermediate parameters
         final T ex2  = ex.square();
         final T exey = ex.multiply(ey);
         final T ey2  = ey.square();
         final T e2   = ex2.add(ey2);
         final T eta  = getOne().subtract(e2).sqrt().add(1);
         final T beta = getOne().divide(eta);
 
         // eccentric longitude argument
         final FieldSinCos<T> scLe = FastMath.sinCos(lE);
         final T cLe    = scLe.cos();
         final T sLe    = scLe.sin();
 
         // coordinates of position and velocity in the orbital plane
         final T x      = a.multiply(getOne().subtract(beta.multiply(ey2)).multiply(cLe).add(beta.multiply(exey).multiply(sLe)).subtract(ex));
         final T y      = a.multiply(getOne().subtract(beta.multiply(ex2)).multiply(sLe).add(beta .multiply(exey).multiply(cLe)).subtract(ey));
 
         return new FieldVector3D<>(x.multiply(ux).add(y.multiply(vx)),
                                    x.multiply(uy).add(y.multiply(vy)),
                                    x.multiply(uz).add(y.multiply(vz)));
 
     }
 
     /** {@inheritDoc} */
     @Override
     protected TimeStampedFieldPVCoordinates<T> initPVCoordinates() {
 
         // position and velocity
         computePVWithoutA();
 
         // acceleration
         final T r2 = partialPV.getPosition().getNormSq();
         final FieldVector3D<T> keplerianAcceleration = new FieldVector3D<>(r2.multiply(r2.sqrt()).reciprocal().multiply(getMu().negate()),
                                                                            partialPV.getPosition());
         final FieldVector3D<T> acceleration = hasNonKeplerianRates() ?
                                               keplerianAcceleration.add(nonKeplerianAcceleration()) :
                                               keplerianAcceleration;
 
         return new TimeStampedFieldPVCoordinates<>(getDate(), partialPV.getPosition(), partialPV.getVelocity(), acceleration);
 
     }
 
     /** {@inheritDoc} */
     @Override
     public FieldEquinoctialOrbit<T> withFrame(final Frame inertialFrame) {
         if (hasNonKeplerianAcceleration()) {
             return new FieldEquinoctialOrbit<>(getPVCoordinates(inertialFrame), inertialFrame, getMu());
         } else {
             final FieldKinematicTransform<T> transform = getFrame().getKinematicTransformTo(inertialFrame, getDate());
             return new FieldEquinoctialOrbit<>(transform.transformOnlyPV(getPVCoordinates()), inertialFrame, getDate(), getMu());
         }
     }
 
     /** {@inheritDoc} */
     @Override
     public FieldEquinoctialOrbit<T> shiftedBy(final double dt) {
         return shiftedBy(getZero().newInstance(dt));
     }
 
     /** {@inheritDoc} */
     @Override
     public FieldEquinoctialOrbit<T> shiftedBy(final T dt) {
 
         // use Keplerian-only motion
         final FieldEquinoctialOrbit<T> keplerianShifted = new FieldEquinoctialOrbit<>(a, ex, ey, hx, hy,
                                                                                       getLM().add(getKeplerianMeanMotion().multiply(dt)),
                                                                                       PositionAngleType.MEAN, cachedPositionAngleType, getFrame(),
                                                                                       getDate().shiftedBy(dt), getMu());
 
         if (hasNonKeplerianRates()) {
 
             // extract non-Keplerian acceleration from first time derivatives
             final FieldVector3D<T> nonKeplerianAcceleration = nonKeplerianAcceleration();
 
