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, which states that those trajectories that make the action stationary, i.e. , are those that yield the true motion. We see from the above that functions that satisfy the boundary conditions
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 statesno dissipative forces acting, the action principle gives us the appropriate equation of motion. When there are dissipative forces, the action defines for us the kinetic and conservative force terms that are to be balanced by the dissipative terms. It also defines for us the momentum.
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 isa priori definition of the system.
To summarize: The variational argument leads to equations of motion of the form
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