# Euler method for unconstrained Rigid Body Dynamics

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Well, I wrote my rigid body class and a class to implement the euler method. I've chosen for my rigid body class the following state variables: position, orientation, linear and angular velocities( more intuitive ). So the state derivate is given by: dx(t)/dt = v(t) dR(t)/dt = w(t) * R(t) dv(t)/dt = F(t)/M dw(t)/dt = invI(t) * ( T(t) - w(t) x ( I(t) * w(t) ) ) x:position v:linear velocity R:orientation w:angular velocity F(t): force applied to the body at time t T(t): torque applied to the rigid body at time t invI: inverse of the world inertia tensor I: world inertia tensor *: product x: cross product I update the rigid body state like this: x(t+dt) = x(t) + v(t)dt; R(t+dt) = R(t) + w(t)*R(t)dt v(t+dt) = v(t) + F(t)dt/M w(t+dt) = w(t) + invI(t) * ( T(t) - w(t) x ( I(t) * w(t) ) )dt I was wondering if theoretically, updating the state that way v(t+dt) = v(t) + F(t)dt/M x(t+dt) = x(t) + v(t+dt)dt; w(t+dt) = w(t) + invI(t) * ( T(t) - w(t) x ( I(t) * w(t) ) )dt R(t+dt) = R(t) + w(t+dt) * R(t)dt was better. I had another idea: actually, we obtain by integrating the motion of a particle( a point mass ), when all quantities( functions ) are continuous: x(t) = xo + vo.dt + a.t.t/2 v(t) = vo + a.t xo and vo being respectively the initial position and speed, and a, a constant acceleration. and hence, x(t+dt) = x(t) + v(t)dt + a(t).t.t/2 v(t+dt) = v(t) + a(t)t could be an integration method. I then wondered if actually such a method is used in games: v(t+dt) = v(t) + F(t)dt/M x(t+dt) = x(t) + v(t)dt + F(t)dtdt/(2M); w(t+dt) = w(t) + invI(t) * ( T(t) - w(t) x ( I(t) * w(t) ) )dt R(t+dt) = R(t) + w(t)*R(t)dtdt/2 This would simply be analoguous to the precedent integration method( x(t+dt) = x(t) + v(t)dt + a(t).t.t/2 ... ) Is such a "method" used in games( or are there similar methods? ) , and if so, is it really better? Thanks for the replies.

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