The notation can be a little confusing. The super scripts are not exponents, but rather the value of x at the time specified by the superscripts. The dot above the x refers the to derivative of x with respects to time, and two dots would be the second time derivative.
colinsteidtmann
To add on to @brandonlouie because I was confused as well:
x represents the position of the particle at a given time t.
x_dot (ẋ) represents the velocity of the particle at time t.
x_double_dot (ẍ) represents the acceleration of the particle at time t.
colinsteidtmann
How come the position equation doesn't add on 1/2 * acceleration * delta t?
RishSharma7
To slightly answer Colin's question (hopefully), I think this has to do with iterative step sizes. You're given an initial value and can compute the next step with the information you have. I actually would check out Wiki's page on 'Euler Method' if you want to clear things up; it's very descriptive.
j-nn
Euler's method allows the simulation of dynamic systems over time. By iteratively updating object positions and velocities based on discrete time steps and the system's differential equations,it allows the animation of complex motions, such as particle systems, fluid simulations, and more. Thiough it seems to allow a room for error.
SudhanvaKulkarni123
I wonder if we also use more advanced methods such linear multistep methods that we study in math128A. For more on LMMs I suggest reading the wiki. But the idea is essentially that we use information from previous time steps as well to make our soon more accurate
Refangs
In calculus class in high school I always wondered what Euler's method could be used for given that it's quite inaccurate. Cool to see it has an application here (albeit the better version makes some modifications)
anavmehta12
Euler's method would seem very inaccurate as we are just sort of guessing where the particles will be at time t from a previous time step which for a fast moving or unpredictable object would be very inaccurate.
agao25
To add on to @anavmehta12, Euler's method can get very unstable very quickly because as the initial time steps deviate, the error only accumulates and gets worse and worse. I wonder if we'll be considering the implicit method, or backward Euler, because that solution method is guaranteed to be stable but uses future time steps to solve for the present state.
The notation can be a little confusing. The super scripts are not exponents, but rather the value of x at the time specified by the superscripts. The dot above the x refers the to derivative of x with respects to time, and two dots would be the second time derivative.
To add on to @brandonlouie because I was confused as well:
x represents the position of the particle at a given time t.
x_dot (ẋ) represents the velocity of the particle at time t.
x_double_dot (ẍ) represents the acceleration of the particle at time t.
How come the position equation doesn't add on 1/2 * acceleration * delta t?
To slightly answer Colin's question (hopefully), I think this has to do with iterative step sizes. You're given an initial value and can compute the next step with the information you have. I actually would check out Wiki's page on 'Euler Method' if you want to clear things up; it's very descriptive.
Euler's method allows the simulation of dynamic systems over time. By iteratively updating object positions and velocities based on discrete time steps and the system's differential equations,it allows the animation of complex motions, such as particle systems, fluid simulations, and more. Thiough it seems to allow a room for error.
I wonder if we also use more advanced methods such linear multistep methods that we study in math128A. For more on LMMs I suggest reading the wiki. But the idea is essentially that we use information from previous time steps as well to make our soon more accurate
In calculus class in high school I always wondered what Euler's method could be used for given that it's quite inaccurate. Cool to see it has an application here (albeit the better version makes some modifications)
Euler's method would seem very inaccurate as we are just sort of guessing where the particles will be at time t from a previous time step which for a fast moving or unpredictable object would be very inaccurate.
To add on to @anavmehta12, Euler's method can get very unstable very quickly because as the initial time steps deviate, the error only accumulates and gets worse and worse. I wonder if we'll be considering the implicit method, or backward Euler, because that solution method is guaranteed to be stable but uses future time steps to solve for the present state.