Known problem appears difficult

[neelakash]

Homework Statement

A question asks to show for conservative forces,KE=f(v^2) and PE is independent of time.

Homework Equations

The Attempt at a Solution

My attempts:
from work energy theorem,increase in KE=work done
=int F.dl=int m(dv/dt).(vdt)
=(m/2) int d(v^2)

Does this show kinetic energy is a quadratic function of v?

Potential energy is a point function,i.e. it depends only on points.Once you get the value it will remain the same for all future instants...therefore potential energy does not depend on time.

Please say if I am correct.

 

[Reshma]

Your second question can be answered easily if you are familiar with the Lagrangian and Hamilton's equations.

The independence of the Lagrangian/Hamiltonian with respect to the time coordinate corresponds to a process in which energy is conserved. If the total energy is conserved, then the work done on the particle must be converted to potential energy, conventionally denoted by V, which must be purely a function of the spatial coordinates x,y,z, but independent of the time t.

If the potential depends on the derivatives of the position coordinates it is said to be a velocity-dependent potential.

 

[neelakash]

That is a problem...in my undergraduate course I do not have the exposure to Lagrange or Hamilton's formulation of classical mechanics.

 

[neelakash]

I thank you very much for your help.
What else could be done?

 

[siddharth]

My attempts:
from work energy theorem,increase in KE=work done
=int F.dl=int m(dv/dt).(vdt)
=(m/2) int d(v^2)

Yes, and integrating this, you get mv^2/2.

PE is independent of time.

How is that true? Take a spring system as an example. The force that a spring exerts is a conservative force. If you stretch the spring horizontally and let it oscillate, the PE of the spring will change as a function of time.

 

[neelakash]

OK,I might have been wrong to say that PE is independent of time.The question asked "no explicit time dependence" which I could not understand that time.Now I understand.

Can you suggest a way to show this?Was I correct?

 

[Reshma]

How is that true? Take a spring system as an example. The force that a spring exerts is a conservative force. If you stretch the spring horizontally and let it oscillate, the PE of the spring will change as a function of time.

Well, the PE will depend on the position of the spring after a time interval not time itself. For conservative force the line integral:
$\oint \vec F \cdot d\vec r = 0$ & $\vec F = -\vec \nabla V$ and V is function of position only with no explicit time dependence.

 

[neelakash]

I again stress on the word "explicit dependence".In case of harmonic oscillator,the position is a function of time and that way PE is time dependent.It is an implicit dependence for x=x(t)