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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.
Traditionally the Newton–Euler equations is the grouping together of Euler's two laws of motion for a rigid body into a single equation with 6 components, using column vectors and matrices. These laws relate the motion of the center of gravity of a rigid body with the sum of forces and torques (or synonymously moments) acting on the rigid body.
The dynamics of a rigid body system is described by the laws of kinematics and by the application of Newton's second law or their derivative form, Lagrangian mechanics. The solution of these equations of motion provides a description of the position, the motion and the acceleration of the individual components of the system, and overall the ...
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.
A body at rest will remain at rest, and a body in motion will remain in motion unless it is acted upon by an external force. (This is known as the law of inertia .) Force ( F → {\displaystyle {\vec {F}}} ) is equal to the change in momentum per change in time ( Δ m v → Δ t {\displaystyle {\frac {\Delta m{\vec {v}}}{\Delta t}}} ).
Action principles can be directly applied to many problems in classical mechanics, e.g. the shape of elastic rods under load, [23]: 9 the shape of a liquid between two vertical plates (a capillary), [23]: 22 or the motion of a pendulum when its support is in motion.
Euler's second axiom or law (law of balance of angular momentum or balance of torques) states that in an inertial frame the time rate of change of angular momentum L of an arbitrary portion of a continuous body is equal to the total applied torque M acting on that portion, and it is expressed as
The action corresponding to the various paths is used to calculate the path integral, which gives the probability amplitudes of the various outcomes. Although equivalent in classical mechanics with Newton's laws, the action principle is better suited for generalizations and plays an important role in modern physics. Indeed, this principle is ...