             // add quadratic effect of non-Keplerian acceleration to Keplerian-only shift
             keplerianShifted.computePVWithoutA();
             final FieldVector3D<T> fixedP   = new FieldVector3D<>(getOne(), keplerianShifted.partialPV.getPosition(),
                                                                   dt.square().multiply(0.5), nonKeplerianAcceleration);
             final T   fixedR2 = fixedP.getNormSq();
             final T   fixedR  = fixedR2.sqrt();
             final FieldVector3D<T> fixedV  = new FieldVector3D<>(getOne(), keplerianShifted.partialPV.getVelocity(),
                                                                  dt, nonKeplerianAcceleration);
             final FieldVector3D<T> fixedA  = new FieldVector3D<>(fixedR2.multiply(fixedR).reciprocal().multiply(getMu().negate()),
                                                                  keplerianShifted.partialPV.getPosition(),
                                                                  getOne(), nonKeplerianAcceleration);
 
             // build a new orbit, taking non-Keplerian acceleration into account
             return new FieldEquinoctialOrbit<>(new TimeStampedFieldPVCoordinates<>(keplerianShifted.getDate(),
                                                                                    fixedP, fixedV, fixedA),
                                                keplerianShifted.getFrame(), keplerianShifted.getMu());
 
         } else {
             // Keplerian-only motion is all we can do
             return keplerianShifted;
         }
 
     }
 
     /** {@inheritDoc} */
     @Override
     protected T[][] computeJacobianMeanWrtCartesian() {
 
         final T[][] jacobian = MathArrays.buildArray(getField(), 6, 6);
 
         // compute various intermediate parameters
         computePVWithoutA();
         final FieldVector3D<T> position = partialPV.getPosition();
         final FieldVector3D<T> velocity = partialPV.getVelocity();
         final T r2         = position.getNormSq();
         final T r          = r2.sqrt();
         final T r3         = r.multiply(r2);
 
         final T mu         = getMu();
         final T sqrtMuA    = a.multiply(mu).sqrt();
         final T a2         = a.square();
 
         final T e2         = ex.square().add(ey.square());
         final T oMe2       = getOne().subtract(e2);
         final T epsilon    = oMe2.sqrt();
         final T beta       = getOne().divide(epsilon.add(1));
         final T ratio      = epsilon.multiply(beta);
 
         final T hx2        = hx.square();
         final T hy2        = hy.square();
         final T hxhy       = hx.multiply(hy);
 
         // precomputing equinoctial frame unit vectors (f, g, w)
         final FieldVector3D<T> f  = new FieldVector3D<>(hx2.subtract(hy2).add(1), hxhy.multiply(2), hy.multiply(-2)).normalize();
         final FieldVector3D<T> g  = new FieldVector3D<>(hxhy.multiply(2), hy2.add(1).subtract(hx2), hx.multiply(2)).normalize();
         final FieldVector3D<T> w  = FieldVector3D.crossProduct(position, velocity).normalize();
 
         // coordinates of the spacecraft in the equinoctial frame
         final T x    = FieldVector3D.dotProduct(position, f);
         final T y    = FieldVector3D.dotProduct(position, g);
         final T xDot = FieldVector3D.dotProduct(velocity, f);
         final T yDot = FieldVector3D.dotProduct(velocity, g);
 
         // drDot / dEx = dXDot / dEx * f + dYDot / dEx * g
         final T c1  = a.divide(sqrtMuA.multiply(epsilon));
         final T c1N = c1.negate();
         final T c2  = a.multiply(sqrtMuA).multiply(beta).divide(r3);
         final T c3  = sqrtMuA.divide(r3.multiply(epsilon));
         final FieldVector3D<T> drDotSdEx = new FieldVector3D<>(c1.multiply(xDot).multiply(yDot).subtract(c2.multiply(ey).multiply(x)).subtract(c3.multiply(x).multiply(y)), f,
                                                                c1N.multiply(xDot).multiply(xDot).subtract(c2.multiply(ey).multiply(y)).add(c3.multiply(x).multiply(x)), g);
 
