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As is known to all, the variational principle in Hamiltonian mechanics has Lagrangian $L=K-V$. Equilibrium requires that

(1) the variation of $L$ be 0

or

(2) $L$ be minimized.

$K-V$ or $V-K$ doesn't matter for (1) but makes a difference for (2). When inserting kinetic energy and potential energy terms into $L$ in specific examples, one can see that $K-V$ is convex and can be minimized ($V-K$ can't). But how to show that $L$ should be $K-V$ rather than $V-K$ in a general context when having no specific energy terms to check the convexity of $L$?

Qmechanic
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feynman
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1 Answers1

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There is no intrinsic preference of $K - V$ versus $V - K$, the choice between the two is a convention.

The criterion is: the true trajectory corresponds to the point in variation space such that the derivative of Hamilton's action is zero.

The notion of minimizing is an overinterpretation. It is sufficient to apply the constraint that the derivative of Hamilton's action is zero.


When you set out to find the minimum of a function you capitalize on the following property: at the minimum the derivative of the function is zero. The algorithm to identify the point where a function is at a minimum is to identify the point where the derivative is zero.

For centuries physicists were under the impression that identifying the point derivative-is-zero was means to an end, believing the end goal was to identify a minimum.

But all along the minimum notion was superfluous; it is sufficient to apply the criterion that the derivative of Hamilton's action is zero.



There is an esthetic consideration, should one be so inclined. The kinetic energy, being proportional to the square of velocity, is intrinsically a positive value. If you would define the action as $V - K$ you would be putting a minus sign in front of the kinetic energy, and that's kind of ugly.

In the absence of any accelerating agent an object will move in a straight line. That is, you can have cases with no term for potential energy, but the kinetic energy terms is always there. (Principle of relativity of inertial motion: you must always be able to transform to another inertial coordinate system.) Having that single term not have a minus sign in front of it looks nicer.

Cleonis
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