What is Trajectories: Definition and 127 Discussions

A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete trajectory is defined by position and momentum, simultaneously.
The mass might be a projectile or a satellite. For example, it can be an orbit — the path of a planet, asteroid, or comet as it travels around a central mass.
In control theory, a trajectory is a time-ordered set of states of a dynamical system (see e.g. Poincaré map). In discrete mathematics, a trajectory is a sequence








{\displaystyle (f^{k}(x))_{k\in \mathbb {N} }}
of values calculated by the iterated application of a mapping


{\displaystyle f}
to an element


{\displaystyle x}
of its source.

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  1. lavendersmell

    I [Questions] Modeling a Baseball Pitch Trajectory in 3D Space

    I am currently taking some time off of college (I am a sophomore), and I'm trying to continue coding and experimenting with Calculus-y math as I'll be going into Calculus II and then III when I go back. I am currently trying to develop a 3D baseball pitch visualizer for my own purposes. I am not...
  2. atyy

    I Relativistic Bohmian trajectories of photons via weak measurements

    I found out about this interesting paper through a Tweet by Steven Thomson. https://arxiv.org/abs/2108.05169 https://www.nature.com/articles/s41467-022-31608-6 Relativistic Bohmian trajectories of photons via weak measurements Joshua Foo, Estelle Asmodelle, Austin P. Lund, Timothy C. Ralph...
  3. sophiatev

    A Characterizing GR Traj in Minkowski Space

    In Minkowski space, with line element $$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$$ (and ##c = 1##) we take spacelike trajectories to have ##ds^2 > 0##, null trajectories to have ##ds^2 = 0##, and timelike trajectories to have ##ds^2 < 0##. This makes sense given our definition of the line element...
  4. Mathieu Rouaud

    A Einstein's Elevator Trajectories: Desloge & Philpott 1987, Hamilton 1978

    Hello, Some papers describe the vertical motion of a ray of light or a non-zero mass particle in a uniformly accelerated reference frame in special relativity: Desloge, E. A., & Philpott, R. J. (1987). Uniformly accelerated reference frames in special relativity. American Journal of Physics...
  5. Buckethead

    B Dual slit with controlled electron trajectories

    In the traditional single electron duel slit experiment, I assume a cathode emits electrons in an unfocused direction spreading across the dual slits like a flashlight beam, but one electron at a time. Electrons however can be finely focused and controlled using magnetic or electric fields...
  6. S

    I Wilson's RG trajectories, Lagrangians and many worlds?

    In this article [1] we can read an explanation about Wilson's approach to renormalization I have read that Kenneth G Wilson favoured the path integral/many histories interpretation of Feynman in quantum mechanics to explain it. I was wondering if he did also consider that multiple worlds...
  7. patric44

    Orthogonal trajectories in polar coordinates

    there is a problem in a book that asks to find the orthogonal trajectories to the curves described by the equation : $$r^{2} = a^{2}\cos(\theta)$$ the attempt of a solution is as following : 1- i defferntiate with respect to ##\theta## : $$2r \frac{dr}{d\theta} = -a^{2}\;\sin(\theta)$$ 2- i...
  8. S

    Bohmian Trajectories with New Slit

    Hi, I've seen these depictions of Bohmian trajectories and I was wondering what would happen to these trajectories once another slit is opened. Do they get "reconfigured"? Do they all change and adapt to accommodate the trajectories coming from the newly opened slit?
  9. G

    How does the Earth move through in 3 separate trajectories?

    Hello ,my name is Damon and I'm new to this site so please forgive me if I'm not following the rules to the letter. Could someone please provide a link or CGI model of our Earth' three separate curving, spinning,(while corkscrewing) ellliptical, linear paths, while traveling at speeds of...
  10. A

    A What are Bohmian trajectories for a free electron?

    A free electron, or any other quantum particle, has an uncertain position/momentum, according to Heisenberg uncertainty principle. The squared amplitude of the wavefunction determines the probability of finding the electron at any point of the space. Accordingly, atomic orbitals are attributed...
  11. Asawira Emaan

    MHB What are the orthogonal trajectories of e^{x}(xcosy - ysiny) = c?