         // drDot / dEy = dXDot / dEy * f + dYDot / dEy * g
         final FieldVector3D<T> drDotSdEy = new FieldVector3D<>(c1.multiply(yDot).multiply(yDot).add(c2.multiply(ex).multiply(x)).subtract(c3.multiply(y).multiply(y)), f,
                                                                c1N.multiply(xDot).multiply(yDot).add(c2.multiply(ex).multiply(y)).add(c3.multiply(x).multiply(y)), g);
 
         // da
         final FieldVector3D<T> vectorAR = new FieldVector3D<>(a2.multiply(2).divide(r3), position);
         final FieldVector3D<T> vectorARDot = new FieldVector3D<>(a2.multiply(2).divide(mu), velocity);
         fillHalfRow(getOne(), vectorAR,    jacobian[0], 0);
         fillHalfRow(getOne(), vectorARDot, jacobian[0], 3);
 
         // dEx
         final T d1 = a.negate().multiply(ratio).divide(r3);
         final T d2 = (hy.multiply(xDot).subtract(hx.multiply(yDot))).divide(sqrtMuA.multiply(epsilon));
         final T d3 = hx.multiply(y).subtract(hy.multiply(x)).divide(sqrtMuA);
         final FieldVector3D<T> vectorExRDot =
             new FieldVector3D<>(x.multiply(2).multiply(yDot).subtract(xDot.multiply(y)).divide(mu), g, y.negate().multiply(yDot).divide(mu), f, ey.negate().multiply(d3).divide(epsilon), w);
         fillHalfRow(ex.multiply(d1), position, ey.negate().multiply(d2), w, epsilon.divide(sqrtMuA), drDotSdEy, jacobian[1], 0);
         fillHalfRow(getOne(), vectorExRDot, jacobian[1], 3);
 
         // dEy
         final FieldVector3D<T> vectorEyRDot =
             new FieldVector3D<>(xDot.multiply(2).multiply(y).subtract(x.multiply(yDot)).divide(mu), f, x.negate().multiply(xDot).divide(mu), g, ex.multiply(d3).divide(epsilon), w);
         fillHalfRow(ey.multiply(d1), position, ex.multiply(d2), w, epsilon.negate().divide(sqrtMuA), drDotSdEx, jacobian[2], 0);
         fillHalfRow(getOne(), vectorEyRDot, jacobian[2], 3);
 
         // dHx
         final T h = (hx2.add(1).add(hy2)).divide(sqrtMuA.multiply(2).multiply(epsilon));
         fillHalfRow( h.negate().multiply(xDot), w, jacobian[3], 0);
         fillHalfRow( h.multiply(x),    w, jacobian[3], 3);
 
        // dHy
         fillHalfRow( h.negate().multiply(yDot), w, jacobian[4], 0);
         fillHalfRow( h.multiply(y),    w, jacobian[4], 3);
 
         // dLambdaM
         final T l = ratio.negate().divide(sqrtMuA);
         fillHalfRow(getOne().negate().divide(sqrtMuA), velocity, d2, w, l.multiply(ex), drDotSdEx, l.multiply(ey), drDotSdEy, jacobian[5], 0);
         fillHalfRow(getZero().newInstance(-2).divide(sqrtMuA), position, ex.multiply(beta), vectorEyRDot, ey.negate().multiply(beta), vectorExRDot, d3, w, jacobian[5], 3);
 
         return jacobian;
 
     }
 
     /** {@inheritDoc} */
     @Override
     protected T[][] computeJacobianEccentricWrtCartesian() {
 
         // start by computing the Jacobian with mean angle
         final T[][] jacobian = computeJacobianMeanWrtCartesian();
 
         // Differentiating the Kepler equation lM = lE - ex sin lE + ey cos lE leads to:
         // dlM = (1 - ex cos lE - ey sin lE) dE - sin lE dex + cos lE dey
         // which is inverted and rewritten as:
         // dlE = a/r dlM + sin lE a/r dex - cos lE a/r dey
         final FieldSinCos<T> scLe = FastMath.sinCos(getLE());
         final T cosLe = scLe.cos();
         final T sinLe = scLe.sin();
         final T aOr   = getOne().divide(getOne().subtract(ex.multiply(cosLe)).subtract(ey.multiply(sinLe)));
 
