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Kinetic energy T is the energy of the system's motion and is a function only of the velocities v k, not the positions r k, nor time t, so T = T(v 1, v 2, ...). V , the potential energy of the system, reflects the energy of interaction between the particles, i.e. how much energy any one particle has due to all the others, together with any ...
In the center of mass frame the kinetic energy is the lowest and the total energy becomes = ˙ + The coordinates x 1 and x 2 can be expressed as = = and in a similar way the energy E is related to the energies E 1 and E 2 that separately contain the kinetic energy of each body: = = ˙ + = = ˙ + = +
The total kinetic energy of a system depends on the inertial frame of reference: it is the sum of the total kinetic energy in a center of momentum frame and the kinetic energy the total mass would have if it were concentrated in the center of mass.
Classical mechanics utilises many equations—as well as other mathematical concepts—which relate various physical quantities to one another. These include differential equations, manifolds, Lie groups, and ergodic theory. [4] This article gives a summary of the most important of these.
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.
The formula for the acceleration A P can now be obtained as: = ˙ + + (), or = / + / +, where α is the angular acceleration vector obtained from the derivative of the angular velocity vector; / =, is the relative position vector (the position of P relative to the origin O of the moving frame M); and = ¨ is the acceleration of the origin of ...
Hamilton's principle states that the true evolution q(t) of a system described by N generalized coordinates q = (q 1, q 2, ..., q N) between two specified states q 1 = q(t 1) and q 2 = q(t 2) at two specified times t 1 and t 2 is a stationary point (a point where the variation is zero) of the action functional [] = ((), ˙ (),) where (, ˙,) is the Lagrangian function for the system.
The kinetic energy is , and since the particle is constrained to move along a curve, its velocity is simply /, where is the distance measured along the curve. Likewise, the gravitational potential energy gained in falling from an initial height y 0 {\displaystyle y_{0}} to a height y {\displaystyle y} is m g ( y 0 − y ) {\displaystyle mg(y_{0 ...