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The backward Euler method is an implicit method, meaning that the formula for the backward Euler method has + on both sides, so when applying the backward Euler method we have to solve an equation. This makes the implementation more costly.
In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by [1] [2] = or equivalently + + =, where and denote the circumradius and inradius respectively (the radii of the circumscribed circle and inscribed circle respectively).
This is the Euler method (or forward Euler method, in contrast with the backward Euler method, to be described below). The method is named after Leonhard Euler who described it in 1768. The Euler method is an example of an explicit method. This means that the new value y n+1 is defined in terms of things that are already known, like y n.
According to the fundamental lemma of calculus of variations, the part of the integrand in parentheses is zero, i.e. ′ = which is called the Euler–Lagrange equation. The left hand side of this equation is called the functional derivative of J [ f ] {\displaystyle J[f]} and is denoted δ J {\displaystyle \delta J} or δ f ( x ...
To summarize: the pendulum example shows that, besides the explicit and implicit Euler methods not being good choices of method to solve the problem, the symplectic Euler method and implicit midpoint rule agree well with the exact flow of the system, with the midpoint rule agreeing more closely.
A rotation represented by an Euler axis and angle. In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two ...
The explicit midpoint method is sometimes also known as the modified Euler method, [1] the implicit method is the most simple collocation method, and, applied to Hamiltonian dynamics, a symplectic integrator. Note that the modified Euler method can refer to Heun's method, [2] for further clarity see List of Runge–Kutta methods.
However, the semi-implicit Euler method is a symplectic integrator, unlike the standard method. As a consequence, the semi-implicit Euler method almost conserves the energy (when the Hamiltonian is time-independent). Often, the energy increases steadily when the standard Euler method is applied, making it far less accurate.
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