         // update longitude row
         final T[] rowEx = jacobian[1];
         final T[] rowEy = jacobian[2];
         final T[] rowL  = jacobian[5];
 
         for (int j = 0; j < 6; ++j) {
             rowL[j] = aOr.multiply(rowL[j].add(sinLe.multiply(rowEx[j])).subtract(cosLe.multiply(rowEy[j])));
         }
         return jacobian;
 
     }
 
     /** {@inheritDoc} */
     @Override
     protected T[][] computeJacobianTrueWrtCartesian() {
 
         // start by computing the Jacobian with eccentric angle
         final T[][] jacobian = computeJacobianEccentricWrtCartesian();
 
         // Differentiating the eccentric longitude equation
         // tan((lV - lE)/2) = [ex sin lE - ey cos lE] / [sqrt(1-ex^2-ey^2) + 1 - ex cos lE - ey sin lE]
         // leads to
         // cT (dlV - dlE) = cE dlE + cX dex + cY dey
         // with
         // cT = [d^2 + (ex sin lE - ey cos lE)^2] / 2
         // d  = 1 + sqrt(1-ex^2-ey^2) - ex cos lE - ey sin lE
         // cE = (ex cos lE + ey sin lE) (sqrt(1-ex^2-ey^2) + 1) - ex^2 - ey^2
         // cX =  sin lE (sqrt(1-ex^2-ey^2) + 1) - ey + ex (ex sin lE - ey cos lE) / sqrt(1-ex^2-ey^2)
         // cY = -cos lE (sqrt(1-ex^2-ey^2) + 1) + ex + ey (ex sin lE - ey cos lE) / sqrt(1-ex^2-ey^2)
         // which can be solved to find the differential of the true longitude
         // dlV = (cT + cE) / cT dlE + cX / cT deX + cY / cT deX
         final FieldSinCos<T> scLe = FastMath.sinCos(getLE());
         final T cosLe     = scLe.cos();
         final T sinLe     = scLe.sin();
         final T eSinE     = ex.multiply(sinLe).subtract(ey.multiply(cosLe));
         final T ecosE     = ex.multiply(cosLe).add(ey.multiply(sinLe));
         final T e2        = ex.square().add(ey.square());
         final T epsilon   = getOne().subtract(e2).sqrt();
         final T onePeps   = epsilon.add(1);
         final T d         = onePeps.subtract(ecosE);
         final T cT        = d.multiply(d).add(eSinE.multiply(eSinE)).divide(2);
         final T cE        = ecosE.multiply(onePeps).subtract(e2);
         final T cX        = ex.multiply(eSinE).divide(epsilon).subtract(ey).add(sinLe.multiply(onePeps));
         final T cY        = ey.multiply(eSinE).divide(epsilon).add( ex).subtract(cosLe.multiply(onePeps));
         final T factorLe  = cT.add(cE).divide(cT);
         final T factorEx  = cX.divide(cT);
         final T factorEy  = cY.divide(cT);
 
         // update longitude row
         final T[] rowEx = jacobian[1];
         final T[] rowEy = jacobian[2];
         final T[] rowL = jacobian[5];
         for (int j = 0; j < 6; ++j) {
             rowL[j] = factorLe.multiply(rowL[j]).add(factorEx.multiply(rowEx[j])).add(factorEy.multiply(rowEy[j]));
         }
 
         return jacobian;
 
     }
 
     /** {@inheritDoc} */
     @Override
     public void addKeplerContribution(final PositionAngleType type, final T gm,
                                       final T[] pDot) {
         pDot[5] = pDot[5].add(computeKeplerianLDot(type, a, ex, ey, gm, cachedL, cachedPositionAngleType));
     }
 
