Integration help, Kepler's problem Lagrangian dynamics

In summary, the conversation discusses carrying out an integration using a substitution and the book's given answer for the integral. The problem involves energy, potential, and angular momentum, and the conversation suggests completing the square to find the solution.
  • #1
black_hole
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Homework Statement



Carry out the integration ψ = ∫[M(dr/r2)] / √(2m(E-U(r)) - (M2/r2))

E = energy, U = potential, M = angular momentum

using the substitution: u = 1/r for U = -α/r

Homework Equations





The Attempt at a Solution



This is as far as I've gotten: -∫ (Mdu) / √(2m(E + αu) - (M2u2))
I have no idea how to take this integral by hand which seems to be what the question is implying. Wolfram gives me something crazy looking.

My book gives the answer as ψ = arccos( (M/r - mα/M) / √(2mE + m2α2/M2) ) ?




 
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  • #2
Try "completing the square" of the expression inside the square root in the denominator. Factor out the coefficient of u2 from the square root beforehand.
 
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FAQ: Integration help, Kepler's problem Lagrangian dynamics

What is the Kepler's problem?

The Kepler's problem is a mathematical model that describes the motion of a point mass under the influence of a central force. It is named after the German astronomer Johannes Kepler who first described the laws of planetary motion in the 17th century.

What is integration help in relation to Kepler's problem?

Integration help refers to the mathematical techniques used to solve the equations of motion for the Kepler's problem. These techniques involve finding numerical or analytical solutions to the equations of motion, which can then be used to understand the behavior of the system.

What is Lagrangian dynamics and how is it related to Kepler's problem?

Lagrangian dynamics is a mathematical framework that describes the motion of a system using a function called the Lagrangian. In the case of Kepler's problem, the Lagrangian is defined in terms of the position and velocity of the point mass, and can be used to derive the equations of motion for the system.

How does Kepler's problem relate to celestial mechanics?

Kepler's problem is an important part of celestial mechanics, which is the branch of astrophysics that studies the motion of objects in space. By understanding the laws of planetary motion described by Kepler's problem, scientists can make predictions about the behavior of celestial bodies such as planets, comets, and asteroids.

What are some applications of Kepler's problem and Lagrangian dynamics?

Kepler's problem and Lagrangian dynamics have many practical applications in fields such as aerospace engineering, astrodynamics, and satellite navigation. They are used to calculate trajectories of spacecraft, predict the motion of celestial bodies, and design efficient orbital maneuvers for space missions.

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