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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.
The second term in the above equation, plays the role of a gravitational force. If f f α {\displaystyle f_{f}^{\alpha }} is the correct expression for force in a freely falling frame ξ α {\displaystyle \xi ^{\alpha }} , we can use then the equivalence principle to write the four-force in an arbitrary coordinate x μ {\displaystyle x^{\mu }} :
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.
This constitutive equation is also called the Newton law of viscosity. The total stress tensor σ {\displaystyle {\boldsymbol {\sigma }}} can always be decomposed as the sum of the isotropic stress tensor and the deviatoric stress tensor ( σ ′ {\displaystyle {\boldsymbol {\sigma }}'} ):
The Onsager reciprocal relations have been considered the fourth law of thermodynamics. [15] [16] [17] They describe the relation between thermodynamic flows and forces in non-equilibrium thermodynamics, under the assumption that thermodynamic variables can be defined locally in a condition of local equilibrium.
This appears to simply be an expression of Newton's second law (F = ma) in terms of body forces instead of point forces. Each term in any case of the Navier–Stokes equations is a body force. A shorter though less rigorous way to arrive at this result would be the application of the chain rule to acceleration:
Long ago, Sir Isaac Newton gave us three laws of motion, which were the work of genius. But Sir Isaac’s talents didn’t extend to investing: He lost a bundle in the South Sea Bubble, explaining ...
Newton's approximation for the impact depth for projectiles at high velocities is based only on momentum considerations. Nothing is said about where the impactor's kinetic energy goes, nor what happens to the momentum after the projectile is stopped. The basic idea is simple: The impactor carries a given momentum.