    Asalamoalaikum, help me with this. I can solve it but it goes very lengthy. Determine the equations of the orthogonal trajectories of the following family of curve; e^{x}(xcosy - ysiny) = c
  12. P

    A Bohmian Trajectories: Intersections & Young Experiments

    i read that a bohmian trajectory (in this interpretation) cannot intersect itself because the speed depends on the position. there is no visualization problem in a Young experiment with trajectories from the slits to the screen. it becomes harder when a particle is trapped in a small region by a...
  13. A

    Phase space trajectories can't intersect...

    Phase space trajectories can't intersect each other is it due to the fact that at the intersection point there will be more than one possible path for the system to evolve with time??
  14. W

    Finding Orthogonal Trajectories (differential equations)

    Homework Statement Find Orthogonal Trajectories of ##\frac{x^2}{a}-\frac{y^2}{a-1}=1## Hint Substitute a new independent variable w ##x^2=w## and an new dependent variable z ##y^2=z## Homework EquationsThe Attempt at a Solution substituting ##x## and ##y## I get...
  15. M

    How to measure arm trajectory?

    I am not sure in which section this question should belong so I am placing it here for now. I have no experience in physics and body motion so appreciate some patience with my question. the information i have are pixel coordinates of the head, shoulder, elbow, hand, knee, hips and leg of 2...
  16. A

    A Use of the Optical Theorem and Regge trajectories

    Cutkosky rule states that: $$2Im \big(A_{ab}\big)=(2\pi)^4\sum_c \delta\Big(\sum_c p^{\mu}_{c}-\sum_a p^{\mu}_{a}\Big)|A_{cb}|^2\hspace{0.5cm} (1)$$ putting ##a=b=p## in Cutkosky rule we deduce the Optical Theorem for ##pp## scattering: $$2Im \big(A_{pp}\big)=(2\pi)^4\sum_c \delta\Big(\sum_c...
  17. S

    Getting the average of several typhoon trajectories

    Hi I am curious to know if there are methods of getting the average of several typhoon trajectories, with the average trajectory represented as straight red lines (for simplicity) as shown in the image. I am assuming that this average wouldn't be a straight line but be represented as a range...
  18. stevendaryl

    Insights Rindler Motion in Special Relativity: Hyperbolic Trajectories - Comments

    Greg Bernhardt submitted a new PF Insights post Rindler Motion in Special Relativity: Hyperbolic Trajectories Continue reading the Original PF Insights Post.
  19. B

    Velocities when generating an Archimedean spiral trajectory

    hey, I just want to know, if I am to send a velocity commands to generate a spiral trajectory, What would be these velocities (angular and linear)?? Thanks in advance
  20. 10Exahertz

    I Taking Projectile Motion to make Planetary Trajectories

    Hi all I have been trying to find A set of equations that can allow me to map the movement of a planetary body on a polar coordinate sheet (a 2-D Problem). As well as allow me to find out about information such as the perigee and apogee radaii. So far i have been using Keplers equations and...
  21. Delta Force

    Difference Between Musket and Rifle Trajectories?

    Wikipedia's article on rifles in the American Civil War mentions this: Is it true that smoothbore guns have a flatter trajectory than rifles? Can someone explain the physics of why that would happen?
  22. T

    I Effect of planetary layer densities on trajectories of a satellite

    According to Cornell university's ask an astronomer site, we figure out the layers and their density of planets by checking the trajectory of a satellite /space probe or something like that, near the planets. But planets are quite spherical. then their density profile shouldn't be effecting the...
  23. J

    Closed trajectories for Kepler with spin-orbit corrections?

