# 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:

${\ displaystyle y '= f (y), \ quad y (0) = y_ {0}}$ The solution of this initial value problem is assumed to be on a (maximum) existence interval . ${\ displaystyle y (t, y_ {0})}$ ${\ displaystyle I _ {\ mathrm {max}} (t)}$ The image is then used as a trajectory or phase curve of the system of equations${\ displaystyle y_ {0}}$ ${\ displaystyle T (y_ {0}): = \ lbrace y (t, y_ {0}) | t \ in I _ {\ mathrm {max}} (y_ {0}) \ rbrace}$ 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. ${\ displaystyle y (t, y_ {0})}$ ${\ displaystyle y_ {0}}$ ## example

The following system of linear differential equations is given:

${\ displaystyle y '(t) = {\ begin {pmatrix} u' (t) \\ v '(t) \ end {pmatrix}} = {\ begin {pmatrix} a & 1-a \\ a-1 & a \ end {pmatrix}} y (t)}$ A general solution of the system is the following linear combination:

${\ displaystyle y (t) = {\ begin {pmatrix} u (t) \\ v (t) \ end {pmatrix}} = C_ {1} \ mathrm {e} ^ {at} {\ begin {pmatrix} \ cos ((a-1) t) \\\ sin ((a-1) t) \ end {pmatrix}} + C_ {2} \ mathrm {e} ^ {at} {\ begin {pmatrix} - \ sin ((a-1) t) \\\ cos ((a-1) t) \ end {pmatrix}}}$ 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): ${\ displaystyle a = 1}$ ${\ displaystyle v (u)}$ ${\ displaystyle u (t)}$ ${\ displaystyle v (t)}$ ${\ displaystyle u '(t) = you}$ ${\ displaystyle v '(t) = dv}$ ${\ displaystyle {\ frac {dv} {du}} = {\ frac {(a-1) u + av} {au + (1-a) v}} = {\ frac {(1-1) u + 1v } {1u + (1-1) v}}}$ The solution is:

${\ displaystyle v (u) = C_ {3} u}$ 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. ${\ displaystyle C_ {3}}$ ## 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 .