Go to previous page Go up Go to next page

7 Setting the Context: The Point Particle

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

∫ t ∫ t ( ) I = fdt T = fdt 1m x˙2 , (108 ) ti ti 2
where m is the mass and T the kinetic energy. A first-order variation of the action with respect to x (t) yields
∫ tf δI = − dt(m ¨x) δx + (m ˙xδx)|tf . (109 ) ti ti
If this is all the physics to be incorporated, i.e. if there are no forces acting on the particle, then we impose d’Alembert’s principle of least action [65], which states that those trajectories x (t) that make the action stationary, i.e. δI = 0, are those that yield the true motion. We see from the above that functions x(t) that satisfy the boundary conditions
δx(ti) = 0 = δx(tf) (110 )
and the equation of motion
m ¨x = 0 (111 )
will indeed make δI = 0. This same reasoning applies in the substantially more difficult fluid actions that will be considered later.

But, of course, forces need to be included. First on the list are the so-called conservative forces, describable by a potential V (x), which are placed into the action according to

∫ ∫ ( ) tf tf 1- 2 I = ti dtL (x, ˙x) = ti dt 2m ˙x − V (x) , (112 )
where L = T − V is the so-called Lagrangian. The variation now leads to
( ) ∫ tf ∂V t δI = − dt m ¨x + ---- δx + (m ˙xδx)|fti . (113 ) ti ∂x
Assuming no externally applied forces, d’Alembert’s principle yields the equation of motion
∂V m ¨x + ----= 0. (114 ) ∂x
An alternative way to write this is to introduce the momentum p (not to be confused with the fluid pressure introduced earlier) defined as
∂L- p = ∂x˙ = m ˙x, (115 )
in which case
∂V ˙p + ----= 0. (116 ) ∂x

In the most honest applications, one has the obligation to incorporate dissipative, i.e. non-conservative, forces. Unfortunately, dissipative forces F d cannot be put into action principles. Fortunately, Newton’s second law is of great guidance, since it states

∂V m ¨x + ----= Fd, (117 ) ∂x
when both conservative and dissipative forces act. A crucial observation of Equation (117View Equation) is that the “kinetic” (m ¨x = ˙p) and conservative (∂V∕ ∂x) forces, which enter the left-hand side, do follow from the action, i.e. 
( ) -δI = − m ¨x + ∂V-- . (118 ) δx ∂x
When there are no 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 Fext. The force can be conservative or dissipative, and will enter the equation of motion in the same way as Fd did above, that is

δI − δx- = Fd + Fext. (119 )
Like a dissipative force, the main effect of the external force can be to siphon kinetic energy from the system. Of course, whether a force is considered to be external or not depends on the a priori definition of the system.

To summarize: The variational argument leads to equations of motion of the form

{ δI ∂V ∂V = 0 for conservative forces, − ---= m ¨x + ----= p˙+ ---- ⁄= 0 for dissipative and∕or external forces. (120 ) δx ∂x ∂x

  Go to previous page Go up Go to next page