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However, in mathematics Newton's laws of motion can be generalized to multidimensional and curved spaces. Often the term Newtonian dynamics is narrowed to Newton's second law m a = F {\displaystyle \displaystyle m\,\mathbf {a} =\mathbf {F} } .
Newton's laws are often stated in terms of point or particle masses, that is, bodies whose volume is negligible. This is a reasonable approximation for real bodies when the motion of internal parts can be neglected, and when the separation between bodies is much larger than the size of each.
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
So long as the force acting on a particle is known, Newton's second law is sufficient to describe the motion of a particle. Once independent relations for each force acting on a particle are available, they can be substituted into Newton's second law to obtain an ordinary differential equation, which is called the equation of motion.
Newton’s second law of motion states that the rate of change of momentum of an object is equal to the resultant force F acting on the object: =, so the impulse J delivered by a steady force F acting for time Δ t is: J = F Δ t . {\displaystyle \mathbf {J} =\mathbf {F} \Delta t.}
In 1687 Newton published his Principia which combined his laws of motion with new mathematical analysis to explain Kepler's empirical results. [7]: 134 His explanation was in the form of a law of universal gravitation: any two bodies are attracted by a force proportional to their mass and inversely proportional to their separation squared.
In other words, any first-order perturbation of the path results in (at most) second-order changes in . Note that the abbreviated action S 0 {\displaystyle {\mathcal {S}}_{0}} is a functional (i.e. a function from a vector space into its underlying scalar field), which in this case takes as its input a function (i.e. the paths between the two ...
The first and second laws prohibit two kinds of perpetual motion machines, respectively: the perpetual motion machine of the first kind which produces work with no energy input, and the perpetual motion machine of the second kind which spontaneously converts thermal energy into mechanical work.