The simplest physics problem, i.e. the point particle, has always served as a guide to deep principles that are used in much harder problems. We have used it already to motivate parallel transport as the foundation for the covariant derivative. Let us call upon the point particle again to set the context for the action-based derivation of the fluid field equations. We will simplify the discussion by considering only motion in one dimension. We assure the reader that we have good reasons, and ask for patience while we remind him/her of what may be very basic facts.
Early on we learn that an action appropriate for the point particle is
where But, of course, forces need to be included. First on the list are the so-called conservative forces,
describable by a potential , which are placed into the action according to
In the most honest applications, one has the obligation to incorporate dissipative, i.e. non-conservative,
forces. Unfortunately, dissipative forces cannot be put into action principles. Fortunately, Newton’s
second law is of great guidance, since it states
We should emphasize that this way of using the action to define the kinetic and conservative
pieces of the equation of motion, as well as the momentum, can also be used in a context where
the system experiences an externally applied force . The force can be conservative or
dissipative, and will enter the equation of motion in the same way as
did above, that is
To summarize: The variational argument leads to equations of motion of the form
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