Unraveling the Mystery Behind Energy Equations

[Inertigratus]

I was just wondering, why do the energy equations look so alike?
Refering to kinetic energy, the energy stored by capacitors and inductors, etc.
My teacher in electromagnetism said something about it, that there's a reason for this.
The energy is proportional to some rate of change squared?

 

[Philip Wood]

In all the cases you've mentioned we have an underlying linear relationship (e.g. between charge and voltage for a capacitor, momentum and velocity for a moving body), and the energy is found as the area of a triangle underneath the linear graph, or, algebraically as the integral of kx wrt x. Hence the 1/2.

[Where we don't have the linear relationship (e.g. the non-linear momentum-velocity relationship in Special Relativity) we don't get the (1/2) k x^2 form for energy]

 

[genericusrnme]

Usually you'll find kinetic energy to be quadratic in velocities
Why?
depends where you're coming from really, it could be a 'it just is' matter or it could be because the action in relativity is proportional to the integral of arclength. you could perhaps argue that it must be velocities squared because it shouldn't matter which direction the velocity is in
there are lots of reasons that it should be in the form that it is but would it be possible to say which is the cause of it's form and which are the cause of it's form? I'd guess not
the potential energy doesn't have this same standard form however, it can be any function of coordinates you want it to be

just my two cents