     /**
      * Compute rate of argument of longitude.
      * @param type position angle type of rate
      * @param a semi major axis
      * @param ex ex
      * @param ey ey
      * @param mu mu
      * @param l argument of longitude
      * @param cachedType position angle type of passed l
      * @param <T> field type
      * @return first-order time derivative for l
      * @since 12.2
      */
     private static <T extends CalculusFieldElement<T>> T computeKeplerianLDot(final PositionAngleType type, final T a, final T ex,
                                                                               final T ey, final T mu, final T l, final PositionAngleType cachedType) {
         final T n               = mu.divide(a).sqrt().divide(a);
         if (type == PositionAngleType.MEAN) {
             return n;
         }
         final FieldSinCos<T> sc;
         final T ksi;
         if (type == PositionAngleType.ECCENTRIC) {
             sc = FastMath.sinCos(FieldEquinoctialLongitudeArgumentUtility.convertL(cachedType, l, ex, ey, type));
             ksi = ((ex.multiply(sc.cos())).add(ey.multiply(sc.sin()))).negate().add(1).reciprocal();
             return n.multiply(ksi);
         } else {
             sc = FastMath.sinCos(FieldEquinoctialLongitudeArgumentUtility.convertL(cachedType, l, ex, ey, type));
             final T oMe2 = a.getField().getOne().subtract(ex.square()).subtract(ey.square());
             ksi  =  ex.multiply(sc.cos()).add(1).add(ey.multiply(sc.sin()));
             return n.multiply(ksi).multiply(ksi).divide(oMe2.multiply(oMe2.sqrt()));
         }
     }
 
     /**  Returns a string representation of this equinoctial parameters object.
      * @return a string representation of this object
      */
     public String toString() {
         return new StringBuilder().append("equinoctial parameters: ").append('{').
                                   append("a: ").append(a.getReal()).
                                   append("; ex: ").append(ex.getReal()).append("; ey: ").append(ey.getReal()).
                                   append("; hx: ").append(hx.getReal()).append("; hy: ").append(hy.getReal()).
                                   append("; lv: ").append(FastMath.toDegrees(getLv().getReal())).
                                   append(";}").toString();
     }
 
     /** {@inheritDoc} */
     @Override
     public PositionAngleType getCachedPositionAngleType() {
         return cachedPositionAngleType;
     }
 
     /** {@inheritDoc} */
     @Override
     public boolean hasNonKeplerianRates() {
         return hasNonKeplerianAcceleration();
     }
 
     /** {@inheritDoc} */
     @Override
     public FieldEquinoctialOrbit<T> withKeplerianRates() {
         return new FieldEquinoctialOrbit<>(getA(), getEquinoctialEx(), getEquinoctialEy(), getHx(), getHy(),
                 cachedL, cachedPositionAngleType, getFrame(), getDate(), getMu());
     }
 
     /** {@inheritDoc} */
     @Override
     public EquinoctialOrbit toOrbit() {
         final double cachedPositionAngle = cachedL.getReal();
         if (hasNonKeplerianRates()) {
             return new EquinoctialOrbit(a.getReal(), ex.getReal(), ey.getReal(),
                                         hx.getReal(), hy.getReal(), cachedPositionAngle,
                                         aDot.getReal(), exDot.getReal(), eyDot.getReal(),
                                         hxDot.getReal(), hyDot.getReal(), cachedLDot.getReal(),
                                         cachedPositionAngleType, getFrame(),
                                         getDate().toAbsoluteDate(), getMu().getReal());
         } else {
             return new EquinoctialOrbit(a.getReal(), ex.getReal(), ey.getReal(),
                                         hx.getReal(), hy.getReal(), cachedPositionAngle,
                                         cachedPositionAngleType, getFrame(),
                                         getDate().toAbsoluteDate(), getMu().getReal());
         }
     }
 
 }