## Newton’s Laws of Motion:

**1st Law**: In the absence of a net force, a body either is at rest or moves in a straight line with constant speed.**2nd Law**: A body experiencing a force F experiences an acceleration a related to F by F = ma, where m is the mass of the body. Alternatively, force is equal to the time derivative of momentum.**3rd Law**: Whenever a first body exerts a force F on a second body, the second body exerts a force -F on the first body. F and -F are equal in magnitude and opposite in direction.

## Time Stepping

Newton’s World is based on the following *incremental equations of motion* with smallest unit of time :

- ,

as another way of writing

- ,

which combined with Newton’s 2nd Law assuming , take the form:

- ,

or

- .

These equations can be solved by time-stepping with time step :

- ,

where

- ,

and , and are position, velocity and acceleration at time after successive time steps with time step . By time-stepping forward in time and at time are computed from and already computed at the preceding time .

### Euler’s Method: Forward Euler

With each tick of time, velocity and position are thus updated according to

from given initial values and at initial time , where is the force acting on the body at time . We refer to this update formula as *Forward Euler. *

### Smart Euler

An alternative update formula is obtained by updating first velocity to and using this value when updating to :

- ,

which we will refer to as *Smart-Euler*. You will soon discover the difference between Euler and Smart-Euler.

A variant of Smart-Euler is

- ,

where the mean velocity is used instead of either or .

### Trapezoidal Method

Below we shall meet variants with depending on . The basic method of this form is the *Trapezoidal Method:*

- ,

where and , which requires iteration because depends on ,which depends on .

### Backward Euler

We compare with *Backward Euler*:

which also requires iteration.

### Midpoint Euler

A variant of the Trapezoidal Method is *Midpoint Euler*:

- ,

with evaluated at the midpoint between and , also requiring iteration.

## Explicit vs Implicit Methods

We distinguish between *explicit methods* like Forward Euler and Smart Euler with direct update, and* implicit methods* requiring* iteration,* like the Trapezoidal Method, Backward and Midpoint Euler, where the update formula for and is repeated with latest values inserted in the right hand side.

With a (small) fixed number of iterations, implicit methods can be viewed as explicit direct update methods.

## First and Second Order Accurate Methods

Forward and Backward Euler are *first order accurate* in the sense that the time-stepping error is proportional to the time step , while the Trapezoidal Method and Midpoint Euler are *second order accurate *in the sense that the time stepping error is proportional to the time-step squared , which is much smaller than since is small. Smart Euler is rather second than first order accurate.

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