|The founder of dynamics, the science of "motion" (Sarpi, 1620) or “accelerating or retarding forces, and of the varied motions which they must produce” (Lagrange, 1788), is Galileo, the physics of which being published in his The Two New Sciences (1632), most of which derived from experiments (Ѻ) wherein balls were rolled down moderately-sloped inclined planes, such as shown above, wherein he used pendulums (Ѻ) to time the rate of acceleration.|
In 1543, Nicolaus Copernicus published On the Revolutions of the Celestial Spheres (De revolutionibus orbium coelestium), wherein the heliocentric theory was established, which marks the start of dynamics.
In 1602, Galileo began doing "demonstrations", and developing theorems therefrom, concerning falling bodies and projectiles.
In 1632, Galileo published Dialogue Concerning the Two Chief World Systems (Dialogo sopra i due massimi sistemi del mondo), which compared the then controversial Copernican system with the inconsistencies of the older earth-centric Ptolemaic system, and contains all Galileo had to say on physics; this is said to mark the point when dynamics, as a science, was founded. 
In 1687, Isaac Newton published Principia, which introduced the concepts of force and mass, the laws of motion, and the principle of universal gravitation. 
In 1689, the term “dynamica” was coined by German polymath Gottfried Leibniz, during his Italian journey, referring to his doctrine of forces; during which time he comprised an extensive then-unpublished work entitled Dynamica, some of which found publication outlet as “Specimen Dynamicum” in Acta eruditorum in 1695. Leibniz, in this latter publication, outlined four notions: 
In c.1700, Newton, in a manuscript, however, had the following to say about this:
a) Active primitive force is purely a metaphysical entity expressing the activity of substances and is also called entelechy;
b) Active derivative force is somehow the phenomenal manifestation of an aggregate of metaphysical substances and is measured by living force, or vis viva;
c) Passive primitive force is purely metaphysical and expresses the imperfection of substances;
d) Passive derivative force, which is also called inertia, is its phenomena manifestation.
“Galileo began to consider the effect of gravity upon projectiles. Newton in his Principia improved that consideration into a larger science. Leibniz christened the child by a new name as if it had been his own, calling it dynamica. But his mark must be set upon all new inventions. And if one may judge by the multitude of new names and characters invented by him, he would go for a great inventor.”— Isaac Newton (c.1700), manuscript (Ѻ) note
In 1788, Joseph Lagrange, in his Analytical Mechanics, defined “dynamics” as the science of accelerative forces and the motions they produce; a subject founded by Galileo, perfected by Huygens, and revolutionized by Newton; specifically:
“Dynamics is the science of accelerating or retarding forces, and of the varied motions which they must produce. This science is wholly due to the moderns, and Galileo is the one who threw or made the first foundations.”— Joseph Lagrange (1788), Analytical Mechanics (Volume One, pg. 221) 
Lagrange showed that most varied consequences respecting the motions of systems of bodies may be derived from one radical formula. Namely, using a method of generalized co-ordinates, Lagrange showed that if one determines a system’s configuration by a sufficient number of variables whose number is the same as that of the degrees of freedom possessed by the system, then the kinetic and potential energies of the system can be expressed in terms of those variables, and the differential equations of motion thence deduced by simple differentiation.
In 1835, William Hamilton, in his his two-part 1835 treatise “On a General Method in Dynamics”, extended the work of Lagrange. 
In 1879, Hamilton’s so-called “force function” was referenced by Rudolf Clausius' 1879 Mechanical Theory of Heat, the founding textbook of the newly emerging science of “thermo-dynamics”. 
The descriptive phrase "cultural dynamics", according to Thomas Wallace, is often employed in tracing cultural transitions, but with rare exceptions tends to be devoid of appropriate models that mention a driving force or an energy source necessary to energize such dynamics; thermodynamics, conversely, according to Wallace is what identifies the "dynamics" of a society: 
“A thermodynamic-based economic model identifies the dynamics that drive all human existence including the economic, social, and political activities of a society.”
Other two-cultures namesakes that employ the term "dynamics" include: Henry Adams' physico-chemical social dynamics (1908).
The following are related quotes:
“To give us the science of motion, god and nature have joined hands and created the intellect of Galileo.”— Paolo Sarpi (c.1620) 
● Dynamic psychology
● Physico-chemical social dynamics (Henry Adams, 1908)
● Social dynamics
1. Daintith, John. (2005). Oxford Dictionary of Science. Oxford University Press.
2. Earnshaw, Samuel. (1844). Dynamics – or a Treatise on Motion (title page “Quote” by John Herschel). J. & J. J. Deighton.
3. (a) Hamilton, W.R. (1834). “On a general method in dynamics by which the study of the motions of all free systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function.” Philos. Trans. R. Soc. London, 124:247-308.
(b) Hamilton, W.R. (1835). “A second essay on a general method in dynamics.” Philos. Trans. R. Soc. London, 125:95-144.
4. Goodman, Lawrence E., Goodman, Susan, and Warner, William H. (2001). Dynamics (Historical Introduction). Courier Dover Publications.
5. Clausius, Rudolf. (1879). The Mechanical Theory of Heat (2nd ed). London: Macmillan & Co.
6. Wallace, Thomas P. (2009). Wealth, Energy, and Human Values: the Dynamics of Decaying Civilizations from Ancient Greece to America (pg. 2). AuthorHouse.
7. Applebaum, Wilbur. (2000). Encyclopedia of the Scientific Revolution: from Copernicus to Newton (pg. #). Routledge.
8. Galileo. (1632). Dialogues Concerning the Two New Sciences (translators: Henry Crew and Alfonso Salvio) (Lagrange quote, pg. v; Sarpi quote, pg. ix). Macmillan, 1914.
● Hamill, Patrick. (2013). A Student’s Guide to: Lagrangians and Hamiltonians. Cambridge University Press.
● Dynamics – Wikipedia.