# Equation of motion in polar coordinates

Given those numbers, M is readily calculated for any time t. In polar coordinates r,f describing the satellite's motion in its orbital plane, f is the polar angle. The equation of the orbit is. By Kepler's law of areas, it grows rapidly near perigee point closest to Earth but slowly near apogee most distant point. At perigee, all three anomalies equal zero.

We assume the period T is known this requires the 3rd law and is discussed for circular orbits in sections 20 and 20a.

It can then be shown that the angle E satisfies " Kepler's equation ". How did that number suddenly crop up, you may ask? The fact is, the division of the circle into degrees may be convenient to use we inherited it from the ancient Babylonians but the number has no particular place in mathematics.

It is probably related to the number of days in a year.

## Plane Curvilinear Motion - Polar Coordinates

With angles measured in radians, Kepler's equation simplifies to. No matter which form is used, mathematics knows no formula which gives E in terms of M. However, solutions can often be approximated to any degree of accuracy by iteration --by starting with an approximate solution, then improving it again and again by an appropriate procedure " algorithm "--more about that word, here. If the eccentricity e is not too big--the ellipse not much different from a circle--then M and E are not too different. So an initial guess. Putting this guess into the term sinE gives an improved guess E". One can now insert E" in the sinE term and get an even closer guess, and so on and so forth Computer handle such a process of continuous improvement "iteration of the solution"--one form of an algorithm very rapidly, and other methods also exist, with sufficient speed even when e is not very small.

For instance, one can first derive. To orient the orbit in 3 dimensions requires a reference plane and a reference direction. For satellite orbits, the reference plane --the horizontal plane in the drawing--is usually the Earth's equatorial plane sometimes it is the plane of the ecliptic. The reference direction in either case is the direction from the center of the Earth to the vernal equinox which belongs to both above planes.

We will call it the x directionsince that is its role in x,y,z coordinates used in orbital calculations. Two non-parallel planes always intersect along a line--the way the plane of a door intersects the plane of the wall along the door's hinge.

The orbital plane and the equatorial plane used for reference do so too, and their intersection is called the line of nodes N.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. From here, I am not sure where to go.

I have tried using this with the radial component of acceleration but got not much luck. Any suggestions would be appreciated! Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered.

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Community and Moderator guidelines for escalating issues via new response…. Feedback on Q2 Community Roadmap. Autofilters for Hot Network Questions. Related 0. Hot Network Questions. Question feed. Mathematics Stack Exchange works best with JavaScript enabled.This article describes a particle in planar motion  when observed from non-inertial reference frames.

Those problems fall in the general field of analytical dynamicsthe determination of orbits from given laws of force. General results presented in fictitious forces here are applied to observations of a moving particle as seen from several specific non-inertial frames, for example, a local frame one tied to the moving particle so it appears stationaryand a co-rotating frame one with an arbitrarily located but fixed axis and a rate of rotation that makes the particle appear to have only radial motion and zero azimuthal motion.

The Lagrangian approach to fictitious forces is introduced. Unlike real forces such as electromagnetic forcesfictitious forces do not originate from physical interactions between objects. The appearance of fictitious forces normally is associated with use of a non-inertial frame of referenceand their absence with use of an inertial frame of reference. The connection between inertial frames and fictitious forces also called inertial forces or pseudo-forcesis expressed, for example, by Arnol'd: .

The equations of motion in a non-inertial system differ from the equations in an inertial system by additional terms called inertial forces. This allows us to detect experimentally the non-inertial nature of a system.

A slightly different tack on the subject is provided by Iro: . An additional force due to nonuniform relative motion of two reference frames is called a pseudo-force.

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Fictitious forces do not appear in the equations of motion in an inertial frame of reference : in an inertial frame, the motion of an object is explained by the real impressed forces. In a non-inertial frame such as a rotating frame, however, Newton's first and second laws still can be used to make accurate physical predictions provided fictitious forces are included along with the real forces.

### 11.7 Polar Equations

For solving problems of mechanics in non-inertial reference frames, the advice given in textbooks is to treat the fictitious forces like real forces and to pretend you are in an inertial frame. It should be mentioned that "treating the fictitious forces like real forces" means, in particular, that fictitious forces as seen in a particular non-inertial frame transform as vectors under coordinate transformations made within that frame, that is, like real forces.

