Is there always a potential for conservative motion?

This means that the force is a function of coordinates only, and not time. Thus, there must exist a potential field associated with the force. In summary, for a conservative motion, the force must be a function of coordinates only, which means there is a potential field. To prove this, one can show that the rotational forces equal 0 and do not depend on time. This implies the existence of a potential field.
  • #1
Nusc
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Homework Statement



It is well known that for a conservative motion there is a potential.
This potential is a function of coordinates only.

Prove this.

Homework Equations





The Attempt at a Solution



I think you have to take a definition of conservative motion then for example prove that for such a motion, the rotational forces equals to 0 and they don't depend on time, then it will mean that there is a potential field.


What is the equation for conservative motion?
 
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  • #2


I guess we suppose that the general form of the force field is F=F(q,p,t)
 
  • #3


In general, a force is conservative iff the path integral of F around some closed path is equal to 0. This is equivalent to stating that the curl of F = 0.
 

FAQ: Is there always a potential for conservative motion?

What is the Hamilton-Jacobi homework problem?

The Hamilton-Jacobi homework problem is a mathematical problem in the field of classical mechanics. It involves finding a solution to Hamilton's equations of motion by using a function known as the Hamilton-Jacobi function.

Why is the Hamilton-Jacobi homework problem important?

The Hamilton-Jacobi homework problem is important because it allows for the calculation of the motion of a system using only the initial and final states, without having to track the intermediate steps. This makes it a valuable tool in solving problems in classical mechanics.

What are some applications of the Hamilton-Jacobi homework problem?

The Hamilton-Jacobi homework problem has various applications in physics, including celestial mechanics, fluid dynamics, and quantum mechanics. It is also used in engineering fields such as control theory and robotics.

What is the difference between the Hamilton-Jacobi homework problem and the Hamiltonian mechanics?

The Hamilton-Jacobi homework problem is a specific problem within the field of Hamiltonian mechanics. While Hamiltonian mechanics deals with the motion of a system using Hamilton's equations, the Hamilton-Jacobi homework problem involves finding a solution to these equations using the Hamilton-Jacobi function.

What are some techniques for solving the Hamilton-Jacobi homework problem?

Some common techniques for solving the Hamilton-Jacobi homework problem include separation of variables, the method of characteristics, and the action-angle variables method. These methods can be applied to various types of systems and provide different levels of complexity and accuracy in the solutions.

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