What Exactly Does Equation (2) Mean? (Equations of Motion from PE function)

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SUMMARY

Equation (2) from the discussion provides a method to derive the force vector by differentiating the potential energy function with respect to position. This approach is essential for formulating the Equations of Motion. However, obtaining position as a function of time may not always yield an analytical solution, particularly in complex scenarios such as those involving the inverse square law. Understanding these concepts is crucial for applying potential energy in physics.

PREREQUISITES
  • Understanding of potential energy functions
  • Knowledge of differentiation in calculus
  • Familiarity with vector calculus
  • Basic principles of classical mechanics
NEXT STEPS
  • Study the derivation of force from potential energy in classical mechanics
  • Learn about solving partial differential equations (PDEs) in physics
  • Explore the implications of the inverse square law in motion equations
  • Investigate numerical methods for solving motion equations when analytical solutions are not feasible
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Students of physics, educators teaching mechanics, and researchers interested in the applications of potential energy in motion analysis will benefit from this discussion.

humancentered666
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Homework Statement
(There isn't really one, I'm just confused about the interpretation of the equation.)
Relevant Equations
F=ma (1)
Fᵢ({x})=-∂V({x})/∂xᵢ (2)
V({x}) is a potental in a system.
What exactly is this equation telling me? How can I use it to work out the Equations of Motion given an equation of potential energy? Won't I have to solve a PDE? I'm extremely sorry if this question comes off ignorant.
 
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humancentered666 said:
How can I use it to work out the Equations of Motion given an equation of potential energy? Won't I have to solve a PDE?
If you are given the potential then, as (2) shows, you can differentiate it with respect to position to find the force vector.
If your aim is to find position as a function of time, that is not always solvable analytically. Even the simple case of an inverse square law is nontrivial.
 

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