This is a continuation to the previous post on Lagrange Formulation.
If the Lagrangian does not depend explicitly on time, then becuase the partial time derivative is zero, the change or total time derivative of the Lagrangian can be non-zero only because of motion (change in coordinate values; not to be confused with a coordinate transformation). We will use Dt to denote the total time derivative in order to distinquish it from the other partial derivatives.
Dt (L) = dL/dx (dx/dt) + dL/dv (dv/dt) = dL/dx (v) + dL/dv (dv/dt)
But, we are imposing and solving the equation of motion, dL/dx = d/dt (dL/dv)
Hence, DtL = d/dt ( v*dL/dv)
Rearranging, Dt (L – v*dL/dv) = 0.
This quantity in the parenthesis is called the Hamiltonian and we showed that its change is zero.In other words, it is conserved over time when the Langrangian has no explicit time dependence. Also, the Hamilton itself cannot have explicit time dependence since it has to always conserved, even when the system is not in motion. This is essentially a weak version of the famous Noether’s theorem that proves any invariance of the Langrangian w.r.t a dependent variable results in a conserved quantity. We will now provide a proof outline for Noether’s theorem for coordinate transformations.
Since dL/dx = d/dt (dL/dv), when any linear combination of the LHS is zero, it follows that the associated linear combination of the parenthetical quanitity on the RHS would have zero time derivative (by the given equation). Hence, that quanitity would be invariant and be conserved over time.
Now we will address the teleological aspects of the least action principle using the Hamiltonian. Let us define a quantity, W = S(q) + Et, where S is the action time-integral of the Lagrangian and E is the constant energy. Note that the conserved Hamiltonian when the Lagrangian does not have explicit time dependence is the total energy of the system and S depends only on the coordinate as the velocities are integrated over time and becuase L is not directly dependent on t.
By the stationary action principle, S needs to be minimized or made stationary. From the above definition, it is equivalent to demand energy conservation and minimize W. Note that energy conservation is not a given if one simply minimizes W as both S and W are defined for physically impossible trajectories as well, as noted in the previous post. However, if S itself is directly minimized then energy conservation takes care of itself, from the symmetry of the Lagrangian seen above.
W is a potential function that is only state dependent (on the coordinate) and not on the path taken to arrive at the current point or state.
Cars and Physics 