Hello,
The sum of Kinetic energy Ke(t) and Potential energy Pe(t) in a physical system is referred to as Total energy Te(t)=Ke(t)+Pe(t), what can we say about the difference Ke(t)-Pe(t) and what can we call it?
I asked this question because I have found polynomial decay of the sum and exponential decay of the difference!
Thank you in advance!
Khaldi Said
If memory serves me well, the total energy is also known as Hamiltonian,
whereas the kind of difference you mentioned is known as a Lagrangian,
i.e., the function depending on coordinates and their velocities from which the eqs of motion can be derived (in arbitrary coordinates).
The difference is indeed the Lagrangian, which is basically the work you do along the way of your trajectory.
More correct formulation, though still not sufficiently rigorous: Lagrangean is a function of state of the system whose integral wrt time along the trajectory assumes extremal value if compared to those ways which are infinitesimaly different. This property via Euler's algorythms allows to derive the differential equation of motion (in simplest cases: the Newon type equations)
As per work energy theorem, difference in K.E. and P.E. gives either work done on the system or work one by the system. This work can be calculated by solving Lagrangean equation along the path you choose with constraints.
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