Next, it is observed that time varying coordinates are used in both inertial and non-inertial frames of reference, so the use of time varying coordinates should not be confounded with a change of observer, but is only a change of the observer's choice of description.

Elaboration of this point and some citations on the subject follow. The term frame of reference is used often in a very broad sense, but for the present discussion its meaning is restricted to refer to an observer's state of motionthat is, to either an inertial frame of reference or a non-inertial frame of reference. The term coordinate system is used to differentiate between different possible choices for a set of variables to describe motion, choices available to any observer, regardless of their state of motion.

Examples are Cartesian coordinatespolar coordinates and more generally curvilinear coordinates. Here are two quotes relating "state of motion" and "coordinate system":  . We first introduce the notion of reference frameitself related to the idea of observer : the reference frame is, in some sense, the "Euclidean space carried by the observer".

Let us give a more mathematical definition:… the reference frame is In traditional developments of special and general relativity it has been customary not to distinguish between two quite distinct ideas. The first is the notion of a coordinate system, understood simply as the smooth, invertible assignment of four numbers to events in spacetime neighborhoods.

The second, the frame of reference, refers to an idealized system used to assign such numbers … To avoid unnecessary restrictions, we can divorce this arrangement from metrical notions. This comfortable circumstance ceases immediately once we begin to consider frames of reference in nonuniform motion even within special relativity.

In a general coordinate system, the basis vectors for the coordinates may vary in time at fixed positions, or they may vary with position at fixed times, or both.

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It may be noted that coordinate systems attached to both inertial frames and non-inertial frames can have basis vectors that vary in time, space or both, for example the description of a trajectory in polar coordinates as seen from an inertial frame.

In discussion of a particle moving in a circular orbit,  in an inertial frame of reference one can identify the centripetal and tangential forces.

It then seems to be no problem to switch hats, change perspective, and talk about the fictitious forces commonly called the centrifugal and Euler force. But what underlies this switch in vocabulary is a change of observational frame of reference from the inertial frame where we started, where centripetal and tangential forces make sense, to a rotating frame of reference where the particle appears motionless and fictitious centrifugal and Euler forces have to be brought into play.

That switch is unconscious, but real. Suppose we sit on a particle in general planar motion not just a circular orbit. What analysis underlies a switch of hats to introduce fictitious centrifugal and Euler forces? To explore that question, begin in an inertial frame of reference.Free Newsletter.

### Mechanics of planar particle motion

Sign up below to receive insightful physics related bonus material. It's sent about once a month. Easily unsubscribe at any time. Join me on Patreon and help support this website. Curvilinear Motion General Curvilinear Motion Curvilinear motion is defined as motion that occurs when a particle travels along a curved path.

The curved path can be in two dimensions in a planeor in three dimensions. This type of motion is more complex than rectilinear straight-line motion. Three-dimensional curvilinear motion describes the most general case of motion for a particle.

To find the velocity and acceleration of a particle experiencing curvilinear motion one only needs to know the position of the particle as a function of time. You simply take the first derivative to find the velocity and the second derivative to find the acceleration. The magnitude of the velocity of particle P is given by The magnitude of the acceleration of particle P is given by Note that the direction of velocity of the particle P is always tangent to the curve i.

But the direction of acceleration is generally not tangent to the curve. However, the acceleration component tangent to the curve is equal to the time derivative of the magnitude of velocity of the particle P along the curve. In other words, if v t is the magnitude of the particle velocity tangent to the curvethe acceleration component of the particle tangent to the curve a t is simply In addition, the acceleration component normal to the curve a n is given by where R is the radius of curvature of the curve at a given point on the curve x py pz p.

The figure below illustrates the acceleration components a t and a n at a given point on the curve x py pz p. For example, However, it is usually not necessary to know the radius of curvature R along a curve. But nonetheless, it is informative to understand it on the basis of its relationship to the normal acceleration a n.

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For a particle P defined in polar coordinates as shown belowwe can derive a general equation for its radial velocity v rradial acceleration a rcircumferential velocity v cand circumferential acceleration a c. Note that the circumferential direction is perpendicular to the radial direction.

