Equations of motion

Under an equation of motion is understood to mean a mathematical equation (or a system of equations ) which fully describes the spatial and temporal evolution of the mechanical system under the effect of external influences. In general, there are systems of second order differential equations.

These differential equations are non-linear in many systems, so you have to apply appropriate approximation method for the solution.

Principle

For setting up of equations of motion in classical physics is

• The second Newton's law,
• The Lagrangian formalism or
• The Hamiltonian formalism

Be used. Based on results in the equation of motion of quantum mechanics, the Schrödinger equation

In engineering mechanics

• The principle of virtual work ( D' Alembert's principle)
• The principle of virtual power ( principle of Jourdain )
• The principle of least constraint

Be used.

Solution

Solving the equation of motion of the trajectory on which the system moves. It is, except for some simple cases (see examples below ), usually can not be represented analytically in closed form and must be obtained by numerical methods. This is, for example, to determine the trajectories of three celestial bodies that attract each other gravitationally, is required (see the three-body problem). To solve an N- particle system can be the discrete element method to apply. In simple cases, the closed-form solution is known as the "Railway equation ".

Examples

A general form of the equation of motion in classical physics is, for example,

.

Or known:

On the left side is the inertia term for the particle of mass on the right side of all the forces acting on the particles are summed up.

Equation of motion of a free mass particle

The equation of motion in this case is

With:

• : Force on particles (= 0),
• : Mass of the particle, and
• : ( Time-dependent ) position of the particle

The solution ( path equation ) are obtained by integrating twice the differential equation:

With the integration constants:

• : Speed ​​of the particle,
• : Position of the particle to

The particles thus moves linearly at constant speed. The mass does not matter.

Equation of motion of a particle under the influence of a constant force

A particle of mass is exposed to the force of gravity:

The path equation is

And provides the ballistic parabola throw dar. obtained for the free fall.

Equation of motion of special relativity

In the special theory of relativity the four-force is defined as the derivative of the relativistic momentum p after the proper time, with

Distinguishing between proper time and the time t, the relationship

Applies and refers to the Lorentz factor.

Equation of motion of general relativity

The movement of a body is defined by the geodesic equation of the curved space-time, provided that only gravitational forces act upon it. Then the body moves along a geodesic of spacetime. The geodesic equation is

Wherein a Christoffel symbol second type is that the dependence of the metric tensor of the space-time point (event ), that is the curvature of space-time characterized.

Equation of motion in structural dynamics

In the structural dynamics of the equation of motion of a dynamically loaded bearing structure is the basis for the calculation:

Here, the load of the vector system. and are the mass, damping and stiffness matrices, of the structure. The vector contains the displacement magnitudes. The matrizielle treatment according to the degrees of freedom of a structure is very suitable for computer calculation, for example using the finite element method.

Quantum mechanical potential well

In quantum mechanics, the Schrödinger equation occurs as a motion equation. For the simple problem of a particle in a one-dimensional box potential of length with infinitely high walls is the time-independent Schrödinger equation:

With

• : Wave function of the particle
• : Box potential.

The energy eigenvalues ​​and the corresponding eigenfunctions are:

.

• Classical Mechanics