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Suppose I throw a ball upwards. First it will rise under gravity and then fall under gravity.

During the rising part the kinetic energy gradually decreases and the potential energy increases until the whole of KE gets converted to PE. So during the rising part it seems as if nature is trying to get the action (i.e. KE-PE) as minimum as possible. So it appears as if the it is favouring the principle of least action. The other argument can be that action is decreasing due to principle of conservation of energy. So what is happening here? Action decreasing due to energy conservation or the reverse?

But during the falling part the reverse happens and the action part increases since the (KE-PE) increases.Here only conservation of energy is clearly visible.

So which principle is more fundamental? "Principle of conservation of energy" or "Principle of Least action"?

Qmechanic
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user103515
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1 Answers1

6

  1. What is fundamental and what is derived? are often aesthetic subjective preferences of mankind rather than true statements about Nature.

  2. Not all systems in Newtonian mechanics have a stationary action principle. See e.g. this Phys.SE post. Another reason for no Lagrangian formulation is a failure of d'Alembert's principle, cf. e.g this and this Phys.SE posts.$^1$

  3. If a stationary action principle exists for a certain physical system, it is typically promoted to a first principle of the theory. All dynamical questions can then be derived/answered from the stationary action principle. E.g. equations of motion, i.e. Newton's 2nd law. Or e.g. if the action has time-translation symmetry, one can combine the stationary action principle with Noether's theorem to derive energy conservation.

  4. Total energy conservation, if it holds, is only one equation. E.g. if there is more than one particle or if there is more than one dimension, it doesn't give a complete description of the system. Therefore energy conservation can typically not serve as a lone first principle.

  5. OP writes (v9):

    Suppose I throw a ball upwards. First it will rise under gravity and then fall under gravity.

    During the rising part the kinetic energy gradually decreases and the potential energy increases until the whole of KE gets converted to PE. So during the rising part it seems as if nature is trying to get the action (i.e. KE-PE) as minimum as possible. So it appears as if the it is favouring the principle of least action. [...]

    But during the falling part the reverse happens and the action part increases since the (KE-PE) increases. [...]

    It seems that OP's example is spurred by an incorrect use of the principle of stationary action. It is important to impose appropriate boundary conditions (BCs), e.g. Dirichlet BCs. Else a variational principle is ill-posed. Without BCs, the action is typically unbounded on the set of virtual paths.

    E.g. one should not compare a falling ball with one that's falling even deeper. One should instead compare virtual trajectories that starts in the same initial position at some fixed initial time, and that ends in the same final position at some fixed final time.

    Concerning initial value problems (IVP) vs. boundary value problems (BVP), see also this Phys.SE post.

References:

  1. L. Susskind & G. Hrabovsky, The Theoretical Minimum: What You Need to Know to Start Doing Physics, 2013.

$^1$ Ref. 1 writes on p.122

In a general coordinate system, the equations of motion may be complicated, but the action principle always applies. All systems of classical physics — even waves and fields — are described by a Lagrangian.

and further

Why are all systems described by action principles and Lagrangians? It’s not easy to say, but the reason is very closely related to the quantum origins of classical physics. It is also closely related to the conservation of energy. For now, we are going to it take as given that all known systems of classical physics can be described in terms of the action principle.

Here Ref. 1 is oversimplifying. The stationary action principle does not always apply. However, the counterexamples are typically at the macroscopic level in an effective theory where microscopic degrees of freedom have been integrated out/averaged over. E.g. a sliding friction force is an effective macroscopic force. The microscopic fundamental theory of Nature indeed obeys a stationary action principle. This is probably what Ref. 1 is referring to.

Qmechanic
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