The movement of the link is causing a rod to slide along the curved channel, as shown. The radial velocity of the rod is given by equation 1 : The radial velocity is in the direction of increasing R. The radial acceleration of the rod is given by equation 2 : The radial acceleration is in the direction of decreasing R.

Follow FrancoNormani. I am at least 16 years of age. I have read and accept the privacy policy. I understand that you will use my information to send me a newsletter.Many systems and styles of measure are in common use today. When graphing on a flat surface, the rectangular coordinate system and the polar coordinate system are the two most popular methods for drawing the graphs of relations. Polar coordinates are best used when periodic functions are considered.

Although either system can usually be used, polar coordinates are especially useful under certain conditions. The rectangular coordinate system is the most widely used coordinate system. Second in importance is the polar coordinate system. It consists of a fixed point 0 called the poleor origin. Extending from this point is a ray called the polar axis. This ray usually is situated horizontally and to the right of the pole.

Any point, Pin the plane can be located by specifying an angle and a distance. If the angle is measured in a counterclockwise direction, the angle is positive. If the angle is measured in a clockwise direction, the angle is negative.

The directed distance, ris measured from the pole to point P. If point P is on the opposite side of the pole, then the value of r is negative. The location of a point can be named using many different pairs of polar coordinates. The polar coordinates for P 4, 9 are. Graphs of trigonometric functions in polar coordinates are very distinctive. The variable a in the equations of these curves determines the size scale of the curve. Previous De Moivres Theorem.

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Parametric Equations Introduction, Eliminating The Paremeter t, Graphing Plane Curves, Precalculus

I Thread starter sergiokapone Start date Oct 20, Tags equation of motion lorentz force. Summary: How to solve equations of motion for charged particle in a uniform magnetic field in a polar coordinates?

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But if I tring to solve this equation using only mathematical background without physical reasoning I can't do this due to entaglements of variables. What trick should I know? Related Classical Physics News on Phys. PeroK Science Advisor. Homework Helper. Insights Author. Gold Member. PeroK said:. You could look for a solution with fixed rrr. That might be a quick way.

As I mentioned I know the silution. I want to solve equations and get the constant r. Why do you think those equations yield a constant r? Most circles in polar coordinates do not have constant r. But what if I do not know is a circle? I need to solve equations in a right way, without any hypothesis. Ah, ok, this depend on initial condition. I can always choose so.

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Guessing a solution is perfectly legitimate, even in pure mathematics. But it doesn't have much values, I think. Don't underestimate the value of a well-chosen coordinate transformation. Especially in physics. Well, cylinder coordinates are simply not the best choice of generalized coordinates for this problem. Try again with Cartesian coordinates.In physicsequations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.

These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system.

If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics. 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. However, kinematics is simpler. It concerns only variables derived from the positions of objects and time. In circumstances of constant acceleration, these simpler equations of motion are usually referred to as the SUVAT equationsarising from the definitions of kinematic quantities: displacement sinitial velocity ufinal velocity vacceleration aand time t.

Equations of motion can therefore be grouped under these main classifiers of motion. In all cases, the main types of motion are translationsrotationsoscillationsor any combinations of these. A differential equation of motion, usually identified as some physical law and applying definitions of physical quantitiesis used to set up an equation for the problem.

A particular solution can be obtained by setting the initial valueswhich fixes the values of the constants. Euclidean vectors in 3D are denoted throughout in bold. This is equivalent to saying an equation of motion in r is a second-order ordinary differential equation ODE in r.

Other dynamical variables like the momentum p of the object, or quantities derived from r and p like angular momentumcan be used in place of r as the quantity to solve for from some equation of motion, although the position of the object at time t is by far the most sought-after quantity. Sometimes, the equation will be linear and is more likely to be exactly solvable. In general, the equation will be non-linearand cannot be solved exactly so a variety of approximations must be used. The solutions to nonlinear equations may show chaotic behavior depending on how sensitive the system is to the initial conditions. Kinematics, dynamics and the mathematical models of the universe developed incrementally over three millennia, thanks to many thinkers, only some of whose names we know.

In antiquity, priestsastrologers and astronomers predicted solar and lunar eclipsesthe solstices and the equinoxes of the Sun and the period of the Moon. But they had nothing other than a set of algorithms to guide them.