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

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Equation (2) allows you to derive the force vector by differentiating the potential energy with respect to position. To determine the equations of motion from potential energy, one must recognize that finding position as a function of time may not yield an analytical solution in all cases. For instance, even simple scenarios like the inverse square law can present complexities. The discussion emphasizes the importance of understanding the relationship between potential energy and force in motion analysis. Overall, solving these equations often requires advanced techniques beyond basic differentiation.
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.
 
Thread 'Correct statement about size of wire to produce larger extension'

Thread 'Correct statement about size of wire to produce larger extension'

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