# Trajectory (mathematics)

In mathematics, the **trajectory** (also known as a trajectory) is usually the solution curve of a differential equation with given initial conditions. The differential equation describes the coordinates of a system (in phase space or spatial space) depending on a parameter, which in mechanical applications is usually time. Then the trajectory describes the coordinates of the system as a function of "time".

## definition

We consider the solution of an initial value problem of the following form:

The solution of this initial value problem is assumed to be on a (maximum) existence interval .

The image is then used as a **trajectory** or phase curve of the system of equations

which is defined by this solution.

## Phase space

The common representation of all trajectories of a system is called a phase portrait or phase space . The phase portrait thus contains all trajectories that provide the solutions to the initial value problems when the initial value passes through all values of the definition range.

## example

The following system of linear differential equations is given:

A general solution of the system is the following linear combination:

We want to draw trajectories for . The function to be represented can be found either by transforming the solutions for and or by solving the following differential equation ( and from the given differential equation):

The solution is:

The graph of this function is the trajectory sought, the constant is determined by the initial condition of the DGL system. A phase space consisting of twelve trajectories with different initial conditions is shown here.

## Trajectories in Geometry

In geometry, the term **trajectory is** also used to designate a function graph that intersects a given set of curves isogonally , i.e. always at the same angle . If this angle is 90 °, one speaks of an orthogonal trajectory .

## See also

## Individual evidence

- ↑ See Brockhaus 1996. Vol. 22. p. 304