We have found that when the forces can be described by a potential energy function it is straightforward to use the Lagrangian techniques to find the equations of motion. Given a set of force, how can we determine the potential function? How can we determine whether there is a potential function?
In conservative forces always mechanical energy is conserved ie, Kinetic energy + Potential energy remains constant at every point.For example, spring force gravitational force and electro static force.
Let's do an example,
How much work does the force do moving from to ?
If we look at the two terms, let's try to integrate each one with respect to and .
They are both derivatives of the same function
Let's calculate the total derivative of the function
and substitute it back into the integral
The integral does not depend on path. We could write
- If the forces are such that the work done when the system is moved from one configuration to another only depends on the initial and final configurations, then the force is conservative.
The force of friction between two dry surfaces is proportional to the normal force between the surfaces in a direction that opposes the relative motion between the surfaces. For an object resting on a surface we have that we can resolve into components as
Let's write out the work done by the force,
where is the length of the path. In particular, the work done by the frictional path does not necessarily vanish over a closed path.
The Potential Energy
where counts over the various particles. Specifically, is the force on particle in the direction.
For this to be true,
So let's calculate for example
If the forces are conservative, then the partial derivatives of the forces are equal to each other. This is a quick way to check whether the forces are indeed conservative!
We can make some observativions from this result. First, if the force varies only along the direction that it points, then the force is conservative. An important example of this is the central force. Second, if and only if the curl of the force is zero, the force is conservative.
Why calculate ?
It takes quite a bit of effort to calculate the potential energy. Why don't we just use the forces directly?
- You can use the transformation rules to find the generalized forces
but this is really ugly and could be even more effort than finding the potential energy.
- Calculate in whatever coordinates are convenient and then transform the coordinates so
- Sometimes you can get by inspection.
- The generalized forces for and are the same so you can drop any constant term. For example, we can write
for the potential energy of a pendulum.
Do conservative systems conserve energy?
Conservative systems are those which you can write the force as the gradient for a potential function. We have seen forces where this is not the case: dry friction and a particle in magnetic field. In the first case we have a non-conservative force that does not conserve energy. In the second case, we have a non-conservative force that does conserve energy.
It is straightforward to come up with an example for a conservative force that does not conserve energy. Specifically we could have
Because the Lagrangian now depends on time explicitly, the Hamiltonian is no longer conserved.
Coordinates for the jiggly pendulum
Let's write down the Cartesian coordinates of the bob of mass ,
Taking the time derivative of the position gives the velocity
Using these results we can calculate the kinetic energy of the bob
and the potential energy
For this system we have so ; however, we can define a potential energy so the system is conservative.