Trajectories, equation of motion and forces....

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SUMMARY

The discussion focuses on deriving the equation of the trajectory of an object under the influence of gravitational forces, specifically using the formula F = GMm/r², where G is the gravitational constant, M is the mass of the sun, and m is the mass of the object. The objective is to express the trajectory as an ellipse in a Cartesian coordinate system centered on the sun, represented by the equation x²/a² + y²/b² = 1. The classical central force problem is referenced as a foundational concept for understanding these dynamics.

PREREQUISITES
  • Understanding of classical mechanics and gravitational forces
  • Familiarity with Cartesian coordinate systems
  • Knowledge of conic sections, particularly ellipses
  • Basic proficiency in mathematical derivations and equations of motion
NEXT STEPS
  • Study the classical central force problem in detail
  • Learn about the derivation of elliptical orbits in gravitational fields
  • Explore the implications of Kepler's laws of planetary motion
  • Investigate numerical methods for simulating trajectories under varying forces
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Students and professionals in physics, astrophysics, and engineering, particularly those interested in orbital mechanics and gravitational dynamics.

physics user1
Setting a cartesian system how can i get the equation of the trajectory of an object knowing the forces acting on that object?

Example: If F= GMm/r^2 and let be the sun at the center (point (0,0,0)) of the cartesian system how do i get the equation of an ellipse in this system? ( x^2/a^2 + y^2/b^2 = 1)

Where x and y define the position of the planet in the chosen system centred in the sun
 
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