Math 2.0

Mechanics of particles and systems Orbital mechanics Problem

Lambert's problem

Builds on Kepler orbitKepler problem

Problem

Let μ>0\mu > 0, let r1,r2∈ℝ3∖{0}r_1, r_2 \in \R^3 \setminus \set{0} be two positions that are not on a common line through 00, and let Δt>0\Delta t > 0. Find the Kepler orbits with gravitational parameter μ\mu that go from r1r_1 to r2r_2 in time Δt\Delta t. That is, find the velocities v1v_1 such that the solution of the Kepler problem r′′=−μ r/∥r∥3r'' = -\mu\, r / \norm{r}^3 with r(0)=r1r(0) = r_1 and r′(0)=v1r'(0) = v_1 satisfies r(Δt)=r2r(\Delta t) = r_2.

Remarks

Lambert's theorem states that the time of flight depends only on the semi-major axis aa of the orbit, on the sum ∥r1∥+∥r2∥\norm{r_1} + \norm{r_2} and on the chord ∥r2−r1∥\norm{r_2 - r_1}. This reduces the problem to one scalar equation for aa, or for a related variable. For a fixed direction of motion and a transfer of less than one revolution, the problem has exactly one solution. For longer times of flight there are also solutions that make one or more full revolutions first. The problem is the core of the planning of interplanetary trajectories and of rendezvous, and of the determination of an orbit from two positions. Practical solvers use universal variables and Newton-type iterations.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Conic section, concept, no evidence levelConic sectionconceptEllipse, concept, no evidence levelEllipseconceptFirst integral, concept, no evidence levelFirst integralconceptFocus of a conic, concept, no evidence levelFocus of a conicconceptKepler orbit, concept, no evidence levelKepler orbitconceptKepler problem, example, no evidence levelKepler problemexampleLambert's problem, problem, no evidence levelLambert's problemproblem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Conic section concept
  2. Ellipse concept
  3. First integral concept
  4. Focus of a conic concept
  5. Kepler orbit concept
  6. Kepler problem example
  7. Lambert's problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Lambert%27s_problem
About this page
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Kindproblem
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MSC70F05, 70M20
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AboutKepler orbit
UsesKepler orbit, Kepler problem
Same as
Changed3 November 2026

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