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Thermodynamics is a perfect example of a science which is developed from axioms. Unlike classical mechanics, thermodynamics has withstood the quantum revolution without having its foundations shaken. However we will not adhere rigorously to the axiomatic presentation of science from the smallest possible number of axioms. This was achieved by  Caratheodory   in his proof of the Second Law, but to which we will not bestow  any preference over the Carnot-Clausius proof. The latter derived its concepts from engineering, which is advantageous. Preliminary introduction to statistical mechanics is given with elementary methods. https://plusphysics.wordpress.com/thermodynamics/

In classical mechanics, historically, core concepts – space, time, mass, force, momentum, torque, and angular momentum – were introduced to solve the most famous problem in physics, the motion of the planets. Two alternative approaches are presented: force and conservation laws. Conservation laws involving energy, momentum and angular momentum provide a second parallel approach to solving many of the same problems. Galilean relativity is introduced. The main purpose of the course is to develop a conceptual understanding of the core concepts and understanding of how theoretical science work, a familiarity with the experimental verification of theoretical laws, and an ability to apply the theoretical framework to describe and predict the motions of bodies. The course explores the mathematical physics developed by Isaac Newton (1642–1727) and introduces some familiarity with the formulations of mechanics developed later by Joseph Lagrange (1736–1813) and William Hamilton (1805–1865). The

Physics Open Courseware PlusPhysics offers a wide range of crash courses in the various disciplines of physics, including introductory mathematical sequences. https://plusphysics.wordpress.com/