Suppose we have a mechanical system with 1 degree of freedom, i.e. an ODE (1) \ddot{q}+V'(q)=

Helen Lewis

Helen Lewis

Answered question

2022-01-16

Suppose we have a mechanical system with 1 degree of freedom, i.e. an ODE
(1)q¨+V(q)=0,
where V:RR is some smooth function (potential energy). We then easily see that any solution of this equation must satisfy
q˙22+V(q)=constant
In other words, if we put
E(q,q˙)=q˙22+V(q)
(energy), then the image of every solution of (1) must lie in a level set of E.

Answer & Explanation

Juan Spiller

Juan Spiller

Beginner2022-01-17Added 38 answers

You are essentially asking whether the particle can stop at a point inside the level set. To stop, it needs to have both zero velocity and zero acceleration (these conditions are sufficient because the dynamics is of second order). First condition requires q˙=0 implying also V(q)=E, while the second holds when V(q)=0. So just choose any potential with a local maximum Vmax and choose initial conditions such that E=Vmax.
Physically these correspond to a ball rolling down a hill and then again up a hill. As the ball approaches the top, its kinetic energy decreases as EV and asymptotically it stops. This interval is proper in the level set because you can get another solution approaching the hill from the other side of the hill.
Perhaps the answer is even more obvious in more dimensions (or more degrees of freedom). There the level sets are of dimension 2N1 and the failure of inclusion is more apparent. Just consider a potential with a local maximum (a boulge at the origin) which you can approach from any direction and asymptotically stop there.
intacte87

intacte87

Beginner2022-01-18Added 42 answers

Consider a free particle with speed v0. We have Tu=R×{v0} and Eu=R×{v0,v0}.
alenahelenash

alenahelenash

Expert2022-01-24Added 556 answers

Another positive answer to your question is obtained when the potential energy is V(x)={0,for x0;exp(1/x2),for x>0} In such a case the zero level set of the total energy is [,0], but the trajectories of motion with zero energy are each one of its sigleton.

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