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The most fundamental parts of Lagrangian mechanics involve calculus. The action principle involves an integral and the Euler-Lagrange equation is a partial differential equation. Unless the students are pretty good with calculus it will be quite hard to teach.
If all you are looking for is a basic introduction without the calculus of variations, then the following article (which, however, assumes knowledge of elementary calculus as a prerequisite) may be of help: Hanc, Jozef, Edwin F. Taylor, and Slavomir Tuleja. "Deriving Lagrange’s equations using elementary calculus." American Journal of Physics 72.4 (2004): ...
If I understand the question right, we suppose we want to prove to someone that the earth orbits the sun. I'm not quite sure that' the case from a scientific point of view. Literally speaking, we can choose any reference frame we like and thus prove a heliocentric system or a or a geocentric. Quoting Einstein:" The struggle, so violent in the early days ...
This answer contains some additional resources that may be useful. Please note that answers which simply list resources but provide no details are strongly discouraged by the site's policy on resource recommendation questions. This answer is left here to contain additional links that provide little or no commentary. Staelin's Electromagnetic Waves. This is ...
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