Is it circular reasoning to derive Newton's laws from action minimization?

Question

Usually, a typical example of the use of the action principle that I've read a lot is the derivation of Newton's equation (generalized to coordinate $q(t)$). However, in the classical mechanics interpretation, isn't this like tripping oneself up? Because we've already identified $T$ and $U$ as the kinetic energy and the potential energy, respectively which are derived, in fact, from Newton's equations.

Answer 1

Answering the third question, in any mature branch of mathematics or physics, there are always equivalent or near-equivalent formulations of the same structure (it's pretty much a definition of maturity, that some of the relations of a new topic to older topics have been worked out). The Lagrangian can be regarded as a type of functional anti-derivative of a set of equations of motion (taking functional derivatives generates the equations of motion), which in turn can be regarded as a presentation of the detailed implications of Newton's law in the context of a particular model.

There may be reasons for taking some mathematical formulations to be more fundamental than others, typically because the map between one and another is not 1-1 or not onto, or, in modern times, because some formalisms present the symmetries of the dynamics more transparently. In the case of Newtonian mechanics, there are numerous different ways to specify a dynamics (perhaps as many as a dozen altogether, developed over three centuries, including, for example, the Hamilton-Jacobi formalism, although I couldn't name them all without looking them up).

Ultimately, if the empirical contents of different models are equivalent, a more empirical Physicist will say that the models are equivalent, so that the choice between using one or another rests in questions such as which is more tractable. However, realism about the role of symmetry groups in nature has led to many physicists making claims about the preeminence of Lagrangian approaches that rely to some extent on issues such as mathematical aesthetics. You will have to decide for yourself where on this spectrum you wish to place yourself, insofar as the decision is not based on experiment.

If you have a strong mathematical background, two references that come to mind that might broaden your perspective are Olver, "Applications of Lie groups to differential equations", or, slightly more accessibly, Marsden & Ratiu, "Introduction to Mechanics and Symmetry".

Answer 2

Potential energy is associated with a property of conservative forces that can be expressed as the gradient of a potential function, rather than coming from a definition using Newton's laws. Kinetic energy on the other hand can be derived from the definition of work done and, together with potential energy, expresses the conservation of energy which is why kinetic energy is defined the way it is. You can't derive the conservation of energy from just Newton's laws.

D'Alembert's principle comes partly from Newton's second law with the additional postulate that forces of constraint don't do work, which is true for some classical mechanics problems. You can't use Newton's laws to prove that the force of reaction of a table on a book resting on it is normal to the surface. D'Alembert's principle together with monogenic forces is used to derive the Euler-Lagrange equations and the principle of least action.

We can therefore conclude that the principle of least action is at least compatible with Newton's first and second laws for monogenic forces which includes gravity and electromagnetism.

Answer 3

Yes, it is circular reasoning to derive Newton's laws from the Principle of Least Action.

$τ$ is the constant information of the system or Hamilton's principal function.

$$0=\frac{dτ}{dt}=\frac{∂τ}{∂x}\;\frac{∂x}{∂t}+\frac{∂τ}{dt}\\ W=\frac{∂τ}{dt}=-\frac{∂τ}{∂x}\;\frac{∂x}{∂t}=-H\\ H=pẋ=mẋ²$$

$∂τ$ is a proper time step of the system consisting of observable simultaneous components.

Note, $p = mẋ$ is by observation, i.e. the physical content. Then it is assigned to $\frac{∂τ}{∂x}$ by definition, leading to $H=mẋ²$.

$∂τ=mẋ∂x$ then says that $ẋ$ and $∂x$ contribute independently to a time step $∂τ$.

Splitting off some non-observable part of the system and associating it to the location of the observed part $H(ẋ,x)=T(ẋ)+V(x)$ half-half, makes $T(ẋ)=mẋ²/2$. The key point is to find a $V(x)$.

$W=-H$ stays constant. $∫Hdt$ would count system time to infinity. $L(x)=mẋ²+W=mẋ-H$ oscillates and returns to the same value in a cycle. $J=∫Ldt$ returns to the same value after one or many cycles. This can be minimized to find $V$:

$$0 = \frac{δJ}{δx} = \frac{1}{δx}∫\left(δx\frac{∂L}{∂x}+δẋ\frac{∂L}{∂ẋ}\right)dt = \frac{1}{δx}∫δx\left(\frac{∂L}{∂x}-\frac{d}{dt}\;\frac{∂L}{∂ẋ}\right)dt \\ \frac{∂L}{∂x} = \frac{d}{dt}\;\frac{∂L}{∂ẋ} \\ F=ṗ$$

Note, $F=∂L/∂x$ and $p=∂L/∂ẋ=∂τ/∂x$ are by definition.

• We need to add a physical $p$ or $F$ separately to find $L$ (Newton approach).
• Or we need to add a physical $L$ to get $p$ and $F$ (Lagrange approach).

Conclusion: One cannot derive Newton's laws from the minimization of the action $J$.

Generally: One cannot derive physics. One needs to observe.