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Snap, [6] or jounce, [2] is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time. [4] Equivalently, it is the second derivative of acceleration or the third derivative of velocity, and is defined by any of the following equivalent expressions: = ȷ = = =.
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
Hence, the position element is written as () and the velocity element as (), indicating that they vary with time. The technique to compute the effect of perturbations becomes one of finding expressions, either exact or approximate, for the functions x 0 ( t ) {\displaystyle x_{0}(t)} and v 0 ( t ) {\displaystyle v_{0}(t)} .
Mean motion is used as an approximation of the actual orbital speed in making an initial calculation of the body's position in its orbit, for instance, from a set of orbital elements. This mean position is refined by Kepler's equation to produce the true position.
A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete trajectory is defined by position and momentum, simultaneously. The mass might be a projectile or a ...
As for instance, if the body passes the periastron at coordinates = (), =, at time =, then to find out the position of the body at any time, you first calculate the mean anomaly from the time and the mean motion by the formula = (), then solve the Kepler equation above to get , then get the coordinates from:
which connects all the 3 spatial coordinates of that particle together, so they are not independent. The constraint may change with time, so time t will appear explicitly in the constraint equations. At any instant of time, any one coordinate will be determined from the other coordinates, e.g. if x k and z k are given, then so is y k.
where M 0 is the mean anomaly at the epoch t 0, which may or may not coincide with τ, the time of pericenter passage. The classical method of finding the position of an object in an elliptical orbit from a set of orbital elements is to calculate the mean anomaly by this equation, and then to solve Kepler's equation for the eccentric anomaly.