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A large number of fundamental equations in physics involve first or second time derivatives of quantities. Many other fundamental quantities in science are time derivatives of one another: force is the time derivative of momentum; power is the time derivative of energy; electric current is the time derivative of electric charge; and so on.
The Ehrenfest theorem, named after Austrian theoretical physicist Paul Ehrenfest, relates the time derivative of the expectation values of the position and momentum operators x and p to the expectation value of the force = ′ on a massive particle moving in a scalar potential (), [1]
In this example, the time derivative of q is the velocity, and so the first Hamilton equation means that the particle's velocity equals the derivative of its kinetic energy with respect to its momentum. The time derivative of the momentum p equals the Newtonian force, and so the second Hamilton equation means that the force equals the negative ...
The momentum of the object at time t is therefore p(t) = m(t)v(t). ... (although its time derivative might appear), then p j is constant. This is the generalization ...
This operator occurs in relativistic quantum field theory, such as the Dirac equation and other relativistic wave equations, since energy and momentum combine into the 4-momentum vector above, momentum and energy operators correspond to space and time derivatives, and they need to be first order partial derivatives for Lorentz covariance.
In quantum mechanics the Hamiltonian ^, (generalized) coordinate ^ and (generalized) momentum ^ are all linear operators. The time derivative of a quantum state is represented by the operator ^ / (by the Schrödinger equation).
The left-hand side is the time derivative of the momentum, and the right-hand side is the force, represented in terms of the potential energy. [9]: 737 Landau and Lifshitz argue that the Lagrangian formulation makes the conceptual content of classical mechanics more clear than starting with Newton's laws. [27]
The general statement of d'Alembert's principle mentions "the time derivatives of the momenta of the system." By Newton's second law, the first time derivative of momentum is the force. By Newton's second law, the first time derivative of momentum is the force.