    Kepler problem explains closed elliptic trajectories for planetary systems or in Bohr's classical atomic model - let say two approximately point objects, the central one has practically fixed position, they attract through 1/r^2 Newton's or Coulomb force. Kind of the best motivated expansion we...
  24. M

    Velocity in eliptical trajectories

    Hello everyone, A little question bothering me concerning elliptical trajectories: Say I have a particle at some point with some velocity (both I know - r0, v0) and I need know its minimal velocity so that it reaches some specific r*. I can easily find this velocity considering conservation of...
  25. doktorwho

    Solving for the trajectory in the polar coordinate system

    Homework Statement On the surface of a river at ##t=0## there is a boat 1 (point ##F_0##) at a distance ##r_0## from the point ##O## (the coordinate beginning) which is on the right side of the coast (picture uploaded below). A line ##OF_0## makes an angle ##θ_0=10°## with the ##x-axis## whose...
  26. B

    B Does the Initial Emission in the Double Slit Experiment Have a Trajectory?

    I have 2 questions 1. I'm telling a friend that in the double slit experiment.. the electron has no trajectories between the emission and detection. But she commented the initial emission has trajectory. Is this correct? What is the term for this situation of the initial emission of the...
  27. H

    Trajectories never cross in phase-space

    I heard this statement from time to time, but what does it really mean?
  28. T

    Can a beam of light appear to have different trajectories?

    Homework Statement In an inertial reference frame, a beam of light is shone 30 degrees from the x-axis. What is the speed of another inertial reference frame along the x-axis where the beam of light is 90 degrees from the x-axis? Homework Equations Can't really think of any equations that...
  29. J

    MATLAB Solving Orbital Motion Equations with RK4 Using MATLAB

    I'm trying to plot the solutions of the second order differential equation d^2R/dt^2 = GM/R^2 + Lz^2/R^3. I'm reducing this to a system of first order ODEs and then using RK4 to solve this system. My code is given by function RK4system() Tsim = 10...
  30. B

    B Can the Bohmian Wave Function Influence Thermodynamic Trajectories?

    I know thermodynamics are the macroscopic coarse graining of microscopic degrees of freedom (like temperature and Brownian motion). But is there a case where let's say the bohmian wave function can create trajectories of particles that can control the macroscopic thermodynamics or has...
  31. B

    I Why do particles move along longest proper time trajectories

    Hi, I am working my way thought Hartle's Gravity. In Section 5.4 he states that "The straight lines along which free particles move in spacetime are paths of longest proper time" and proceeds to proof that "in flat space time the proper time is a curve of extremal proper time". Can someone...
  32. P

    B Does the L2 norm of a vector destroy all directional info?

    Sorry I'm a little rusty with my math and proof logic, and this feels like a dumb question, but oh well! The Euclidian norm of a vector in ℝ3 is \|{v}\| = \sqrt{x^2 + y^2 + z^2} where \|{v}\| \geq 0. I'm trying to show that there is always an infinite number of solutions for arbitrary...
  33. Pull and Twist

    MHB HELP: Orthogonal Trajectories

    Trying to figure out the orthogonal trajectory of x^2 + y^2=cx^3 Here's what I got... but it does not match the books answer. I don't know where I am going wrong. I think I was able to differentiate the equation correctly in order to get the inverted reciprocal slope and then I may have flubbed...
  34. S

    Trajectories of planets using reduced mass and CM frame

    In planetary motion, the reduced mass of a system \mu is used in order to study the motion of the planet m in the non-inertial frame of the star M. Using \mu the trajectory of m turns out to be a conic. But this is the trajectory of the planet m as seen from the star M, correct? I read that in...
  35. P

    Trajectories, equation of motion and forces....

    Setting a cartesian system how can i get the equation of the trajectory of an object knowing the forces acting on that object? Example: If F= GMm/r^2 and let be the sun at the center (point (0,0,0)) of the cartesian system how do i get the equation of an ellipse in this system? ( x^2/a^2 +...
  36. StevieTNZ

    "Experimental nonlocal and surreal Bohmian trajectories"

    Thought people would be interested in this recently published paper: http://advances.sciencemag.org/content/2/2/e1501466 It came up on my Facebook feed because I am friends with one of the authors. Picked up by New Scientist...
  37. P

    Gravity Simulator 2D: Calculating Trajectories

    I want to create a simple 2d Gravity simulator where I have a large body i.e. A circle which could be a planet or the sun. I then want to simulate small comets or asteroids traveling past it, crashing into or being pulled into orbit. I know the gravitational force formula but that seems the...
  38. A

    Bohmian trajectories vs. Feynman paths, always continuous?

