# “Principle of least action” and “Principle of conservation of energy”: Which one is fundamental and which one is derived? [closed]

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" ?

• Conservation of energy doesn't imply that potential energy should be minimised. A ball standing still in midair would also conserve energy, I don't know what you are asking. – ACuriousMind Feb 24 '16 at 11:12
• Neither are enough to cover all of classical mechanics. The least action principle only describes an important subset of all possible mechanical systems and energy may or may not be conserved in a particular system, which makes neither of these "fundamental". – CuriousOne Feb 24 '16 at 11:12
• Using principle of least action and time symmetry you can show that if only conservative forces act on a system then its energy will be conserved. But I concur with @CuriousOne. Neither are enough to explain everything. You need to know more about the system in question. – Apoorv Khurasia Feb 24 '16 at 11:28
• We can say with confidence that the Principle of Least Action, which was originally formulated by Maupertuis and Euler in 1744, and later improved by Lagrange and others. With little more than the Calculus of Variations and the Principle of Least Action, one can derive all of Newtonian mechanics. Conservation of Energy was not understood until about 1850 when thermodynamics was being created, and serves as an additional constraint on mechanics. As noted above, actual problems require a complete specification, including suitable boundary conditions. – Peter Diehr Feb 24 '16 at 12:28
• Let me just get out my fundamentometer, here and we'll find out. Ot to be less sarcastic there is a degree of philosophy in this; just exactly what do you mean by "fundamental". – dmckee Feb 24 '16 at 15:14

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'Alemberts 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.

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$^1$ 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.

• Recently I have been reading the classical mechanics book by Leonard Susskinds. He writes that: 1. All systems of classical physics-even waves and fields-are described by lagrangian. 2. All known systems of classical physics can be described in terms of action principle. 3. Total energy of all classical systems is conserved if the forces are conservative. He also writes that "Why are all systems described by action principle and Lagrangian? It is not easy to say but the reason is very closely related to quantum origin of classical physics and to conservation of energy" @dmckee – user103515 Mar 3 '16 at 8:17
• So, finally what can be precisely concluded about the principle of least action and conservation of energy? Which one is the cause and which one is the effect? Or they are not related at all? – user103515 Mar 4 '16 at 7:36
• I updated the answer. – Qmechanic Mar 4 '16 at 10:39