The Lagrangian L = T-V, where T is the kinetic energy and V the potential energy. Can be re-written in terms of a variable q, which might as well be x. Action is denoted S.

The hamiltonian is the total energy of the system; {\cal H}=T+V

U is internal energy; defined as the sum of all kinetic and potential energies present in the system.
Q is heat
W is work

dU = \delta Q + \delta W; this is called the “total differential of” U

Both Q (heat) and W (work) are path variables.

A subscript variable like \left(\frac{dS}{dT}\right)_{V} means “held constant”.

A “harmonic” potential has V \approx a x^2; that is, F \approx bx.

Extensive: internal energy depends linearly on the size of the system:

Likewise for the density of a homogeneous system; if the system is divided in half, the extensive properties, such as the mass and the volume, are each divided in half, and the intensive property, the density, remains the same in each subsystem.