Talk:Kinetic Energy of Classical Particle

Result or definition

It is not so easy to define the "derivation of kinetic energy". The current broad definition of kinetic energy is actually quite good, since it only relates to motion. However, why should we choose quadratic dependence? One can in principle have any power of velocity multiplied by an arbitrary coefficient called "inertial mass" and call the whole term the kinetic energy. Actually, in theoretical physics there are examples of phantom fields with negative kinetic energy and arbitrary powers of velocities (so to speak). I believe it is better to state a certain equivalence between Newton's laws and Lagrangian (something like equivalence of definitions). As for kinetic energy, the only way to derive in my opinion is through the "body's ability to do work". But it is a bit vague in mathematical sense. I would suggest that first we define a certain function called classical kinetic energy and then show that in certain processes some other functions are generated like a function of work, whose value in absolute magnitude approaches or becomes equal to that of kinetic energy. My logic in Lagrangian mechanics is that kinetic energy is a certain auxiliary function which is quite useful in classical cases. --Julius (talk) 16:52, 9 September 2019 (EDT)

We already have a definition of kinetic energy. My thought was that the classical case was derivable from the definition of energy as the integral of velocity with respect to time (or whatever it is, based on whatever it was that I've mostly forgotten from many years ago), in that the "classical" case is the simple one where mass is constant with veolocity. Can't remember the detail now, but I do remember that K.E. is a derived quantity rather than an absolute definition. --prime mover (talk) 17:12, 9 September 2019 (EDT)

What we have is a definition of a concept of kinetic energy, i.e. some scalar function of some aspect of motion and other parameters like mass (although this is not always necessary). The abstractness of kinetic energy will be really important when we switch from particles to fields. Integration of velocity by itself does not result in kinetic energy. Somebody has to decide that exactly this integral and not its square or logarithm is the kinetic energy, because our general definition only corresponds to the concept of motion, not some derivative squared (and if we are planning to include fields and pseudo-particles, this freedom will be necessary). We can simply write a theorem that an integration of velocity provides a function, which matches the one of my definition. (A counterexample would be an integral squared, which broadly speaking is still energy, but does not correspond to a standard classical particle).

Another question is the assumptions about the physical system. In high school and first year university courses, dynamics of the system are defined by masses and forces. Whenever one switches to Lagrangian mechanics, now the assumptions are (standard) kinetic energy and potential energy functions. In the latter context the force is a derived quantity. Hence concept of work does not exist yet. So what we end up with is a problem of equivalence of definitions, but now we have to compare different starting points with different accompanying notions.

It could also be that I am limited only to mathematical literature of Lagrangian approach. If I ever find mathematical literature on Newtonian approach, I would be happy to exercise the possibility of multiple sets of axioms of physics. In some sense, we are trying to solve one of Hilbert's problems. --Julius (talk) 04:29, 10 September 2019 (EDT)

Different approaches require different axioms. What is axiomatic for one approach is not axiomatic for another. I would hope that we could implement a straightforward entry-level approach to physics and applied mathematics that did not require the advanced concepts that you are discussing here. (Of course, if these concepts are not advanced but considered elementary, then I need to be replaced.) --prime mover (talk) 05:20, 10 September 2019 (EDT)

Probably something can be done. I simply wanted to set up a few basic placeholders to continue with the material of my book. The issue of equivalence of systems of axioms, with some of them sharing names, is really nasty. Also, these systems are not exactly equivalent, their overlap is only partial. I believe the problem will be mostly be terminological, and we will adjust accordingly.

Lagrangian mechanics is usually taught during the 2nd or 3rd year of undergraduate studies of physics (not necessarily theoretical or mathematical) and mathematics. Also. most of scientific articles are written primarily in Lagrangian/Hamiltonian formulation, with an exception of dissipative systems. --Julius (talk) 17:18, 10 September 2019 (EDT)

Never touched on it myself in my masters degree. I expect it's idiosyncratic. --prime mover (talk) 17:46, 10 September 2019 (EDT)