    After reading some of the other posts on the Forum, I'm clear on the fact that Bohmian trajectories (of the de Broglie Bohm formulation) and the paths of the Feynman path integral formulation are very different things. I'm wondering (and it's a naive question, no doubt), when talking about...
  39. Logic Cloud

    Can Quantum Mechanics Have 'Elegant' Trajectories in Hilbert Space?

    State-space trajectories in classical mechanics can be used to nicely represent the time evolution of a given system. In the case of the harmonic oscillator, for instance, we get ellipses. How does this situation carry over to quantum mechanics? Can the time evolution of, say, the quantum...
  40. SU403RUNFAST

    Differential equations, orthogonal trajectories

    Homework Statement you are given a family of curves, in this case i was given a bunch of circles x^2+y^2=cx, sketch these curves for c=0,2,4,6, both positive and negative, solve the equation for c and differentiate both sides with respect to x and solve for dy/dx. You obtain an ODE in the form...
  41. bongobl

    Projectile motion trajectories differing from 45 degrees

    Hi guys, I am stuck with a problem here. First, It is given that for 2-dimensional projectile motion, a trajectory of 45 degrees will yield the greatest range. However, how do I show that angles that differ from 45 degrees by the same amount will yield the same range? For example, the range of...
  42. bcrowell

    Computing trajectories in Schwarzschild spacetime

    I want to produce some realistic figures showing the spatial trajectories of test particles in a Schwarzschild spacetime. For instance, I'd like to start a massive test particle at aponegricon (how often do you get to use that word!?) in an orbit that Kepler and Newton would have predicted to be...
  43. S

    Meteor trajectories -- Interested in computing them?

    Is anyone in this forum interested in computing meteor trajectories?
  44. M

    Find the equations of the trajectories of y"+y^3=0

    Homework Statement Find the equations of the trajectories of y"+y^3=0. Homework Equations None. The Attempt at a Solution y"+p(y)=0 v(dv/dy)+p(y)=0 integrate v^2/2+P(y)=C so I got v^2/2+y^4/4=C. Is v^2/2+y^4/4=C the correct answer?
  45. M

    Find the equations of the trajectories?

    Homework Statement Find the equations of the trajectories of y"+y^3=0. Homework Equations None. The Attempt at a Solution The Undamped Case: y"+p(y)=0 v(dv/dy)+p(y)=0 integrate v^2/2+P(y)=C --------------------------------- So following the formula above: The answer I got is v^2/2+y^4/4=C...
  46. T

    In depth Mars flight trajectory studies

    Can anyone point me in the direction of any published scientific work involving the study of Earth to Mars flight trajectories? I am thinking of researching this topic for a project and I would like to know about the work that has already been done.
  47. P

    First order ODE, orthogonal trajectories

    1. The problem statement, all variables and given/known da ##\frac{x^{2}}{k^{2}} + \frac{y^{2}}{\frac{k^{4}}{4}} = 1## with k != 0 this can be simplified to ##x^{2} + 4y^{2} = k^{2}## Find dy/dx implicitly, then find the new dy/dx if you want orthogonal trajectories to the ellipse. Lastly solve...
  48. F

    Analyzing Space-Time Trajectories: Rectilinear & Positive Velocity

    There's this graphic of space (s) versus time (t) of a particle. How would I know if the trajectory is always rectilinear and if the velocity is always positive? Observation: none of these "sentences" are true in this graphic. But I want to know how would I know these things if they were...
  49. ChrisVer

    Quantum Trajectories: Bubble Chamber & Detector Meaning

    If quantum mechanics don't allow the term trajectory for particles, then what do we see in bubble chambers, or what's the meaning of trying to "reconstruct" particle trajectories within a detector?